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do this numerically by calculating each residual and comparing it to twice the standard deviation. On the TI-83, 83+, or 84+, the graphical approach is easier. The graphical procedure is shown first, followed by the numerical calculations. You would generally need to use only one of these methods. Example 12.12 In the third exam/final exam example, you can determine if there is an outlier or not. If there is an outlier, as an exercise, delete it and fit the remaining data to a new line. For this example, the new line ought to fit the remaining data better. This means the SSE should be smaller and the correlation coefficient ought to be closer to 1 or โ1. Solution 12.12 652 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION Graphical Identification of Outliers With the TI-83, 83+, 84+ graphing calculators, it is easy to identify the outliers graphically and visually. If we were to measure the vertical distance from any data point to the corresponding point on the line of best fit and that distance were equal to 2s or more, then we would consider the data point to be "too far" from the line of best fit. We need to find and graph the lines that are two standard deviations below and above the regression line. Any points that are outside these two lines are outliers. We will call these lines Y2 and Y3: As we did with the equation of the regression line and the correlation coefficient, we will use technology to calculate this standard deviation for us. Using the LinRegTTest with this data, scroll down through the output screens to find s = 16.412. Line Y2 = โ173.5 + 4.83x โ2(16.4) and line Y3 = โ173.5 + 4.83x + 2(16.4) where ลท = โ173.5 + 4.83x is the line of best fit. Y2 and Y3 have the same slope as the line of best fit. Graph the scatterplot with the best fit line in equation Y1, then enter the two extra lines as Y2 and Y3 in the "Y="equation editor and press ZOOM 9. You will find that the only data point that is not between lines Y2 and Y3 is the point x = 65, y = 175. On the calculator screen it is just barely outside these lines. The outlier is the student who |
had a grade of 65 on the third exam and 175 on the final exam; this point is further than two standard deviations away from the best-fit line. Sometimes a point is so close to the lines used to flag outliers on the graph that it is difficult to tell if the point is between or outside the lines. On a computer, enlarging the graph may help; on a small calculator screen, zooming in may make the graph clearer. Note that when the graph does not give a clear enough picture, you can use the numerical comparisons to identify outliers. Figure 12.18 12.12 Identify the potential outlier in the scatter plot. The standard deviation of the residuals or errors is approximately 8.6. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 653 Figure 12.19 Numerical Identification of Outliers In Table 12.5, the first two columns are the third-exam and final-exam data. The third column shows the predicted ลท values calculated from the line of best fit: ลท = โ173.5 + 4.83x. The residuals, or errors, have been calculated in the fourth column of the table: observed y valueโpredicted y value = y โ ลท. s is the standard deviation of all the y โ ลท = ฮต values where n = the total number of data points. If each residual is calculated and squared, and the results are added, we get the SSE. The standard deviation of the residuals is calculated from the SSE as: s = SSE n โ 2 NOTE We divide by (n โ 2) because the regression model involves two estimates. Rather than calculate the value of s ourselves, we can find s using the computer or calculator. For this example, the calculator function LinRegTTest found s = 16.4 as the standard deviation of the residuals 35; โ17; 16; โ6; โ19; 9; 3; โ1; โ10; โ9; โ1. x y ลท y โ ลท 65 175 140 175 โ 140 = 35 67 133 150 133 โ 150= โ17 71 185 169 185 โ 169 = 16 71 163 169 163 โ 169 = โ6 66 126 145 126 โ 145 = โ19 75 198 |
189 198 โ 189 = 9 67 153 150 153 โ 150 = 3 70 163 164 163 โ 164 = โ1 71 159 169 159 โ 169 = โ10 Table 12.5 654 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION x y ลท y โ ลท 69 151 160 151 โ 160 = โ9 69 159 160 159 โ 160 = โ1 Table 12.5 We are looking for all data points for which the residual is greater than 2s = 2(16.4) = 32.8 or less than โ32.8. Compare these values to the residuals in column four of the table. The only such data point is the student who had a grade of 65 on the third exam and 175 on the final exam; the residual for this student is 35. How does the outlier affect the best fit line? Numerically and graphically, we have identified the point (65, 175) as an outlier. We should re-examine the data for this point to see if there are any problems with the data. If there is an error, we should fix the error if possible, or delete the data. If the data is correct, we would leave it in the data set. For this problem, we will suppose that we examined the data and found that this outlier data was an error. Therefore we will continue on and delete the outlier, so that we can explore how it affects the results, as a learning experience. Compute a new best-fit line and correlation coefficient using the ten remaining points: On the TI-83, TI-83+, TI-84+ calculators, delete the outlier from L1 and L2. Using the LinRegTTest, the new line of best fit and the correlation coefficient are: ลท = โ355.19 + 7.39x and r = 0.9121 The new line with r = 0.9121 is a stronger correlation than the original (r = 0.6631) because r = 0.9121 is closer to one. This means that the new line is a better fit to the ten remaining data values. The line can better predict the final exam score given the third exam score. Numerical Identification of Outliers: Calculating s and Finding Outliers Manually If you do not have the function LinRegTTest, then you can calculate the outlier in the first example by doing the following. First, square each |y โ ลท| |
The squares are 352; 172; 162; 62; 192; 92; 32; 12; 102; 92; 12 Then, add (sum) all the |y โ ลท| squared terms using the formula 11 โ ฮฃ i = 1 โ|yi โ y^ 2 โ i|โ = 11 ฮฃ i = 1 ฮตi 2 (Recall that yi โ ลทi = ฮตi.) = 352 + 172 + 162 + 62 + 192 + 92 + 32 + 12 + 102 + 92 + 12 = 2440 = SSE. The result, SSE is the Sum of Squared Errors. Next, calculate s, the standard deviation of all the y โ ลท = ฮต values where n = the total number of data points. The calculation is s = SSE n โ 2. For the third exam/final exam problem, s = 2440 11 โ 2 = 16.47. Next, multiply s by 1.9: (1.9)(16.47) = 31.29 31.29 is almost 2 standard deviations away from the mean of the y โ ลท values. If we were to measure the vertical distance from any data point to the corresponding point on the line of best fit and that distance is at least 1.9s, then we would consider the data point to be "too far" from the line of best fit. We call that point a potential outlier. For the example, if any of the |y โ ลท| values are at least 31.29, the corresponding (x, y) data point is a potential outlier. For the third exam/final exam problem, all the |y โ ลท|'s are less than 31.29 except for the first one which is 35. 35 > 31.29 That is, |y โ ลท| โฅ (1.9)(s) This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 The point which corresponds to |y โ ลท| = 35 is (65, 175). Therefore, the data point (65,175) is a potential outlier. For this example, we will delete it. (Remember, we do not always delete an outlier.) CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 655 NOTE When outliers are deleted |
, the researcher should either record that data was deleted, and why, or the researcher should provide results both with and without the deleted data. If data is erroneous and the correct values are known (e.g., student one actually scored a 70 instead of a 65), then this correction can be made to the data. The next step is to compute a new best-fit line using the ten remaining points. The new line of best fit and the correlation coefficient are: ลท = โ355.19 + 7.39x and r = 0.9121 Example 12.13 Using this new line of best fit (based on the remaining ten data points in the third exam/final exam example (http://cnx.org/content/m47117/1.3/##element-22) ), what would a student who receives a 73 on the third exam expect to receive on the final exam? Is this the same as the prediction made using the original line? Solution 12.13 Using the new line of best fit, ลท = โ355.19 + 7.39(73) = 184.28. A student who scored 73 points on the third exam would expect to earn 184 points on the final exam. The original line predicted ลท = โ173.51 + 4.83(73) = 179.08 so the prediction using the new line with the outlier eliminated differs from the original prediction. 12.13 The data points for the graph from the third exam/final exam example (http://cnx.org/content/ m47117/1.3/##element-22) are as follows: (1, 5), (2, 7), (2, 6), (3, 9), (4, 12), (4, 13), (5, 18), (6, 19), (7, 12), and (7, 21). Remove the outlier and recalculate the line of best fit. Find the value of ลท when x = 10. 656 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION Example 12.14 The Consumer Price Index (CPI) measures the average change over time in the prices paid by urban consumers for consumer goods and services. The CPI affects nearly all Americans because of the many ways it is used. One of its biggest uses is as a measure of inflation. By providing information about price changes in the Nation's economy to government, business, and labor, the CPI helps them to make economic decisions. |
The President, Congress, and the Federal Reserve Board use the CPI's trends to formulate monetary and fiscal policies. In the following table, x is the year and y is the CPI. x y x y 1915 10.1 1969 36.7 1926 17.7 1975 49.3 1935 13.7 1979 72.6 1940 14.7 1980 82.4 1947 24.1 1986 109.6 1952 26.5 1991 130.7 1964 31.0 1999 166.6 Table 12.6 Data a. Draw a scatterplot of the data. b. Calculate the least squares line. Write the equation in the form ลท = a + bx. c. Draw the line on the scatterplot. d. Find the correlation coefficient. Is it significant? e. What is the average CPI for the year 1990? Solution 12.14 a. See Figure 12.19. b. ลท = โ3204 + 1.662x is the equation of the line of best fit. c. r = 0.8694 d. The number of data points is n = 14. Use the 95% Critical Values of the Sample Correlation Coefficient table at the end of Chapter 12. n โ 2 = 12. The corresponding critical value is 0.532. Since 0.8694 > 0.532, r is significant. ลท = โ3204 + 1.662(1990) = 103.4 CPI e. Using the calculator LinRegTTest, we find that s = 25.4 ; graphing the lines Y2 = โ3204 + 1.662X โ 2(25.4) and Y3 = โ3204 + 1.662X + 2(25.4) shows that no data values are outside those lines, identifying no outliers. (Note that the year 1999 was very close to the upper line, but still inside it.) Figure 12.20 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 657 NOTE In the example, notice the pattern of the points compared to the line. Although the correlation coefficient is significant, the pattern in the scatterplot indicates that a curve would be a more appropriate model to use than a line. In this example, a statistician should prefer to use other methods to fit a curve to this data |
, rather than model the data with the line we found. In addition to doing the calculations, it is always important to look at the scatterplot when deciding whether a linear model is appropriate. If you are interested in seeing more years of data, visit the Bureau of Labor Statistics CPI website ftp://ftp.bls.gov/ pub/special.requests/cpi/cpiai.txt; our data is taken from the column entitled "Annual Avg." (third column from the right). For example you could add more current years of data. Try adding the more recent years: 2004: CPI = 188.9; 2008: CPI = 215.3; 2011: CPI = 224.9. See how it affects the model. (Check: ลท = โ4436 + 2.295x; r = 0.9018. Is r significant? Is the fit better with the addition of the new points?) 12.14 The following table shows economic development measured in per capita income PCINC. Year PCINC Year PCINC 1870 1880 1890 1900 1910 340 499 592 757 927 Table 12.7 1920 1050 1930 1170 1940 1364 1950 1836 1960 2132 a. What are the independent and dependent variables? b. Draw a scatter plot. c. Use regression to find the line of best fit and the correlation coefficient. d. e. Interpret the significance of the correlation coefficient. Is there a linear relationship between the variables? f. Find the coefficient of determination and interpret it. g. What is the slope of the regression equation? What does it mean? h. Use the line of best fit to estimate PCINC for 1900, for 2000. i. Determine if there are any outliers. 95% Critical Values of the Sample Correlation Coefficient Table Degrees of Freedom: n โ 2 Critical Values: (+ and โ) 1 2 3 Table 12.8 0.997 0.950 0.878 658 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION Degrees of Freedom: n โ 2 Critical Values: (+ and โ) 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 60 70 80 90 100 Table 12.8 0.811 0.754 0.707 0.666 0.632 0.602 0.576 0.555 0.532 0.514 0.497 0.482 0.468 0. |
456 0.444 0.433 0.423 0.413 0.404 0.396 0.388 0.381 0.374 0.367 0.361 0.355 0.349 0.304 0.273 0.250 0.232 0.217 0.205 0.195 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 659 12.1 Regression (Distance from School) Class Time: Names: Student Learning Outcomes โข The student will calculate and construct the line of best fit between two variables. โข The student will evaluate the relationship between two variables to determine if that relationship is significant. Collect the Data Use eight members of your class for the sample. Collect bivariate data (distance an individual lives from school, the cost of supplies for the current term). 1. Complete the table. Distance from school Cost of supplies this term Table 12.9 2. Which variable should be the dependent variable and which should be the independent variable? Why? 3. Graph โdistanceโ vs. โcost.โ Plot the points on the graph. Label both axes with words. Scale both axes. Figure 12.21 Analyze the Data Enter your data into your calculator or computer. Write the linear equation, rounding to four decimal places. 660 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 1. Calculate the following: a. a = ______ b. b = ______ c. correlation = ______ d. n = ______ e. equation: ลท = ______ f. Is the correlation significant? Why or why not? (Answer in one to three complete sentences.) 2. Supply an answer for the following senarios: a. For a person who lives eight miles from campus, predict the total cost of supplies this term: b. For a person who lives eighty miles from campus, predict the total cost of supplies this term: 3. Obtain the graph on your calculator or computer. Sketch the regression line. Figure 12.22 Discussion Questions 1. Answer each question in complete sentences. a. Does the line seem to fit the data? Why? b. What does the correlation imply about the relationship between the distance and the cost? 2. Are there any outliers? If so, which point is an outlier? 3. Should the outlier, |
if it exists, be removed? Why or why not? This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 661 12.2 Regression (Textbook Cost) Class Time: Names: Student Learning Outcomes โข The student will calculate and construct the line of best fit between two variables. โข The student will evaluate the relationship between two variables to determine if that relationship is significant. Collect the Data Survey ten textbooks. Collect bivariate data (number of pages in a textbook, the cost of the textbook). 1. Complete the table. Number of pages Cost of textbook Table 12.10 2. Which variable should be the dependent variable and which should be the independent variable? Why? 3. Graph โpagesโ vs. โcost.โ Plot the points on the graph in Analyze the Data. Label both axes with words. Scale both axes. Analyze the Data Enter your data into your calculator or computer. Write the linear equation, rounding to four decimal places. 1. Calculate the following: a. a = ______ b. b = ______ c. correlation = ______ d. n = ______ e. equation: y = ______ f. Is the correlation significant? Why or why not? (Answer in complete sentences.) 2. Supply an answer for the following senarios: a. For a textbook with 400 pages, predict the cost. b. For a textbook with 600 pages, predict the cost. 3. Obtain the graph on your calculator or computer. Sketch the regression line. 662 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION Figure 12.23 Discussion Questions 1. Answer each question in complete sentences. a. Does the line seem to fit the data? Why? b. What does the correlation imply about the relationship between the number of pages and the cost? 2. Are there any outliers? If so, which point(s) is an outlier? 3. Should the outlier, if it exists, be removed? Why or why not? This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 663 12.3 Regression ( |
Fuel Efficiency) Class Time: Names: Student Learning Outcomes โข The student will calculate and construct the line of best fit between two variables. โข The student will evaluate the relationship between two variables to determine if that relationship is significant. Collect the Data Use the most recent April issue of Consumer Reports. It will give the total fuel efficiency (in miles per gallon) and weight (in pounds) of new model cars with automatic transmissions. We will use this data to determine the relationship, if any, between the fuel efficiency of a car and its weight. 1. Using your random number generator, randomly select 20 cars from the list and record their weights and fuel efficiency into Table 12.11. Weight Fuel Efficiency Table 12.11 2. Which variable should be the dependent variable and which should be the independent variable? Why? 3. By hand, do a scatterplot of โweightโ vs. โfuel efficiencyโ. Plot the points on graph paper. Label both axes with words. Scale both axes accurately. 664 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION Figure 12.24 Analyze the Data Enter your data into your calculator or computer. Write the linear equation, rounding to 4 decimal places. 1. Calculate the following: a. a = ______ b. b = ______ c. correlation = ______ d. n = ______ e. equation: ลท = ______ 2. Obtain the graph of the regression line on your calculator. Sketch the regression line on the same axes as your scatter plot. Discussion Questions 1. Is the correlation significant? Explain how you determined this in complete sentences. 2. 3. Is the relationship a positive one or a negative one? Explain how you can tell and what this means in terms of weight and fuel efficiency. In one or two complete sentences, what is the practical interpretation of the slope of the least squares line in terms of fuel efficiency and weight? 4. For a car that weighs 4,000 pounds, predict its fuel efficiency. Include units. 5. Can we predict the fuel efficiency of a car that weighs 10,000 pounds using the least squares line? Explain why or why not. 6. Answer each question in complete sentences. a. Does the line seem to fit the data? Why or why not? b. What does the correlation imply about the relationship between fuel efficiency and weight of a car? Is this what you expected? 7. Are there any outliers? If so, which point is an outlier? This content is available |
for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 665 KEY TERMS Coefficient of Correlation a measure developed by Karl Pearson (early 1900s) that gives the strength of association between the independent variable and the dependent variable; the formula is: r = nโ xy โ (โ x)(โ y) 2 [nโ x2 โ (โ x) ][nโ y2 โ (โ y) 2 ] where n is the number of data points. The coefficient cannot be more then 1 and less then โ1. The closer the coefficient is to ยฑ1, the stronger the evidence of a significant linear relationship between x and y. Outlier an observation that does not fit the rest of the data CHAPTER REVIEW 12.1 Linear Equations The most basic type of association is a linear association. This type of relationship can be defined algebraically by the equations used, numerically with actual or predicted data values, or graphically from a plotted curve. (Lines are classified as straight curves.) Algebraically, a linear equation typically takes the form y = mx + b, where m and b are constants, x is the independent variable, y is the dependent variable. In a statistical context, a linear equation is written in the form y = a + bx, where a and b are the constants. This form is used to help readers distinguish the statistical context from the algebraic context. In the equation y = a + bx, the constant b that multiplies the x variable (b is called a coefficient) is called as the slope. The slope describes the rate of change between the independent and dependent variables; in other words, the rate of change describes the change that occurs in the dependent variable as the independent variable is changed. In the equation y = a + bx, the constant a is called as the y-intercept. Graphically, the y-intercept is the y coordinate of the point where the graph of the line crosses the y axis. At this point x = 0. The slope of a line is a value that describes the rate of change between the independent and dependent variables. The slope tells us how the dependent variable (y) changes for every one unit increase in the independent (x) variable, on average. The y- |
intercept is used to describe the dependent variable when the independent variable equals zero. Graphically, the slope is represented by three line types in elementary statistics. 12.2 Scatter Plots Scatter plots are particularly helpful graphs when we want to see if there is a linear relationship among data points. They indicate both the direction of the relationship between the x variables and the y variables, and the strength of the relationship. We calculate the strength of the relationship between an independent variable and a dependent variable using linear regression. 12.3 The Regression Equation A regression line, or a line of best fit, can be drawn on a scatter plot and used to predict outcomes for the x and y variables in a given data set or sample data. There are several ways to find a regression line, but usually the least-squares regression line is used because it creates a uniform line. Residuals, also called โerrors,โ measure the distance from the actual value of y and the estimated value of y. The Sum of Squared Errors, when set to its minimum, calculates the points on the line of best fit. Regression lines can be used to predict values within the given set of data, but should not be used to make predictions for values outside the set of data. The correlation coefficient r measures the strength of the linear association between x and y. The variable r has to be between โ1 and +1. When r is positive, the x and y will tend to increase and decrease together. When r is negative, x will increase and y will decrease, or the opposite, x will decrease and y will increase. The coefficient of determination r2, is equal to the square of the correlation coefficient. When expressed as a percent, r2 represents the percent of variation in the dependent variable y that can be explained by variation in the independent variable x using the regression line. 12.4 Testing the Significance of the Correlation Coefficient Linear regression is a procedure for fitting a straight line of the form ลท = a + bx to data. The conditions for regression are: โข Linear In the population, there is a linear relationship that models the average value of y for different values of x. 666 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION โข Independent The residuals are assumed to be independent. โข Normal The y values are distributed normally for any value of x. โข Equal variance The standard deviation of the y values is equal for each x value. โข Random The data are |
produced from a well-designed random sample or randomized experiment. The slope b and intercept a of the least-squares line estimate the slope ฮฒ and intercept ฮฑ of the population (true) regression line. To estimate the population standard deviation of y, ฯ, use the standard deviation of the residuals, s. s = SEE n โ 2. The variable ฯ (rho) is the population correlation coefficient. To test the null hypothesis H0: ฯ = hypothesized value, use a linear regression t-test. The most common null hypothesis is H0: ฯ = 0 which indicates there is no linear relationship between x and y in the population. The TI-83, 83+, 84, 84+ calculator function LinRegTTest can perform this test (STATS TESTS LinRegTTest). 12.5 Prediction After determining the presence of a strong correlation coefficient and calculating the line of best fit, you can use the least squares regression line to make predictions about your data. 12.6 Outliers To determine if a point is an outlier, do one of the following: 1. Input the following equations into the TI 83, 83+,84, 84+: y1 = a + bx y2 = (2s)a + bx y3 = โ (2s)a + bx where s is the standard deviation of the residuals If any point is above y2or below y3 then the point is considered to be an outlier. 2. Use the residuals and compare their absolute values to 1.9s where s is the standard deviation of the residuals. If the absolute value of any residual is greater than or equal to 1.9s, then the corresponding point is an outlier. 3. Note: The calculator function LinRegTTest (STATS TESTS LinRegTTest) calculates s. To determine if a point is an influential point, graph the least-squares line with the point included, then graph the leastsquares line with the point excluded. If the graph changes by a considerable amount, the point is influential. FORMULA REVIEW 12.1 Linear Equations y = a + bx where a is the y-intercept and b is the slope. The variable x is the independent variable and y is the dependent variable. 12.4 Testing the Significance of the Correlation Coefficient Least Squares Line or Line of Best Fit: a = y-intercept b = slope |
Standard deviation of the residuals: s = SEE n โ 2. where SSE = sum of squared errors n = the number of data points y^ = a + bx where PRACTICE 12.1 Linear Equations This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 667 Use the following information to answer the next three exercises. A vacation resort rents SCUBA equipment to certified divers. The resort charges an up-front fee of $25 and another fee of $12.50 an hour. 1. What are the dependent and independent variables? 2. Find the equation that expresses the total fee in terms of the number of hours the equipment is rented. 3. Graph the equation from Exercise 12.2. Use the following information to answer the next two exercises. A credit card company charges $10 when a payment is late, and $5 a day each day the payment remains unpaid. 4. Find the equation that expresses the total fee in terms of the number of days the payment is late. 5. Graph the equation from Exercise 12.4. 6. Is the equation y = 10 + 5x โ 3x2 linear? Why or why not? 7. Which of the following equations are linear? a. y = 6x + 8 b. y + 7 = 3x c. y โ x = 8x2 d. 4y = 8 8. Does the graph show a linear equation? Why or why not? Figure 12.25 Table 12.12 contains real data for the first two decades of AIDS reporting. Year # AIDS cases diagnosed # AIDS deaths Pre-1981 91 1981 1982 1983 1984 1985 1986 1987 1988 319 1,170 3,076 6,240 11,776 19,032 28,564 35,447 29 121 453 1,482 3,466 6,878 11,987 16,162 20,868 Table 12.12 Adults and Adolescents only, United States 668 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 42,674 48,634 59,660 78,530 78,834 71,874 68,505 59,347 47,149 38,393 25,174 25,522 25,643 26,464 |
27,591 31,335 36,560 41,055 44,730 49,095 49,456 38,510 20,736 19,005 18,454 17,347 17,402 16,371 Total 802,118 489,093 Table 12.12 Adults and Adolescents only, United States 9. Use the columns "year" and "# AIDS cases diagnosed. Why is โyearโ the independent variable and โ# AIDS cases diagnosed.โ the dependent variable (instead of the reverse)? Use the following information to answer the next two exercises. A specialty cleaning company charges an equipment fee and an hourly labor fee. A linear equation that expresses the total amount of the fee the company charges for each session is y = 50 + 100x. 10. What are the independent and dependent variables? 11. What is the y-intercept and what is the slope? Interpret them using complete sentences. Use the following information to answer the next three questions. Due to erosion, a river shoreline is losing several thousand pounds of soil each year. A linear equation that expresses the total amount of soil lost per year is y = 12,000x. 12. What are the independent and dependent variables? 13. How many pounds of soil does the shoreline lose in a year? 14. What is the y-intercept? Interpret its meaning. Use the following information to answer the next two exercises. The price of a single issue of stock can fluctuate throughout the day. A linear equation that represents the price of stock for Shipment Express is y = 15 โ 1.5x where x is the number of hours passed in an eight-hour day of trading. 15. What are the slope and y-intercept? Interpret their meaning. 16. If you owned this stock, would you want a positive or negative slope? Why? 12.2 Scatter Plots 17. Does the scatter plot appear linear? Strong or weak? Positive or negative? This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 669 Figure 12.26 18. Does the scatter plot appear linear? Strong or weak? Positive or negative? Figure 12.27 19. Does the scatter plot appear linear? Strong or weak? Positive or negative? 670 CHAPTER 12 | LINEAR REGRESSION AND |
CORRELATION Figure 12.28 12.3 The Regression Equation Use the following information to answer the next five exercises. A random sample of ten professional athletes produced the following data where x is the number of endorsements the player has and y is the amount of money made (in millions of dollars). 12 9 9 3 13 4 10 Table 12.13 20. Draw a scatter plot of the data. 21. Use regression to find the equation for the line of best fit. 22. Draw the line of best fit on the scatter plot. 23. What is the slope of the line of best fit? What does it represent? 24. What is the y-intercept of the line of best fit? What does it represent? 25. What does an r value of zero mean? 26. When n = 2 and r = 1, are the data significant? Explain. 27. When n = 100 and r = -0.89, is there a significant correlation? Explain. 12.4 Testing the Significance of the Correlation Coefficient 28. When testing the significance of the correlation coefficient, what is the null hypothesis? 29. When testing the significance of the correlation coefficient, what is the alternative hypothesis? 30. If the level of significance is 0.05 and the p-value is 0.04, what conclusion can you draw? This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 671 12.5 Prediction Use the following information to answer the next two exercises. An electronics retailer used regression to find a simple model to predict sales growth in the first quarter of the new year (January through March). The model is good for 90 days, where x is the day. The model can be written as follows: ลท = 101.32 + 2.48x where ลท is in thousands of dollars. 31. What would you predict the sales to be on day 60? 32. What would you predict the sales to be on day 90? Use the following information to answer the next three exercises. A landscaping company is hired to mow the grass for several large properties. The total area of the properties combined is 1,345 acres. The rate at which one person can mow is as follows: ลท = 1350 โ 1.2x where x is the number of |
hours and ลท represents the number of acres left to mow. 33. How many acres will be left to mow after 20 hours of work? 34. How many acres will be left to mow after 100 hours of work? 35. How many hours will it take to mow all of the lawns? (When is ลท = 0?) Table 12.14 contains real data for the first two decades of AIDS reporting. Year # AIDS cases diagnosed # AIDS deaths Pre-1981 91 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 319 1,170 3,076 6,240 11,776 19,032 28,564 35,447 42,674 48,634 59,660 78,530 78,834 71,874 68,505 59,347 47,149 38,393 25,174 25,522 25,643 26,464 29 121 453 1,482 3,466 6,878 11,987 16,162 20,868 27,591 31,335 36,560 41,055 44,730 49,095 49,456 38,510 20,736 19,005 18,454 17,347 17,402 16,371 Table 12.14 Adults and Adolescents only, United States 672 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION Total 802,118 489,093 Table 12.14 Adults and Adolescents only, United States 36. Graph โyearโ versus โ# AIDS cases diagnosedโ (plot the scatter plot). Do not include pre-1981 data. 37. Perform linear regression. What is the linear equation? Round to the nearest whole number. 38. Write the equations: a. Linear equation: __________ b. a = ________ c. b = ________ d. r = ________ e. n = ________ 39. Solve. a. When x = 1985, ลท = _____ b. When x = 1990, ลท =_____ c. When x = 1970, ลท =______ Why doesnโt this answer make sense? 40. Does the line seem to fit the data? Why or why not? 41. What does the correlation imply about the relationship between time (years) and the number of diagnosed AIDS cases reported in the U.S.? 42. Plot the two given points on the following graph. Then, connect the two points to form |
the regression line. Figure 12.29 Obtain the graph on your calculator or computer. 43. Write the equation: ลท= ____________ 44. Hand draw a smooth curve on the graph that shows the flow of the data. 45. Does the line seem to fit the data? Why or why not? 46. Do you think a linear fit is best? Why or why not? 47. What does the correlation imply about the relationship between time (years) and the number of diagnosed AIDS cases reported in the U.S.? 48. Graph โyearโ vs. โ# AIDS cases diagnosed.โ Do not include pre-1981. Label both axes with words. Scale both axes. 49. Enter your data into your calculator or computer. The pre-1981 data should not be included. Why is that so? Write the linear equation, rounding to four decimal places: 50. Calculate the following: a. a = _____ b. b = _____ c. correlation = _____ d. n = _____ This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 673 12.6 Outliers Use the following information to answer the next four exercises. The scatter plot shows the relationship between hours spent studying and exam scores. The line shown is the calculated line of best fit. The correlation coefficient is 0.69. Figure 12.30 51. Do there appear to be any outliers? 52. A point is removed, and the line of best fit is recalculated. The new correlation coefficient is 0.98. Does the point appear to have been an outlier? Why? 53. What effect did the potential outlier have on the line of best fit? 54. Are you more or less confident in the predictive ability of the new line of best fit? 55. The Sum of Squared Errors for a data set of 18 numbers is 49. What is the standard deviation? 56. The Standard Deviation for the Sum of Squared Errors for a data set is 9.8. What is the cutoff for the vertical distance that a point can be from the line of best fit to be considered an outlier? HOMEWORK 12.1 Linear Equations 57. For each of the following situations, state the independent variable and the dependent variable. a. |
A study is done to determine if elderly drivers are involved in more motor vehicle fatalities than other drivers. The number of fatalities per 100,000 drivers is compared to the age of drivers. Insurance companies base life insurance premiums partially on the age of the applicant. b. A study is done to determine if the weekly grocery bill changes based on the number of family members. c. d. Utility bills vary according to power consumption. e. A study is done to determine if a higher education reduces the crime rate in a population. 58. Piece-rate systems are widely debated incentive payment plans. In a recent study of loan officer effectiveness, the following piece-rate system was examined: % of goal reached < 80 80 100 120 Incentive n/ a $4,000 with an additional $125 added per percentage point from 81โ99% $6,500 with an additional $125 added per percentage point from 101โ119% $9,500 with an additional $125 added per percentage point starting at 121% Table 12.15 674 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION If a loan officer makes 95% of his or her goal, write the linear function that applies based on the incentive plan table. In context, explain the y-intercept and slope. 12.2 Scatter Plots 59. The Gross Domestic Product Purchasing Power Parity is an indication of a countryโs currency value compared to another country. Table 12.16 shows the GDP PPP of Cuba as compared to US dollars. Construct a scatter plot of the data. Year Cubaโs PPP Year Cubaโs PPP 1999 1,700 2006 4,000 2000 1,700 2007 11,000 2002 2,300 2008 9,500 2003 2,900 2009 9,700 2004 3,000 2010 9,900 2005 3,500 Table 12.16 60. The following table shows the poverty rates and cell phone usage in the United States. Construct a scatter plot of the data Year Poverty Rate Cellular Usage per Capita 2003 12.7 2005 12.6 2007 2009 12 12 Table 12.17 54.67 74.19 84.86 90.82 61. Does the higher cost of tuition translate into higher-paying jobs? The table lists the top ten colleges based on mid-career salary and the associated yearly tuition costs. Construct a scatter plot of the data. Mid-Career Salary (in thousands) Yearly Tuition School Princeton Harvey Mudd CalTech 137 135 127 |
US Naval Academy 122 West Point MIT Lehigh University NYU-Poly Babson College Stanford Table 12.18 120 118 118 117 117 114 28,540 40,133 39,900 0 0 42,050 43,220 39,565 40,400 54,506 62. If the level of significance is 0.05 and the p-value is 0.06, what conclusion can you draw? 63. If there are 15 data points in a set of data, what is the number of degree of freedom? This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 675 12.3 The Regression Equation 64. What is the process through which we can calculate a line that goes through a scatter plot with a linear pattern? 65. Explain what it means when a correlation has an r2 of 0.72. 66. Can a coefficient of determination be negative? Why or why not? 12.4 Testing the Significance of the Correlation Coefficient 67. If the level of significance is 0.05 and the p-value is 0.06, what conclusion can you draw? 68. If there are 15 data points in a set of data, what is the number of degree of freedom? 12.5 Prediction 69. Recently, the annual number of driver deaths per 100,000 for the selected age groups was as follows: Age Number of Driver Deaths per 100,000 17.5 22 29.5 44.5 64.5 80 38 36 24 20 18 28 Table 12.19 a. For each age group, pick the midpoint of the interval for the x value. (For the 75+ group, use 80.) b. Using โagesโ as the independent variable and โNumber of driver deaths per 100,000โ as the dependent variable, make a scatter plot of the data. c. Calculate the least squares (bestโfit) line. Put the equation in the form of: ลท = a + bx d. Find the correlation coefficient. Is it significant? e. Predict the number of deaths for ages 40 and 60. f. Based on the given data, is there a linear relationship between age of a driver and driver fatality rate? g. What is the slope of the least squares (best-fit) line? Interpret the slope |
. 70. Table 12.20 shows the life expectancy for an individual born in the United States in certain years. Year of Birth Life Expectancy 1930 1940 1950 1965 1973 1982 1987 1992 2010 Table 12.20 59.7 62.9 70.2 69.7 71.4 74.5 75 75.7 78.7 a. Decide which variable should be the independent variable and which should be the dependent variable. b. Draw a scatter plot of the ordered pairs. 676 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION c. Calculate the least squares line. Put the equation in the form of: ลท = a + bx d. Find the correlation coefficient. Is it significant? e. Find the estimated life expectancy for an individual born in 1950 and for one born in 1982. f. Why arenโt the answers to part e the same as the values in Table 12.20 that correspond to those years? g. Use the two points in part e to plot the least squares line on your graph from part b. h. Based on the data, is there a linear relationship between the year of birth and life expectancy? i. Are there any outliers in the data? j. Using the least squares line, find the estimated life expectancy for an individual born in 1850. Does the least squares line give an accurate estimate for that year? Explain why or why not. k. What is the slope of the least-squares (best-fit) line? Interpret the slope. 71. The maximum discount value of the Entertainmentยฎ card for the โFine Diningโ section, Edition ten, for various pages is given in Table 12.21 Page number Maximum value ($) 4 14 25 32 43 57 72 85 90 Table 12.21 16 19 15 17 19 15 16 15 17 a. Decide which variable should be the independent variable and which should be the dependent variable. b. Draw a scatter plot of the ordered pairs. c. Calculate the least-squares line. Put the equation in the form of: ลท = a + bx d. Find the correlation coefficient. Is it significant? e. Find the estimated maximum values for the restaurants on page ten and on page 70. f. Does it appear that the restaurants giving the maximum value are placed in the beginning of the โFine Diningโ section? How did you arrive at your answer? g. Suppose that there were 200 pages of restaurants. What do you estimate |
to be the maximum value for a restaurant listed on page 200? Is the least squares line valid for page 200? Why or why not? h. i. What is the slope of the least-squares (best-fit) line? Interpret the slope. 72. Table 12.22 gives the gold medal times for every other Summer Olympics for the womenโs 100-meter freestyle (swimming). Year Time (seconds) 1912 1924 1932 1952 1960 1968 1976 Table 12.22 82.2 72.4 66.8 66.8 61.2 60.0 55.65 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 677 Year Time (seconds) 1984 1992 2000 2008 Table 12.22 55.92 54.64 53.8 53.1 a. Decide which variable should be the independent variable and which should be the dependent variable. b. Draw a scatter plot of the data. c. Does it appear from inspection that there is a relationship between the variables? Why or why not? d. Calculate the least squares line. Put the equation in the form of: ลท = a + bx. e. Find the correlation coefficient. Is the decrease in times significant? f. Find the estimated gold medal time for 1932. Find the estimated time for 1984. g. Why are the answers from part f different from the chart values? h. Does it appear that a line is the best way to fit the data? Why or why not? i. Use the least-squares line to estimate the gold medal time for the next Summer Olympics. Do you think that your answer is reasonable? Why or why not? # letters in name Year entered the Union Rank for entering the Union Area (square miles) 73. State Alabama Colorado Hawaii Iowa Maryland Missouri 7 8 6 4 8 8 New Jersey 9 Ohio South Carolina Utah Wisconsin Table 12.23 4 13 4 9 1819 1876 1959 1846 1788 1821 1787 1803 1788 1896 1848 22 38 50 29 7 24 3 17 8 45 30 52,423 104,100 10,932 56,276 12,407 69,709 8,722 44,828 32,008 84,904 65,499 We are interested in whether or not the number of letters in a state |
name depends upon the year the state entered the Union. a. Decide which variable should be the independent variable and which should be the dependent variable. b. Draw a scatter plot of the data. c. Does it appear from inspection that there is a relationship between the variables? Why or why not? d. Calculate the least-squares line. Put the equation in the form of: ลท = a + bx. e. Find the correlation coefficient. What does it imply about the significance of the relationship? f. Find the estimated number of letters (to the nearest integer) a state would have if it entered the Union in 1900. Find the estimated number of letters a state would have if it entered the Union in 1940. g. Does it appear that a line is the best way to fit the data? Why or why not? h. Use the least-squares line to estimate the number of letters a new state that enters the Union this year would have. Can the least squares line be used to predict it? Why or why not? 12.6 Outliers 678 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 74. The height (sidewalk to roof) of notable tall buildings in America is compared to the number of stories of the building (beginning at street level). Height (in feet) Stories 1,050 428 362 529 790 401 380 1,454 1,127 700 Table 12.24 57 28 26 40 60 22 38 110 100 46 a. Using โstoriesโ as the independent variable and โheightโ as the dependent variable, make a scatter plot of the data. b. Does it appear from inspection that there is a relationship between the variables? c. Calculate the least squares line. Put the equation in the form of: ลท = a + bx d. Find the correlation coefficient. Is it significant? e. Find the estimated heights for 32 stories and for 94 stories. f. Based on the data in Table 12.24, is there a linear relationship between the number of stories in tall buildings and the height of the buildings? g. Are there any outliers in the data? If so, which point(s)? h. What is the estimated height of a building with six stories? Does the least squares line give an accurate estimate of height? Explain why or why not. i. Based on the least squares line, adding an extra story is predicted to add about how many feet to a building |
? j. What is the slope of the least squares (best-fit) line? Interpret the slope. 75. Ornithologists, scientists who study birds, tag sparrow hawks in 13 different colonies to study their population. They gather data for the percent of new sparrow hawks in each colony and the percent of those that have returned from migration. Percent return:74; 66; 81; 52; 73; 62; 52; 45; 62; 46; 60; 46; 38 Percent new:5; 6; 8; 11; 12; 15; 16; 17; 18; 18; 19; 20; 20 a. Enter the data into your calculator and make a scatter plot. b. Use your calculatorโs regression function to find the equation of the least-squares regression line. Add this to your scatter plot from part a. c. Explain in words what the slope and y-intercept of the regression line tell us. d. How well does the regression line fit the data? Explain your response. e. Which point has the largest residual? Explain what the residual means in context. Is this point an outlier? An influential point? Explain. f. An ecologist wants to predict how many birds will join another colony of sparrow hawks to which 70% of the adults from the previous year have returned. What is the prediction? 76. The following table shows data on average per capita wine consumption and heart disease rate in a random sample of 10 countries. Yearly wine consumption in liters 2.5 3.9 2.9 2.4 2.9 0.8 9.1 2.7 0.8 0.7 Death from heart diseases 221 167 131 191 220 297 71 172 211 300 Table 12.25 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 679 a. Enter the data into your calculator and make a scatter plot. b. Use your calculatorโs regression function to find the equation of the least-squares regression line. Add this to your scatter plot from part a. c. Explain in words what the slope and y-intercept of the regression line tell us. d. How well does the regression line fit the data? Explain your response. e. Which point has the largest residual? Explain what the |
residual means in context. Is this point an outlier? An influential point? Explain. f. Do the data provide convincing evidence that there is a linear relationship between the amount of alcohol consumed and the heart disease death rate? Carry out an appropriate test at a significance level of 0.05 to help answer this question. 77. The following table consists of one student athleteโs time (in minutes) to swim 2000 yards and the studentโs heart rate (beats per minute) after swimming on a random sample of 10 days: Swim Time Heart Rate 34.12 35.72 34.72 34.05 34.13 35.73 36.17 35.57 35.37 35.57 Table 12.26 144 152 124 140 152 146 128 136 144 148 a. Enter the data into your calculator and make a scatter plot. b. Use your calculatorโs regression function to find the equation of the least-squares regression line. Add this to your scatter plot from part a. c. Explain in words what the slope and y-intercept of the regression line tell us. d. How well does the regression line fit the data? Explain your response. e. Which point has the largest residual? Explain what the residual means in context. Is this point an outlier? An influential point? Explain. 78. A researcher is investigating whether non-white minorities commit a disproportionate number of homicides. He uses demographic data from Detroit, MI to compare homicide rates and the number of the population that are white males. White Males Homicide rate per 100,000 people 558,724 538,584 519,171 500,457 482,418 465,029 448,267 432,109 416,533 Table 12.27 8.6 8.9 8.52 8.89 13.07 14.57 21.36 28.03 31.49 680 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION White Males Homicide rate per 100,000 people 401,518 387,046 373,095 359,647 Table 12.27 37.39 46.26 47.24 52.33 a. Use your calculator to construct a scatter plot of the data. What should the independent variable be? Why? b. Use your calculatorโs regression function to find the equation of the least-squares regression line. Add this to your scatter plot. c. Discuss what the following mean in context. i. The slope of the regression equation ii. |
The y-intercept of the regression equation iii. The correlation r iv. The coefficient of determination r2. d. Do the data provide convincing evidence that there is a linear relationship between the number of white males in the population and the homicide rate? Carry out an appropriate test at a significance level of 0.05 to help answer this question. Mid-Career Salary (in thousands) Yearly Tuition 79. School Princeton Harvey Mudd CalTech 137 135 127 US Naval Academy 122 West Point MIT Lehigh University NYU-Poly Babson College Stanford Table 12.28 120 118 118 117 117 114 28,540 40,133 39,900 0 0 42,050 43,220 39,565 40,400 54,506 Using the data to determine the linear-regression line equation with the outliers removed. Is there a linear correlation for the data set with outliers removed? Justify your answer. REFERENCES 12.1 Linear Equations Data from the Centers for Disease Control and Prevention. Data from the National Center for HIV, STD, and TB Prevention. 12.5 Prediction Data from the Centers for Disease Control and Prevention. Data from the National Center for HIV, STD, and TB Prevention. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 Data from the United States Census Bureau. Available online at http://www.census.gov/compendia/statab/cats/ transportation/motor_vehicle_accidents_and_fatalities.html CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 681 Data from the National Center for Health Statistics. 12.6 Outliers Data from the House Ways and Means Committee, the Health and Human Services Department. Data from Microsoft Bookshelf. Data from the United States Department of Labor, the Bureau of Labor Statistics. Data from the Physicianโs Handbook, 1990. Data from the United States Department of Labor, the Bureau of Labor Statistics. BRINGING IT TOGETHER: HOMEWORK 80. The average number of people in a family that received welfare for various years is given in Table 12.29. Year Welfare family size 1969 1973 1975 1979 1983 1988 1991 Table 12.29 4.0 3.6 3.2 3.0 3.0 3.0 2.9 a. Using โyearโ as the |
independent variable and โwelfare family sizeโ as the dependent variable, draw a scatter plot of the data. b. Calculate the least-squares line. Put the equation in the form of: ลท = a + bx c. Find the correlation coefficient. Is it significant? d. Pick two years between 1969 and 1991 and find the estimated welfare family sizes. e. Based on the data in Table 12.29, is there a linear relationship between the year and the average number of people in a welfare family? f. Using the least-squares line, estimate the welfare family sizes for 1960 and 1995. Does the least-squares line give an accurate estimate for those years? Explain why or why not. g. Are there any outliers in the data? h. What is the estimated average welfare family size for 1986? Does the least squares line give an accurate estimate for that year? Explain why or why not. i. What is the slope of the least squares (best-fit) line? Interpret the slope. 81. The percent of female wage and salary workers who are paid hourly rates is given in Table 12.30 for the years 1979 to 1992. Year Percent of workers paid hourly rates 1979 1980 1981 1982 Table 12.30 61.2 60.7 61.3 61.3 682 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION Year Percent of workers paid hourly rates 1983 1984 1985 1986 1987 1990 1992 Table 12.30 61.8 61.7 61.8 62.0 62.7 62.8 62.9 a. Using โyearโ as the independent variable and โpercentโ as the dependent variable, draw a scatter plot of the data. b. Does it appear from inspection that there is a relationship between the variables? Why or why not? c. Calculate the least-squares line. Put the equation in the form of: ลท = a + bx d. Find the correlation coefficient. Is it significant? e. Find the estimated percents for 1991 and 1988. f. Based on the data, is there a linear relationship between the year and the percent of female wage and salary earners who are paid hourly rates? g. Are there any outliers in the data? h. What is the estimated percent for the year 2050? Does the least-squares line give an accurate estimate for that year? Explain why or why not. i. What is the slope of the least-squ |
ares (best-fit) line? Interpret the slope. Use the following information to answer the next two exercises. The cost of a leading liquid laundry detergent in different sizes is given in Table 12.31. Size (ounces) Cost ($) Cost per ounce 16 32 64 200 Table 12.31 3.99 4.99 5.99 10.99 82. 83. a. Using โsizeโ as the independent variable and โcostโ as the dependent variable, draw a scatter plot. b. Does it appear from inspection that there is a relationship between the variables? Why or why not? c. Calculate the least-squares line. Put the equation in the form of: ลท = a + bx d. Find the correlation coefficient. Is it significant? e. f. g. Does it appear that a line is the best way to fit the data? Why or why not? h. Are there any outliers in the given data? i. If the laundry detergent were sold in a 40-ounce size, find the estimated cost. If the laundry detergent were sold in a 90-ounce size, find the estimated cost. Is the least-squares line valid for predicting what a 300-ounce size of the laundry detergent would you cost? Why or why not? j. What is the slope of the least-squares (best-fit) line? Interpret the slope. a. Complete Table 12.31 for the cost per ounce of the different sizes. b. Using โsizeโ as the independent variable and โcost per ounceโ as the dependent variable, draw a scatter plot of the data. c. Does it appear from inspection that there is a relationship between the variables? Why or why not? d. Calculate the least-squares line. Put the equation in the form of: ลท = a + bx This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 683 If the laundry detergent were sold in a 40-ounce size, find the estimated cost per ounce. If the laundry detergent were sold in a 90-ounce size, find the estimated cost per ounce. e. Find the correlation coefficient. Is it significant? f. g. h. Does it appear that a line is |
the best way to fit the data? Why or why not? i. Are there any outliers in the the data? j. Is the least-squares line valid for predicting what a 300-ounce size of the laundry detergent would cost per ounce? Why or why not? k. What is the slope of the least-squares (best-fit) line? Interpret the slope. 84. According to a flyer by a Prudential Insurance Company representative, the costs of approximate probate fees and taxes for selected net taxable estates are as follows: Net Taxable Estate ($) Approximate Probate Fees and Taxes ($) 600,000 750,000 1,000,000 1,500,000 2,000,000 2,500,000 3,000,000 Table 12.32 30,000 92,500 203,000 438,000 688,000 1,037,000 1,350,000 a. Decide which variable should be the independent variable and which should be the dependent variable. b. Draw a scatter plot of the data. c. Does it appear from inspection that there is a relationship between the variables? Why or why not? d. Calculate the least-squares line. Put the equation in the form of: ลท = a + bx. e. Find the correlation coefficient. Is it significant? f. Find the estimated total cost for a next taxable estate of $1,000,000. Find the cost for $2,500,000. g. Does it appear that a line is the best way to fit the data? Why or why not? h. Are there any outliers in the data? i. Based on these results, what would be the probate fees and taxes for an estate that does not have any assets? j. What is the slope of the least-squares (best-fit) line? Interpret the slope. 85. The following are advertised sale prices of color televisions at Andersonโs. Size (inches) Sale Price ($) 9 20 27 31 35 40 60 Table 12.33 147 197 297 447 1177 2177 2497 a. Decide which variable should be the independent variable and which should be the dependent variable. b. Draw a scatter plot of the data. c. Does it appear from inspection that there is a relationship between the variables? Why or why not? d. Calculate the least-squares line. Put the equation in the form of: ๏ฟฝ |
๏ฟฝ = a + bx e. Find the correlation coefficient. Is it significant? 684 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION f. Find the estimated sale price for a 32 inch television. Find the cost for a 50 inch television. g. Does it appear that a line is the best way to fit the data? Why or why not? h. Are there any outliers in the data? i. What is the slope of the least-squares (best-fit) line? Interpret the slope. 86. Table 12.34 shows the average heights for American boy s in 1990. Age (years) Height (cm) birth 2 3 5 7 10 14 Table 12.34 50.8 83.8 91.4 106.6 119.3 137.1 157.5 a. Decide which variable should be the independent variable and which should be the dependent variable. b. Draw a scatter plot of the data. c. Does it appear from inspection that there is a relationship between the variables? Why or why not? d. Calculate the least-squares line. Put the equation in the form of: ลท = a + bx e. Find the correlation coefficient. Is it significant? f. Find the estimated average height for a one-year-old. Find the estimated average height for an eleven-year-old. g. Does it appear that a line is the best way to fit the data? Why or why not? h. Are there any outliers in the data? i. Use the least squares line to estimate the average height for a sixty-two-year-old man. Do you think that your answer is reasonable? Why or why not? j. What is the slope of the least-squares (best-fit) line? Interpret the slope. # letters in name Year entered the Union Ranks for entering the Union Area (square miles) 87. State Alabama Colorado Hawaii Iowa Maryland Missouri 7 8 6 4 8 8 New Jersey 9 Ohio South Carolina Utah Wisconsin Table 12.35 4 13 4 9 1819 1876 1959 1846 1788 1821 1787 1803 1788 1896 1848 22 38 50 29 7 24 3 17 8 45 30 52,423 104,100 10,932 56,276 12,407 69,709 8,722 44,828 32,008 84,904 65,499 We are interested in whether there is a relationship between the ranking of a state and the area |
of the state. a. What are the independent and dependent variables? This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 685 b. What do you think the scatter plot will look like? Make a scatter plot of the data. c. Does it appear from inspection that there is a relationship between the variables? Why or why not? d. Calculate the least-squares line. Put the equation in the form of: ลท = a + bx e. Find the correlation coefficient. What does it imply about the significance of the relationship? f. Find the estimated areas for Alabama and for Colorado. Are they close to the actual areas? g. Use the two points in part f to plot the least-squares line on your graph from part b. h. Does it appear that a line is the best way to fit the data? Why or why not? i. Are there any outliers? j. Use the least squares line to estimate the area of a new state that enters the Union. Can the least-squares line be used to predict it? Why or why not? k. Delete โHawaiiโ and substitute โAlaskaโ for it. Alaska is the forty-ninth, state with an area of 656,424 square miles. l. Calculate the new least-squares line. m. Find the estimated area for Alabama. Is it closer to the actual area with this new least-squares line or with the previous one that included Hawaii? Why do you think thatโs the case? n. Do you think that, in general, newer states are larger than the original states? SOLUTIONS 1 dependent variable: fee amount; independent variable: time 3 Figure 12.31 5 Figure 12.32 7 y = 6x + 8, 4y = 8, and y + 7 = 3x are all linear equations. 686 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 9 The number of AIDS cases depends on the year. Therefore, year becomes the independent variable and the number of AIDS cases is the dependent variable. 11 The y-intercept is 50 (a = 50). At the start of the cleaning, the company charges a one-time fee of $50 (this is when x |
= 0). The slope is 100 (b = 100). For each session, the company charges $100 for each hour they clean. 13 12,000 pounds of soil 15 The slope is โ1.5 (b = โ1.5). This means the stock is losing value at a rate of $1.50 per hour. The y-intercept is $15 (a = 15). This means the price of stock before the trading day was $15. 17 The data appear to be linear with a strong, positive correlation. 19 The data appear to have no correlation. 21 ลท = 2.23 + 1.99x 23 The slope is 1.99 (b = 1.99). It means that for every endorsement deal a professional player gets, he gets an average of another $1.99 million in pay each year. 25 It means that there is no correlation between the data sets. 27 Yes, there are enough data points and the value of r is strong enough to show that there is a strong negative correlation between the data sets. 29 Ha: ฯ โ 0 31 $250,120 33 1,326 acres 35 1,125 hours, or when x = 1,125 37 Check studentโs solution. 39 a. When x = 1985, ลท = 25,52 b. When x = 1990, ลท = 34,275 c. When x = 1970, ลท = โ725 Why doesnโt this answer make sense? The range of x values was 1981 to 2002; the year 1970 is not in this range. The regression equation does not apply, because predicting for the year 1970 is extrapolation, which requires a different process. Also, a negative number does not make sense in this context, where we are predicting AIDS cases diagnosed. 41 Also, the correlation r = 0.4526. If r is compared to the value in the 95% Critical Values of the Sample Correlation Coefficient Table, because r > 0.423, r is significant, and you would think that the line could be used for prediction. But the scatter plot indicates otherwise. 43 y^ = 3,448,225 + 1750x 45 There was an increase in AIDS cases diagnosed until 1993. From 1993 through 2002, the number of AIDS cases diagnosed declined each year. It is not appropriate to use a linear regression line to fit to the data. 47 Since there is no linear association between year and # of AIDS cases diagnosed, it is not appropriate to calculate a linear |
correlation coefficient. When there is a linear association and it is appropriate to calculate a correlation, we cannot say that one variable โcausesโ the other variable. 49 We donโt know if the pre-1981 data was collected from a single year. So we donโt have an accurate x value for this figure. Regression equation: ลท (#AIDS Cases) = โ3,448,225 + 1749.777 (year) This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 687 Coefficients Intercept โ3,448,225 X Variable 1 1,749.777 Table 12.36 51 Yes, there appears to be an outlier at (6, 58). 53 The potential outlier flattened the slope of the line of best fit because it was below the data set. It made the line of best fit less accurate is a predictor for the data. 55 s = 1.75 57 a. b. c. d. e. independent variable: age; dependent variable: fatalities independent variable: # of family members; dependent variable: grocery bill independent variable: age of applicant; dependent variable: insurance premium independent variable: power consumption; dependent variable: utility independent variable: higher education (years); dependent variable: crime rates 59 Check studentโs solution. 61 For graph: check studentโs solution. Note that tuition is the independent variable and salary is the dependent variable. 63 13 65 It means that 72% of the variation in the dependent variable (y) can be explained by the variation in the independent variable (x). 67 We do not reject the null hypothesis. There is not sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is not significantly different from zero. 69 a. Age Number of Driver Deaths per 100,000 16โ19 20โ24 25โ34 35โ54 55โ74 75+ Table 12.37 38 36 24 20 18 28 b. Check studentโs solution. c. ลท = 35.5818045 โ 0.19182491x d. r = โ0.57874 For four df and alpha = 0.05, the LinRegTTest gives p-value = 0.2288 so we do not reject the null hypothesis; there is |
not a significant linear relationship between deaths and age. 688 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION Using the table of critical values for the correlation coefficient, with four df, the critical value is 0.811. The correlation coefficient r = โ0.57874 is not less than โ0.811, so we do not reject the null hypothesis. e. if age = 40, ลท (deaths) = 35.5818045 โ 0.19182491(40) = 27.9 if age = 60, ลท (deaths) = 35.5818045 โ 0.19182491(60) = 24.1 f. For entire dataset, there is a linear relationship for the ages up to age 74. The oldest age group shows an increase in deaths from the prior group, which is not consistent with the younger ages. g. slope = โ0.19182491 71 a. We wonder if the better discounts appear earlier in the book so we select page as X and discount as Y. b. Check studentโs solution. c. ลท = 17.21757 โ 0.01412x d. r = โ 0.2752 For seven df and alpha = 0.05, using LinRegTTest p-value = 0.4736 so we do not reject; there is a not a significant linear relationship between page and discount. Using the table of critical values for the correlation coefficient, with seven df, the critical value is 0.666. The correlation coefficient xi = โ0.2752 is not less than 0.666 so we do not reject. e. page 10: 17.08 page 70: 16.23 f. There is not a significant linear correlation so it appears there is no relationship between the page and the amount of the discount. g. page 200: 14.39 h. No, using the regression equation to predict for page 200 is extrapolation. i. slope = โ0.01412 As the page number increases by one page, the discount decreases by $0.01412 73 a. Year is the independent or x variable; the number of letters is the dependent or y variable. b. Check studentโs solution. c. no d. ลท = 47.03 โ 0.0216x e. โ0.4280 f. 6; 5 g. No, the relationship does not appear to be linear; the correlation is not |
significant. h. current year: 2013: 3.55 or four letters; this is not an appropriate use of the least squares line. It is extrapolation. 75 a. and b. Check studentโs solution. c. The slope of the regression line is -0.3179 with a y-intercept of 32.966. In context, the y-intercept indicates that when there are no returning sparrow hawks, there will be almost 31% new sparrow hawks, which doesnโt make sense since if there are no returning birds, then the new percentage would have to be 100% (this is an example of why we do not extrapolate). The slope tells us that for each percentage increase in returning birds, the percentage of new birds in the colony decreases by 0.3179%. d. If we examine r2, we see that only 50.238% of the variation in the percent of new birds is explained by the model and the correlation coefficient, r = 0.71 only indicates a somewhat strong correlation between returning and new percentages. e. The ordered pair (66, 6) generates the largest residual of 6.0. This means that when the observed return percentage is 66%, our observed new percentage, 6%, is almost 6% less than the predicted new value of 11.98%. If we remove this data pair, we see only an adjusted slope of -0.2723 and an adjusted intercept of 30.606. In other words, even though this data generates the largest residual, it is not an outlier, nor is the data pair an influential point. f. If there are 70% returning birds, we would expect to see y = -0.2723(70) + 30.606 = 0.115 or 11.5% new birds in the colony. 77 a. Check studentโs solution. b. Check studentโs solution. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 689 c. We have a slope of โ1.4946 with a y-intercept of 193.88. The slope, in context, indicates that for each additional minute added to the swim time, the heart rate will decrease by 1.5 beats per minute. If the student is not swimming at all |
, the y-intercept indicates that his heart rate will be 193.88 beats per minute. While the slope has meaning (the longer it takes to swim 2,000 meters, the less effort the heart puts out), the y-intercept does not make sense. If the athlete is not swimming (resting), then his heart rate should be very low. d. Since only 1.5% of the heart rate variation is explained by this regression equation, we must conclude that this association is not explained with a linear relationship. e. The point (34.72, 124) generates the largest residual of โ11.82. This means that our observed heart rate is almost 12 beats less than our predicted rate of 136 beats per minute. When this point is removed, the slope becomes 1.6914 with the y-intercept changing to 83.694. While the linear association is still very weak, we see that the removed data pair can be considered an influential point in the sense that the y-intercept becomes more meaningful. 79 If we remove the two service academies (the tuition is $0.00), we construct a new regression equation of y = โ0.0009x + 160 with a correlation coefficient of 0.71397 and a coefficient of determination of 0.50976. This allows us to say there is a fairly strong linear association between tuition costs and salaries if the service academies are removed from the data set. 81 a. Check student's solution. b. yes c. ลท = โ266.8863+0.1656x d. 0.9448; Yes e. 62.8233; 62.3265 f. yes g. yes; (1987, 62.7) h. 72.5937; no i. slope = 0.1656. As the year increases by one, the percent of workers paid hourly rates tends to increase by 0.1656. 83 a. Size (ounces) Cost ($) cents/oz 16 32 64 200 Table 12.38 3.99 4.99 5.99 10.99 24.94 15.59 9.36 5.50 b. Check studentโs solution. c. There is a linear relationship for the sizes 16 through 64, but that linear trend does not continue to the 200-oz size. d. ลท = 20.2368 โ 0.0819x e. r = โ0.8086 f. 40-oz: 16.96 cents/ |
oz g. 90-oz: 12.87 cents/oz h. The relationship is not linear; the least squares line is not appropriate. i. no outliers j. No, you would be extrapolating. The 300-oz size is outside the range of x. k. slope = โ0.08194; for each additional ounce in size, the cost per ounce decreases by 0.082 cents. 690 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 85 a. Size is x, the independent variable, price is y, the dependent variable. b. Check studentโs solution. c. The relationship does not appear to be linear. d. ลท = โ745.252 + 54.75569x e. r = 0.8944, yes it is significant f. 32-inch: $1006.93, 50-inch: $1992.53 g. No, the relationship does not appear to be linear. However, r is significant. h. yes, the 60-inch TV i. For each additional inch, the price increases by $54.76 87 a. Let rank be the independent variable and area be the dependent variable. b. Check studentโs solution. c. There appears to be a linear relationship, with one outlier. d. ลท (area) = 24177.06 + 1010.478x e. r = 0.50047, r is not significant so there is no relationship between the variables. f. Alabama: 46407.576 Colorado: 62575.224 g. Alabama estimate is closer than Colorado estimate. h. If the outlier is removed, there is a linear relationship. i. There is one outlier (Hawaii). j. rank 51: 75711.4; no k. Alabama Colorado Alaska Iowa Maryland Missouri New Jersey Ohio 7 8 6 4 8 8 9 4 1819 1876 1959 1846 1788 1821 1787 1803 South Carolina 13 1788 Utah Wisconsin 4 9 1896 1848 Table 12.39 22 38 51 29 7 24 3 17 8 45 30 52,423 104,100 656,424 56,276 12,407 69,709 8,722 44,828 32,008 84,904 65,499 l. ลท = โ87065.3 + 7828.532x m. Alabama: 85,162.404; the prior estimate was closer. Alaska is an outlier. n |
. yes, with the exception of Hawaii 81 a. Check student's solution. b. yes This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 691 c. ลท = โ266.8863+0.1656x d. 0.9448; Yes e. 62.8233; 62.3265 f. yes g. yes; (1987, 62.7) h. 72.5937; no i. slope = 0.1656. As the year increases by one, the percent of workers paid hourly rates tends to increase by 0.1656. 83 a. Size (ounces) Cost ($) cents/oz 16 32 64 200 Table 12.40 3.99 4.99 5.99 10.99 24.94 15.59 9.36 5.50 b. Check studentโs solution. c. There is a linear relationship for the sizes 16 through 64, but that linear trend does not continue to the 200-oz size. d. ลท = 20.2368 โ 0.0819x e. r = โ0.8086 f. 40-oz: 16.96 cents/oz g. 90-oz: 12.87 cents/oz h. The relationship is not linear; the least squares line is not appropriate. i. no outliers j. No, you would be extrapolating. The 300-oz size is outside the range of x. k. slope = โ0.08194; for each additional ounce in size, the cost per ounce decreases by 0.082 cents. 85 a. Size is x, the independent variable, price is y, the dependent variable. b. Check studentโs solution. c. The relationship does not appear to be linear. d. ลท = โ745.252 + 54.75569x e. r = 0.8944, yes it is significant f. 32-inch: $1006.93, 50-inch: $1992.53 g. No, the relationship does not appear to be linear. However, r is significant. h. yes, the 60-inch TV i. For each additional inch, the price increases by $54.76 87 a. Let rank be the independent variable and area be the dependent variable. |
b. Check studentโs solution. c. There appears to be a linear relationship, with one outlier. 692 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION d. ลท (area) = 24177.06 + 1010.478x e. r = 0.50047, r is not significant so there is no relationship between the variables. f. Alabama: 46407.576 Colorado: 62575.224 g. Alabama estimate is closer than Colorado estimate. h. If the outlier is removed, there is a linear relationship. i. There is one outlier (Hawaii). j. rank 51: 75711.4; no k. Alabama Colorado Alaska Iowa Maryland Missouri New Jersey Ohio 7 8 6 4 8 8 9 4 1819 1876 1959 1846 1788 1821 1787 1803 South Carolina 13 1788 Utah Wisconsin 4 9 1896 1848 Table 12.41 22 38 51 29 7 24 3 17 8 45 30 52,423 104,100 656,424 56,276 12,407 69,709 8,722 44,828 32,008 84,904 65,499 l. ลท = โ87065.3 + 7828.532x m. Alabama: 85,162.404; the prior estimate was closer. Alaska is an outlier. n. yes, with the exception of Hawaii This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 693 13 | F DISTRIBUTION AND ONE-WAY ANOVA Figure 13.1 One-way ANOVA is used to measure information from several groups. Introduction Chapter Objectives By the end of this chapter, the student should be able to: Interpret the F probability distribution as the number of groups and the sample size change. โข โข Discuss two uses for the F distribution: one-way ANOVA and the test of two variances. โข Conduct and interpret one-way ANOVA. โข Conduct and interpret hypothesis tests of two variances. Many statistical applications in psychology, social science, business administration, and the natural sciences involve several groups. For example, an environmentalist is interested in knowing if the average amount of pollution varies in several bodies of water. A sociologist is interested in knowing if the amount of income a person earns varies according |
to his or her upbringing. A consumer looking for a new car might compare the average gas mileage of several models. 694 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA For hypothesis tests comparing averages between more than two groups, statisticians have developed a method called "Analysis of Variance" (abbreviated ANOVA). In this chapter, you will study the simplest form of ANOVA called single factor or one-way ANOVA. You will also study the F distribution, used for one-way ANOVA, and the test of two variances. This is just a very brief overview of one-way ANOVA. You will study this topic in much greater detail in future statistics courses. One-Way ANOVA, as it is presented here, relies heavily on a calculator or computer. 13.1 | One-Way ANOVA The purpose of a one-way ANOVA test is to determine the existence of a statistically significant difference among several group means. The test actually uses variances to help determine if the means are equal or not. In order to perform a oneway ANOVA test, there are five basic assumptions to be fulfilled: 1. Each population from which a sample is taken is assumed to be normal. 2. All samples are randomly selected and independent. 3. The populations are assumed to have equal standard deviations (or variances). 4. The factor is a categorical variable. 5. The response is a numerical variable. The Null and Alternative Hypotheses The null hypothesis is simply that all the group population means are the same. The alternative hypothesis is that at least one pair of means is different. For example, if there are k groups: H0: ฮผ1 = ฮผ2 = ฮผ3 =... = ฮผk Ha: At least two of the group means ฮผ1, ฮผ2, ฮผ3,..., ฮผk are not equal. The graphs, a set of box plots representing the distribution of values with the group means indicated by a horizontal line through the box, help in the understanding of the hypothesis test. In the first graph (red box plots), H0: ฮผ1 = ฮผ2 = ฮผ3 and the three populations have the same distribution if the null hypothesis is true. The variance of the combined data is approximately the same as the variance of each of the populations. If the null hypothesis is false, then the variance of the combined data is larger which is caused by the different means as shown in the second graph (green box plots). Figure |
13.2 (a) H0 is true. All means are the same; the differences are due to random variation. (b) H0 is not true. All means are not the same; the differences are too large to be due to random variation. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 695 13.2 | The F Distribution and the F-Ratio The distribution used for the hypothesis test is a new one. It is called the F distribution, named after Sir Ronald Fisher, an English statistician. The F statistic is a ratio (a fraction). There are two sets of degrees of freedom; one for the numerator and one for the denominator. For example, if F follows an F distribution and the number of degrees of freedom for the numerator is four, and the number of degrees of freedom for the denominator is ten, then F ~ F4,10. NOTE The F distribution is derived from the Student's t-distribution. The values of the F distribution are squares of the corresponding values of the t-distribution. One-Way ANOVA expands the t-test for comparing more than two groups. The scope of that derivation is beyond the level of this course. To calculate the F ratio, two estimates of the variance are made. 1. Variance between samples: An estimate of ฯ2 that is the variance of the sample means multiplied by n (when the sample sizes are the same.). If the samples are different sizes, the variance between samples is weighted to account for the different sample sizes. The variance is also called variation due to treatment or explained variation. 2. Variance within samples: An estimate of ฯ2 that is the average of the sample variances (also known as a pooled variance). When the sample sizes are different, the variance within samples is weighted. The variance is also called the variation due to error or unexplained variation. โข SSbetween = the sum of squares that represents the variation among the different samples โข SSwithin = the sum of squares that represents the variation within samples that is due to chance. To find a "sum of squares" means to add together squared quantities that, in some cases, may be weighted. We used sum of squares to calculate the sample variance and the sample standard deviation in Section 2.. |
MS means " mean square." MSbetween is the variance between groups, and MSwithin is the variance within groups. Calculation of Sum of Squares and Mean Square โข k = the number of different groups โข nj = the size of the jth group โข sj = the sum of the values in the jth group โข n = total number of all the values combined (total sample size: โnj) โข x = one value: โx = โsj โข Sum of squares of all values from every group combined: โx2 โข Between group variability: SStotal = โx2 โ โโ x2โ โ โ n โข Total sum of squares: โx2 โ 2 โ โโ xโ n โ โข Explained variation: (โ s j)2 โก โข(sj)2 n n j โฃ โค โฅ โ โฆ โ sum of squares representing variation among the different samples: SSbetween = โข Unexplained variation: sum of squares representing variation within samples due to chance: SSwithin = SStotal โ SSbetween โข df's for different groups (df's for the numerator): df = k โ 1 โข Equation for errors within samples (df's for the denominator): dfwithin = n โ k โข Mean square (variance estimate) explained by the different groups: MSbetween = SSbetween d fbetween 696 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA โข Mean square (variance estimate) that is due to chance (unexplained): MSwithin = SSwithin d fwithin MSbetween and MSwithin can be written as follows: โข MSbetween = SSbetween d fbetween = SSbetween k โ 1 โข MSwithin = SSwithin d fwithin = SSwithin n โ k The one-way ANOVA test depends on the fact that MSbetween can be influenced by population differences among means of the several groups. Since MSwithin compares values of each group to its own group mean, the fact that group means might be different does not affect MSwithin. The null hypothesis says that all groups are samples from populations having the same normal distribution. The alternate hypothesis says that at least two of the sample groups come from populations with different normal distributions. If the null hypothesis is true, MSbetween and MSwithin should both estimate the same value |
. NOTE The null hypothesis says that all the group population means are equal. The hypothesis of equal means implies that the populations have the same normal distribution, because it is assumed that the populations are normal and that they have equal variances. F-Ratio or F Statistic F = MSbetween MSwithin If MSbetween and MSwithin estimate the same value (following the belief that H0 is true), then the F-ratio should be approximately equal to one. Mostly, just sampling errors would contribute to variations away from one. As it turns out, MSbetween consists of the population variance plus a variance produced from the differences between the samples. MSwithin is an estimate of the population variance. Since variances are always positive, if the null hypothesis is false, MSbetween will generally be larger than MSwithin.Then the F-ratio will be larger than one. However, if the population effect is small, it is not unlikely that MSwithin will be larger in a given sample. The foregoing calculations were done with groups of different sizes. If the groups are the same size, the calculations simplify somewhat and the F-ratio can be written as: F-Ratio Formula when the groups are the same size F = 2 n โ
s xยฏ s2 pooled where... โข n = the sample size โข dfnumerator = k โ 1 โข dfdenominator = n โ k โข โข s2 pooled = the mean of the sample variances (pooled variance) 2 = the variance of the sample means s xยฏ Data are typically put into a table for easy viewing. One-Way ANOVA results are often displayed in this manner by computer software. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 Source of Variation Sum of Squares (SS) Degrees of Freedom (df) Mean Square (MS) F CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 697 SS(Factor) SS(Error) SS(Total MS(Factor) = SS(Factor)/(k โ 1) F = MS(Factor)/MS(Error) MS(Error) = SS(Error)/(n โ k) Factor (Between) Error (Within) Total Table 13.1 Example 13.1 Three different diet plans are to be tested for mean weight loss. The entries in the table are the weight losses for |
the different plans. The one-way ANOVA results are shown in Table 13.2. Plan 1: n1 = 4 Plan 2: n2 = 3 Plan 3: n3 = 3 5 4.5 4 3 Table 13.2 3.5 7 4.5 8 4 3.5 s1 = 16.5, s2 =15, s3 = 15.7 Following are the calculations needed to fill in the one-way ANOVA table. The table is used to conduct a hypothesis test. SS(between) = โ (s j) โ โ โ โโ s j n = 2 s1 4 2 s2 3 + + 2 s3 3 (s1 + s2 + s3)2 10 โ where n1 = 4, n2 = 3, n3 = 3 and n = n1 + n2 + n3 = 10 = (16.5)2 4 + (15)2 3 + (5.5)2 3 โ (16.5 + 15 + 15.5)2 10 SS(between) = 2.2458 โโ xโ n S(total) = โ x2 โ โ โ 2 = โ52 + 4.52 + 42 + 32 + 3.52 + 72 + 4.52 + 82 + 42 + 3.52โ โ โ (5 + 4.5 + 4 + 3 + 3.5 + 7 + 4.5 + 8 + 4 + 3.5)2 10 โ = 244 โ 472 10 = 244 โ 220.9 SS(total) = 23.1 SS(within) = SS(total) โ SS(between) = 23.1 โ 2.2458 SS(within) = 20.8542 698 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA One-Way ANOVA Table: The formulas for SS(Total), SS(Factor) = SS(Between) and SS(Error) = SS(Within) as shown previously. The same information is provided by the TI calculator hypothesis test function ANOVA in STAT TESTS (syntax is ANOVA(L1, L2, L3) where L1, L2, L3 have the data from Plan 1, Plan 2, Plan 3 respectively). Source of Variation Sum of Squares (SS) Degrees of Freedom |
(df) Mean Square (MS) F Factor (Between) SS(Factor) = SS(Between) = 2.2458 k โ 1 = 3 groups โ 1 = 2 SS(Error) = SS(Within) = 20.8542 SS(Total) = 2.2458 + 20.8542 = 23.1 n โ k = 10 total data โ 3 groups = 7 n โ 1 = 10 total data โ 1 = 9 Error (Within) Total Table 13.3 F = MS(Factor)/MS(Error) = 1.1229/2.9792 = 0.3769 MS(Factor) = SS(Factor)/(k โ 1) = 2.2458/2 = 1.1229 MS(Error) = SS(Error)/(n โ k) = 20.8542/7 = 2.9792 13.1 As part of an experiment to see how different types of soil cover would affect slicing tomato production, Marist College students grew tomato plants under different soil cover conditions. Groups of three plants each had one of the following treatments โข bare soil โข a commercial ground cover โข black plastic โข straw โข compost All plants grew under the same conditions and were the same variety. Students recorded the weight (in grams) of tomatoes produced by each of the n = 15 plants: Bare: n1 = 3 Ground Cover: n2 = 3 Plastic: n3 = 3 Straw: n4 = 3 Compost: n5 = 3 6,583 8,560 3,830 7,285 6,897 9,230 6,277 7,818 8,677 2,625 2,997 4,915 Table 13.4 5,348 5,682 5,482 Create the one-way ANOVA table. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 The one-way ANOVA hypothesis test is always right-tailed because larger F-values are way out in the right tail of the F-distribution curve and tend to make us reject H0. CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 699 Notation The notation for the F distribution is F ~ Fdf(num),df(denom) where df(num) = dfbetween and df(denom) = dfwithin The mean for the F distribution is ยต = d f |
(num) d f (denom) โ 1 13.3 | Facts About the F Distribution Here are some facts about the F distribution. 1. The curve is not symmetrical but skewed to the right. 2. There is a different curve for each set of dfs. 3. The F statistic is greater than or equal to zero. 4. As the degrees of freedom for the numerator and for the denominator get larger, the curve approximates the normal. 5. Other uses for the F distribution include comparing two variances and two-way Analysis of Variance. Two-Way Analysis is beyond the scope of this chapter. Figure 13.3 Example 13.2 Letโs return to the slicing tomato exercise in Try It. The means of the tomato yields under the five mulching conditions are represented by ฮผ1, ฮผ2, ฮผ3, ฮผ4, ฮผ5. We will conduct a hypothesis test to determine if all means are the same or at least one is different. Using a significance level of 5%, test the null hypothesis that there is no difference in mean yields among the five groups against the alternative hypothesis that at least one mean is different from the rest. Solution 13.2 The null and alternative hypotheses are: H0: ฮผ1 = ฮผ2 = ฮผ3 = ฮผ4 = ฮผ5 Ha: ฮผi โ ฮผj some i โ j The one-way ANOVA results are shown in Table 13.4 700 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA Sum of Squares (SS) Degrees of Freedom (df) Mean Square (MS) F 36,648,561 4 20,446,726 10 = 9,162,140 9,162,140 2,044,672.6 = 4.4810 = 2,044,672.6 Source of Variation Factor (Between) 36,648,561 5 โ 1 = 4 Error (Within) 20,446,726 15 โ 5 = 10 Total 57,095,287 15 โ 1 = 14 Table 13.5 Distribution for the test: F4,10 df(num) = 5 โ 1 = 4 df(denom) = 15 โ 5 = 10 Test statistic: F = 4.4810 Figure 13.4 Probability Statement: p-value = P(F > 4.481) = 0.0248. Compare ฮฑ and the p-value: ฮฑ = 0.05, p-value = 0.0248 |
Make a decision: Since ฮฑ > p-value, we reject H0. Conclusion: At the 5% significance level, we have reasonably strong evidence that differences in mean yields for slicing tomato plants grown under different mulching conditions are unlikely to be due to chance alone. We may conclude that at least some of mulches led to different mean yields. To find these results on the calculator: Press STAT. Press 1:EDIT. Put the data into the lists L1, L2, L3, L4, L5. Press STAT, and arrow over to TESTS, and arrow down to ANOVA. Press ENTER, and then enter L1, L2, L3, L4, L5). Press ENTER. You will see that the values in the foregoing ANOVA table are easily produced by the calculator, including the test statistic and the p-value of the test. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 701 The calculator displays: F = 4.4810 p = 0.0248 (p-value) Factor df = 4 SS = 36648560.9 MS = 9162140.23 Error df = 10 SS = 20446726 MS = 2044672.6 13.2 MRSA, or Staphylococcus aureus, can cause a serious bacterial infections in hospital patients. Table 13.6 shows various colony counts from different patients who may or may not have MRSA. Conc = 0.6 Conc = 0.8 Conc = 1.0 Conc = 1.2 Conc = 1.4 9 66 98 Table 13.6 16 93 82 22 147 120 30 199 148 27 168 132 Plot of the data for the different concentrations: Figure 13.5 Test whether the mean number of colonies are the same or are different. Construct the ANOVA table (by hand or by using a TI-83, 83+, or 84+ calculator), find the p-value, and state your conclusion. Use a 5% significance level. 702 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA Example 13.3 Four sororities took a random sample of sisters regarding their grade means for the past term. The results are shown in Table 13.7. Sorority 1 Sorority 2 Sorority 3 Sorority 4 |
2.17 1.85 2.83 1.69 3.33 2.63 1.77 3.25 1.86 2.21 2.63 3.78 4.00 2.55 2.45 3.79 3.45 3.08 2.26 3.18 Table 13.7 MEAN GRADES FOR FOUR SORORITIES Using a significance level of 1%, is there a difference in mean grades among the sororities? Solution 13.3 Let ฮผ1, ฮผ2, ฮผ3, ฮผ4 be the population means of the sororities. Remember that the null hypothesis claims that the sorority groups are from the same normal distribution. The alternate hypothesis says that at least two of the sorority groups come from populations with different normal distributions. Notice that the four sample sizes are each five. NOTE This is an example of a balanced design, because each factor (i.e., sorority) has the same number of observations. H0: ฮผ1 = ฮผ2 = ฮผ3 = ฮผ4 Ha: Not all of the means ฮผ1, ฮผ2, ฮผ3, ฮผ4 are equal. Distribution for the test: F3,16 where k = 4 groups and n = 20 samples in total df(num)= k โ 1 = 4 โ 1 = 3 df(denom) = n โ k = 20 โ 4 = 16 Calculate the test statistic: F = 2.23 Graph: This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 703 Figure 13.6 Probability statement: p-value = P(F > 2.23) = 0.1241 Compare ฮฑ and the p-value: ฮฑ = 0.01 p-value = 0.1241 ฮฑ < p-value Make a decision: Since ฮฑ < p-value, you cannot reject H0. Conclusion: There is not sufficient evidence to conclude that there is a difference among the mean grades for the sororities. Put the data into lists L1, L2, L3, and L4. Press STAT and arrow over to TESTS. Arrow down to F:ANOVA. Press ENTER and Enter (L1,L2,L3,L4). The calculator displays the F statistic, the p-value and the values |
for the one-way ANOVA table: F = 2.2303 p = 0.1241 (p-value) Factor df = 3 SS = 2.88732 MS = 0.96244 Error df = 16 SS = 6.9044 MS = 0.431525 704 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 13.3 Four sports teams took a random sample of players regarding their GPAs for the last year. The results are shown in Table 13.8. Basketball Baseball Hockey Lacrosse 3.6 2.9 2.5 3.3 3.8 2.1 2.6 3.9 3.1 3.4 4.0 2.0 2.6 3.2 3.2 2.0 3.6 3.9 2.7 2.5 Table 13.8 GPAs FOR FOUR SPORTS TEAMS Use a significance level of 5%, and determine if there is a difference in GPA among the teams. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 705 Example 13.4 A fourth grade class is studying the environment. One of the assignments is to grow bean plants in different soils. Tommy chose to grow his bean plants in soil found outside his classroom mixed with dryer lint. Tara chose to grow her bean plants in potting soil bought at the local nursery. Nick chose to grow his bean plants in soil from his mother's garden. No chemicals were used on the plants, only water. They were grown inside the classroom next to a large window. Each child grew five plants. At the end of the growing period, each plant was measured, producing the data (in inches) in Table 13.9. Tommy's Plants Tara's Plants Nick's Plants 24 21 23 30 23 Table 13.9 25 31 23 20 28 23 27 22 30 20 Does it appear that the three media in which the bean plants were grown produce the same mean height? Test at a 3% level of significance. Solution 13.4 This time, we will perform the calculations that lead to the F' statistic. Notice that each group has the same number of plants, so we will use the formula F' = 2. n โ
s xยฏ s2 pooled First, calculate the sample mean and sample variance of |
each group. Tommy's Plants Tara's Plants Nick's Plants 24.2 11.7 25.4 18.3 24.4 16.3 Sample Mean Sample Variance Table 13.10 Next, calculate the variance of the three group means (Calculate the variance of 24.2, 25.4, and 24.4). Variance of the group means = 0.413 = s xยฏ 2 Then MSbetween = ns xยฏ 2 = (5)(0.413) where n = 5 is the sample size (number of plants each child grew). Calculate the mean of the three sample variances (Calculate the mean of 11.7, 18.3, and 16.3). Mean of the sample variances = 15.433 = s2 pooled Then MSwithin = s2 pooled = 15.433. The F statistic (or F ratio) is F = MSbetween MSwithin = 2 ns xยฏ s2 pooled = (5)(0.413) 15.433 = 0.134 The dfs for the numerator = the number of groups โ 1 = 3 โ 1 = 2. The dfs for the denominator = the total number of samples โ the number of groups = 15 โ 3 = 12 The distribution for the test is F2,12 and the F statistic is F = 0.134 The p-value is P(F > 0.134) = 0.8759. 706 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA Decision: Since ฮฑ = 0.03 and the p-value = 0.8759, do not reject H0. (Why?) Conclusion: With a 3% level of significance, from the sample data, the evidence is not sufficient to conclude that the mean heights of the bean plants are different. To calculate the p-value: *Press 2nd DISTR *Arrow down to Fcdf(and press ENTER. *Enter 0.134, E99, 2, 12) *Press ENTER The p-value is 0.8759. 13.4 Another fourth grader also grew bean plants, but this time in a jelly-like mass. The heights were (in inches) 24, 28, 25, 30, and 32. Do a one-way ANOVA test on the four groups. Are the heights of the bean plants different? Use the same method as shown in Example 13.4. From the class, create four groups of the same size as follows: men |
under 22, men at least 22, women under 22, women at least 22. Have each member of each group record the number of states in the United States he or she has visited. Run an ANOVA test to determine if the average number of states visited in the four groups are the same. Test at a 1% level of significance. Use one of the solution sheets in Appendix E. 13.4 | Test of Two Variances Another of the uses of the F distribution is testing two variances. It is often desirable to compare two variances rather than two averages. For instance, college administrators would like two college professors grading exams to have the same variation in their grading. In order for a lid to fit a container, the variation in the lid and the container should be the same. A supermarket might be interested in the variability of check-out times for two checkers. In order to perform a F test of two variances, it is important that the following are true: 1. The populations from which the two samples are drawn are normally distributed. 2. The two populations are independent of each other. Unlike most other tests in this book, the F test for equality of two variances is very sensitive to deviations from normality. If the two distributions are not normal, the test can give higher p-values than it should, or lower ones, in ways that are unpredictable. Many texts suggest that students not use this test at all, but in the interest of completeness we include it here. Suppose we sample randomly from two independent normal populations. Let ฯ1 2 and s2 s1 variances, we use the F ratio: 2 be the sample variances. Let the sample sizes be n1 and n2. Since we are interested in comparing the two sample 2 and ฯ2 2 be the population variances and This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 707 F = โก โข โฃ โก โข โฃ (s1)2 (ฯ1)2 (s2)2 (ฯ2)2 โค โฅ โฆ โค โฅ โฆ F has the distribution F ~ F(n1 โ 1, n2 โ 1) where n1 โ 1 |
are the degrees of freedom for the numerator and n2 โ 1 are the degrees of freedom for the denominator. If the null hypothesis is ฯ1 2 = ฯ2 2, then the F Ratio becomes F = โก โข โฃ โก โข โฃ (s1)2 (ฯ1)2 (s2)2 (ฯ2)2 โค โฅ โฆ โค โฅ โฆ = (s1)2 (s2)2. NOTE The F ratio could also be (s2)2 (s1)2. It depends on Ha and on which sample variance is larger. If the two populations have equal variances, then s1 2 and s2 2 are close in value and F = (s1)2 (s2)2 is close to one. But if the two population variances are very different, s1 variance causes the ratio (s1)2 (s2)2 2 and s2 2 tend to be very different, too. Choosing s1 2 as the larger sample to be greater than one. If s1 2 and s2 2 are far apart, then F = (s1)2 (s2)2 is a large number. Therefore, if F is close to one, the evidence favors the null hypothesis (the two population variances are equal). But if F is much larger than one, then the evidence is against the null hypothesis. A test of two variances may be left, right, or two-tailed. Example 13.5 Two college instructors are interested in whether or not there is any variation in the way they grade math exams. They each grade the same set of 30 exams. The first instructor's grades have a variance of 52.3. The second instructor's grades have a variance of 89.9. Test the claim that the first instructor's variance is smaller. (In most colleges, it is desirable for the variances of exam grades to be nearly the same among instructors.) The level of significance is 10%. Solution 13.5 Let 1 and 2 be the subscripts that indicate the first and second instructor, respectively. n1 = n2 = 30. H0: ฯ1 2 = ฯ2 2 and Ha: ฯ1 2 2 < ฯ2 Calculate the test statistic: By the null hypothesis (ฯ1 2 = ฯ2 2 ), the F statistic iss1) |
2 (ฯ1)2 (s2)2 (ฯ2)2 โค โฅ โฆ โค โฅ โฆ = (s1)2 (s2)2 = 52.3 89.9 = 0.5818 Distribution for the test: F29,29 where n1 โ 1 = 29 and n2 โ 1 = 29. Graph: This test is left tailed. 708 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA Draw the graph labeling and shading appropriately. Figure 13.7 Probability statement: p-value = P(F < 0.5818) = 0.0753 Compare ฮฑ and the p-value: ฮฑ = 0.10 ฮฑ > p-value. Make a decision: Since ฮฑ > p-value, reject H0. Conclusion: With a 10% level of significance, from the data, there is sufficient evidence to conclude that the variance in grades for the first instructor is smaller. Press STAT and arrow over to TESTS. Arrow down to D:2-SampFTest. Press ENTER. Arrow to Stats (89.9), and 30. Press ENTER after and press ENTER. For Sx1, n1, Sx2, and n2, enter each. Arrow to ฯ1: and < ฯ2. Press ENTER. Arrow down to Calculate and press ENTER. F = 0.5818 and p-value = 0.0753. Do the procedure again and try Draw instead of Calculate. (52.3), 30, This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 709 13.5 The New York Choral Society divides male singers up into four categories from highest voices to lowest: Tenor1, Tenor2, Bass1, Bass2. In the table are heights of the men in the Tenor1 and Bass2 groups. One suspects that taller men will have lower voices, and that the variance of height may go up with the lower voices as well. Do we have good evidence that the variance of the heights of singers in each of these two groups (Tenor1 and Bass2) are different? Tenor1 Bass2 Tenor 1 Bass 2 Tenor 1 Bass 2 69 |
72 71 66 76 74 71 66 68 72 75 67 75 74 72 72 74 72 Table 13.11 67 70 65 72 70 68 64 73 66 72 74 70 66 68 75 68 70 72 68 67 64 67 70 70 69 72 71 74 75 710 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 13.1 One-Way ANOVA Class Time: Names: Student Learning Outcome โข The student will conduct a simple one-way ANOVA test involving three variables. Collect the Data 1. Record the price per pound of eight fruits, eight vegetables, and eight breads in your local supermarket. Fruits Vegetables Breads Table 13.12 2. Explain how you could try to collect the data randomly. Analyze the Data and Conduct a Hypothesis Test 1. Compute the following: a. Fruit: i. ii. xยฏ = ______ s x = ______ iii. n = ______ b. Vegetables: i. ii. xยฏ = ______ s x = ______ iii. n = ______ c. Bread: i. ii. xยฏ = ______ s x = ______ iii. n = ______ 2. Find the following: a. df(num) = ______ b. df(denom) = ______ This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 3. State the approximate distribution for the test. 4. Test statistic: F = ______ 5. Sketch a graph of this situation. CLEARLY, label and scale the horizontal axis and shade the region(s) CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 711 corresponding to the p-value. 6. p-value = ______ 7. Test at ฮฑ = 0.05. State your decision and conclusion. 8. a. Decision: Why did you make this decision? b. Conclusion (write a complete sentence). c. Based on the results of your study, is there a need to investigate any of the food groupsโ prices? Why or why not? 712 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA KEY TERMS Analysis of Variance also referred to as ANOVA, is a method of testing whether or not the means of three or more populations are equal. The method is applicable if: โข all populations of interest are normally distributed. โข โข the populations have equal standard deviations. samples (not |
necessarily of the same size) are randomly and independently selected from each population. The test statistic for analysis of variance is the F-ratio. One-Way ANOVA a method of testing whether or not the means of three or more populations are equal; the method is applicable if: โข all populations of interest are normally distributed. โข โข the populations have equal standard deviations. samples (not necessarily of the same size) are randomly and independently selected from each population. The test statistic for analysis of variance is the F-ratio. Variance mean of the squared deviations from the mean; the square of the standard deviation. For a set of data, a deviation can be represented as x โ xยฏ where x is a value of the data and xยฏ variance is equal to the sum of the squares of the deviations divided by the difference of the sample size and one. is the sample mean. The sample CHAPTER REVIEW 13.1 One-Way ANOVA Analysis of variance extends the comparison of two groups to several, each a level of a categorical variable (factor). Samples from each group are independent, and must be randomly selected from normal populations with equal variances. We test the null hypothesis of equal means of the response in every group versus the alternative hypothesis of one or more group means being different from the others. A one-way ANOVA hypothesis test determines if several population means are equal. The distribution for the test is the F distribution with two different degrees of freedom. Assumptions: 1. Each population from which a sample is taken is assumed to be normal. 2. All samples are randomly selected and independent. 3. The populations are assumed to have equal standard deviations (or variances). 13.2 The F Distribution and the F-Ratio Analysis of variance compares the means of a response variable for several groups. ANOVA compares the variation within each group to the variation of the mean of each group. The ratio of these two is the F statistic from an F distribution with (number of groups โ 1) as the numerator degrees of freedom and (number of observations โ number of groups) as the denominator degrees of freedom. These statistics are summarized in the ANOVA table. 13.3 Facts About the F Distribution The graph of the F distribution is always positive and skewed right, though the shape can be mounded or exponential depending on the combination of numerator and denominator degrees of freedom. The F statistic is the ratio of a measure of the variation in the group means to a similar measure |
of the variation within the groups. If the null hypothesis is correct, then the numerator should be small compared to the denominator. A small F statistic will result, and the area under the F curve to the right will be large, representing a large p-value. When the null hypothesis of equal group means is incorrect, then the numerator should be large compared to the denominator, giving a large F statistic and a small area (small p-value) to the right of the statistic under the F curve. When the data have unequal group sizes (unbalanced data), then techniques from Section 13.2 need to be used for hand calculations. In the case of balanced data (the groups are the same size) however, simplified calculations based on group This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 713 means and variances may be used. In practice, of course, software is usually employed in the analysis. As in any analysis, graphs of various sorts should be used in conjunction with numerical techniques. Always look of your data! 13.4 Test of Two Variances The F test for the equality of two variances rests heavily on the assumption of normal distributions. The test is unreliable if this assumption is not met. If both distributions are normal, then the ratio of the two sample variances is distributed as an F statistic, with numerator and denominator degrees of freedom that are one less than the samples sizes of the corresponding two groups. A test of two variances hypothesis test determines if two variances are the same. The distribution for the hypothesis test is the F distribution with two different degrees of freedom. Assumptions: 1. The populations from which the two samples are drawn are normally distributed. 2. The two populations are independent of each other. FORMULA REVIEW 13.2 The F Distribution and the F-Ratio SSbetween = โ (s j) โ โ โ โโ s j n SStotal = โ x2 โ 2 โ โโ xโ โ n SSwithin = SStotal โ SSbetween dfbetween = df(num) = k โ 1 dfwithin = df(denom) = n โ k MSbetween = SSbetween d f |
between MSwithin = SSwithin d fwithin F = MSbetween MSwithin F ratio when the groups are the same size: F = Mean of the F distribution: ยต = d f (num) d f (denom) โ 1 PRACTICE 13.1 One-Way ANOVA where: โข k = the number of groups โข nj = the size of the jth group โข sj = the sum of the values in the jth group โข n = the total number of all values (observations) combined โข x = one value (one observation) from the data โข โข s xยฏ s2 2 = the variance of the sample means pooled = the mean of the sample variances (pooled variance) 13.4 Test of Two Variances F has the distribution F ~ F(n1 โ 1, n2 โ 1 ns xยฏ s2 pooled If ฯ1 = ฯ2, then F = 2 2 s1 s2 Use the following information to answer the next five exercises. There are five basic assumptions that must be fulfilled in order to perform a one-way ANOVA test. What are they? 1. Write one assumption. 2. Write another assumption. 3. Write a third assumption. 4. Write a fourth assumption. 714 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 5. Write the final assumption. 6. State the null hypothesis for a one-way ANOVA test if there are four groups. 7. State the alternative hypothesis for a one-way ANOVA test if there are three groups. 8. When do you use an ANOVA test? 13.2 The F Distribution and the F-Ratio Use the following information to answer the next eight exercises. Groups of men from three different areas of the country are to be tested for mean weight. The entries in the table are the weights for the different groups. The one-way ANOVA results are shown in Table 13.13. Group 1 Group 2 Group 3 216 198 240 187 176 202 213 284 228 210 Table 13.13 170 165 182 197 201 9. What is the Sum of Squares Factor? 10. What is the Sum of Squares Error? 11. What is the df for the numerator? 12. What is the df for the denominator? 13. What is the Mean Square Factor? 14. What is the Mean Square Error? 15. What is the F statistic? Use the following information to answer the next eight exercises. Girls |
from four different soccer teams are to be tested for mean goals scored per game. The entries in the table are the goals per game for the different teams. The one-way ANOVA results are shown in Table 13.14. Team 1 Team 2 Team 3 Team Table 13.14 16. What is SSbetween? 17. What is the df for the numerator? 18. What is MSbetween? 19. What is SSwithin? 20. What is the df for the denominator? 21. What is MSwithin? 22. What is the F statistic? This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 23. Judging by the F statistic, do you think it is likely or unlikely that you will reject the null hypothesis? CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 715 13.3 Facts About the F Distribution 24. An F statistic can have what values? 25. What happens to the curves as the degrees of freedom for the numerator and the denominator get larger? Use the following information to answer the next seven exercise. Four basketball teams took a random sample of players regarding how high each player can jump (in inches). The results are shown in Table 13.15. Team 1 Team 2 Team 3 Team 4 Team 5 36 42 51 32 35 38 48 50 39 38 44 46 41 39 40 Table 13.15 26. What is the df(num)? 27. What is the df(denom)? 28. What are the Sum of Squares and Mean Squares Factors? 29. What are the Sum of Squares and Mean Squares Errors? 30. What is the F statistic? 31. What is the p-value? 32. At the 5% significance level, is there a difference in the mean jump heights among the teams? Use the following information to answer the next seven exercises. A video game developer is testing a new game on three different groups. Each group represents a different target market for the game. The developer collects scores from a random sample from each group. The results are shown in Table 13.16 Group A Group B Group C 101 108 98 107 111 151 149 160 112 126 101 109 198 186 160 Table 13.16 33. What is the df(num)? 34. What is the df(denom)? 35. What are the SSbetween and MSbetween? 36. What are the |
SSwithin and MSwithin? 37. What is the F Statistic? 38. What is the p-value? 39. At the 10% significance level, are the scores among the different groups different? Use the following information to answer the next three exercises. Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country. Suppose that the following data are randomly collected from five teenagers in each region of the country. The numbers represent the age at which teenagers obtained their drivers licenses. 716 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA Northeast South West Central East 16.3 16.1 16.4 16.5 16.9 16.5 16.4 16.2 16.4 16.5 16.6 16.1 16.2 16.6 16.5 16.4 17.1 17.2 16.6 16.8 xยฏ = ________ s2 = ________ Table 13.17 ________ ________ ________ ________ ________ ________ ________ ________ Enter the data into your calculator or computer. 40. p-value = ______ State the decisions and conclusions (in complete sentences) for the following preconceived levels of ฮฑ. 41. ฮฑ = 0.05 a. Decision: ____________________________ b. Conclusion: ____________________________ 42. ฮฑ = 0.01 a. Decision: ____________________________ b. Conclusion: ____________________________ 13.4 Test of Two Variances Use the following information to answer the next two exercises. There are two assumptions that must be true in order to perform an F test of two variances. 43. Name one assumption that must be true. 44. What is the other assumption that must be true? Use the following information to answer the next five exercises. Two coworkers commute from the same building. They are interested in whether or not there is any variation in the time it takes them to drive to work. They each record their times for 20 commutes. The first workerโs times have a variance of 12.1. The second workerโs times have a variance of 16.9. The first worker thinks that he is more consistent with his commute times and that his commute time is shorter. Test the claim at the 10% level. 45. State the null and alternative hypotheses. 46. What is s1 in this problem? 47. What is s2 in this problem? 48. What is n? 49. What is the F statistic? |
50. What is the p-value? 51. Is the claim accurate? Use the following information to answer the next four exercises. Two students are interested in whether or not there is variation in their test scores for math class. There are 15 total math tests they have taken so far. The first studentโs grades have a standard deviation of 38.1. The second studentโs grades have a standard deviation of 22.5. The second student thinks his scores are lower. 52. State the null and alternative hypotheses. 53. What is the F Statistic? 54. What is the p-value? 55. At the 5% significance level, do we reject the null hypothesis? This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 717 Use the following information to answer the next three exercises. Two cyclists are comparing the variances of their overall paces going uphill. Each cyclist records his or her speeds going up 35 hills. The first cyclist has a variance of 23.8 and the second cyclist has a variance of 32.1. The cyclists want to see if their variances are the same or different. 56. State the null and alternative hypotheses. 57. What is the F Statistic? 58. At the 5% significance level, what can we say about the cyclistsโ variances? HOMEWORK 13.1 One-Way ANOVA 59. Three different traffic routes are tested for mean driving time. The entries in the table are the driving times in minutes on the three different routes. The one-way ANOVA results are shown in Table 13.18. Route 1 Route 2 Route 3 30 32 27 35 27 29 28 36 Table 13.18 16 41 22 31 State SSbetween, SSwithin, and the F statistic. 60. Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country. Suppose that the following data are randomly collected from five teenagers in each region of the country. The numbers represent the age at which teenagers obtained their drivers licenses. Northeast South West Central East 16.3 16.1 16.4 16.5 16.9 16.5 16.4 16.2 16.4 16.5 16.6 16.1 16.2 16.6 16.5 16. |
4 17.1 17.2 16.6 16.8 xยฏ = ________ s2 = ________ Table 13.19 ________ ________ ________ ________ ________ ________ ________ ________ State the hypotheses. H0: ____________ Ha: ____________ 13.2 The F Distribution and the F-Ratio Use the following information to answer the next three exercises. Suppose a group is interested in determining whether teenagers obtain their drivers licenses at approximately the same average age across the country. Suppose that the following data are randomly collected from five teenagers in each region of the country. The numbers represent the age at which teenagers obtained their drivers licenses. 718 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA Northeast South West Central East 16.3 16.1 16.4 16.5 16.9 16.5 16.4 16.2 16.4 16.5 16.6 16.1 16.2 16.6 16.5 16.4 17.1 17.2 16.6 16.8 xยฏ = ________ s2 = ________ Table 13.20 ________ ________ ________ ________ ________ ________ ________ ________ H0: ยต1 = ยต2 = ยต3 = ยต4 = ยต5 Hฮฑ: At least any two of the group means ยต1, ยต2, โฆ, ยต5 are not equal. 61. degrees of freedom โ numerator: df(num) = _________ 62. degrees of freedom โ denominator: df(denom) = ________ 63. F statistic = ________ 13.3 Facts About the F Distribution DIRECTIONS Use a solution sheet to conduct the following hypothesis tests. The solution sheet can be found in Appendix E. 64. Three students, Linda, Tuan, and Javier, are given five laboratory rats each for a nutritional experiment. Each rat's weight is recorded in grams. Linda feeds her rats Formula A, Tuan feeds his rats Formula B, and Javier feeds his rats Formula C. At the end of a specified time period, each rat is weighed again, and the net gain in grams is recorded. Using a significance level of 10%, test the hypothesis that the three formulas produce the same mean weight gain. Linda's rats Tuan's rats Javier's rats 43.5 39.4 41.3 46.0 38.2 47.0 40.5 38.9 46.3 44.2 51.2 40.9 37.9 45 |
.0 48.6 Table 13.21 Weights of Student Lab Rats 65. A grassroots group opposed to a proposed increase in the gas tax claimed that the increase would hurt working-class people the most, since they commute the farthest to work. Suppose that the group randomly surveyed 24 individuals and asked them their daily one-way commuting mileage. The results are in Table 13.22. Using a 5% significance level, test the hypothesis that the three mean commuting mileages are the same. working-class professional (middle incomes) professional (wealthy) 17.8 26.7 49.4 Table 13.22 16.5 17.4 22.0 8.5 6.3 4.6 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 working-class professional (middle incomes) professional (wealthy) CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 719 9.4 65.4 47.1 19.5 51.2 Table 13.22 7.4 9.4 2.1 6.4 13.9 12.6 11.0 28.6 15.4 9.3 66. Examine the seven practice laps from Table 13.1. Determine whether the mean lap time is statistically the same for the seven practice laps, or if there is at least one lap that has a different mean time from the others. Use the following information to answer the next two exercises. Table 13.23 lists the number of pages in four different types of magazines. home decorating news health computer 172 286 163 205 197 Table 13.23 87 94 123 106 101 82 153 87 103 96 104 136 98 207 146 67. Using a significance level of 5%, test the hypothesis that the four magazine types have the same mean length. 68. Eliminate one magazine type that you now feel has a mean length different from the others. Redo the hypothesis test, testing that the remaining three means are statistically the same. Use a new solution sheet. Based on this test, are the mean lengths for the remaining three magazines statistically the same? 69. A researcher wants to know if the mean times (in minutes) that people watch their favorite news station are the same. Suppose that Table 13.24 shows the results of a study. CNN FOX Local 72 37 56 60 51 45 12 18 38 23 35 15 43 68 50 31 22 |
Table 13.24 Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05. 70. Are the means for the final exams the same for all statistics class delivery types? Table 13.25 shows the scores on final exams from several randomly selected classes that used the different delivery types. 720 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA Online Hybrid Face-to-Face 83 73 84 81 72 84 77 80 81 Table 13.25 80 78 84 81 86 79 82 Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05. 71. Are the mean number of times a month a person eats out the same for whites, blacks, Hispanics and Asians? Suppose that Table 13.26 shows the results of a study. White Black Hispanic Asian 4 1 5 2 6 8 2 4 6 Table 13.26 Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05. 72. Are the mean numbers of daily visitors to a ski resort the same for the three types of snow conditions? Suppose that Table 13.27 shows the results of a study. Powder Machine Made Hard Packed 1,210 1,080 1,537 941 Table 13.27 2,107 1,149 862 1,870 1,528 1,382 2,846 1,638 2,019 1,178 2,233 Assume that all distributions are normal, the four population standard deviations are approximately the same, and the data were collected independently and randomly. Use a level of significance of 0.05. 73. Sanjay made identical paper airplanes out of three different weights of paper, light, medium and heavy. He made four airplanes from each of the weights, and launched them himself across the room. Here are the distances (in meters) that his planes flew. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 721 Paper Type/Trial Trial 1 Trial 2 Trial 3 Trial 4 Heavy Medium Light Table 13.28 5. |
1 meters 3.1 meters 4.7 meters 5.3 meters 4 meters 3.5 meters 4.5 meters 6.1 meters 3.1 meters 3.3 meters 2.1 meters 1.9 meters Figure 13.8 a. Take a look at the data in the graph. Look at the spread of data for each group (light, medium, heavy). Does it seem reasonable to assume a normal distribution with the same variance for each group? Yes or No. b. Why is this a balanced design? c. Calculate the sample mean and sample standard deviation for each group. d. Does the weight of the paper have an effect on how far the plane will travel? Use a 1% level of significance. Complete the test using the method shown in the bean plant example in Example 13.4. โฆ variance of the group means __________ โฆ MSbetween= ___________ โฆ mean of the three sample variances ___________ โฆ MSwithin = _____________ โฆ F statistic = ____________ โฆ df(num) = __________, df(denom) = ___________ โฆ number of groups _______ โฆ number of observations _______ โฆ p-value = __________ (P(F > _______) = __________) โฆ Graph the p-value. โฆ decision: _______________________ โฆ conclusion: _______________________________________________________________ 74. DDT is a pesticide that has been banned from use in the United States and most other areas of the world. It is quite effective, but persisted in the environment and over time became seen as harmful to higher-level organisms. Famously, egg shells of eagles and other raptors were believed to be thinner and prone to breakage in the nest because of ingestion of DDT in the food chain of the birds. An experiment was conducted on the number of eggs (fecundity) laid by female fruit flies. There are three groups of flies. One group was bred to be resistant to DDT (the RS group). Another was bred to be especially susceptible to DDT (SS). Finally there was a control line of non-selected or typical fruitflies (NS). Here are the data: RS SS NS RS SS NS 12.8 38.4 35.4 22.4 23.1 22.6 Table 13.29 722 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA RS SS NS RS SS NS 21.6 32. |
9 27.4 27.5 29.4 40.4 14.8 48.5 19.3 20.3 16 34.4 23.1 20.9 41.8 38.7 20.1 30.4 34.6 11.6 20.3 26.4 23.3 14.9 19.7 22.3 37.6 23.7 22.9 51.8 22.6 30.2 36.9 26.1 22.5 33.8 29.6 33.4 37.3 29.5 15.1 37.9 16.4 26.7 28.2 38.6 31 29.5 20.3 39 23.4 44.4 16.9 42.4 29.3 12.8 33.7 23.2 16.1 36.6 14.9 14.6 29.2 23.6 10.8 47.4 27.3 12.2 41.7 Table 13.29 The values are the average number of eggs laid daily for each of 75 flies (25 in each group) over the first 14 days of their lives. Using a 1% level of significance, are the mean rates of egg selection for the three strains of fruitfly different? If so, in what way? Specifically, the researchers were interested in whether or not the selectively bred strains were different from the nonselected line, and whether the two selected lines were different from each other. Here is a chart of the three groups: Figure 13.9 75. The data shown is the recorded body temperatures of 130 subjects as estimated from available histograms. Traditionally we are taught that the normal human body temperature is 98.6 F. This is not quite correct for everyone. Are the mean temperatures among the four groups different? Calculate 95% confidence intervals for the mean body temperature in each group and comment about the confidence intervals. FL FH ML MH FL FH ML MH 96.4 96.8 96.3 96.9 98.4 98.6 98.1 98.6 96.7 97.7 96.7 97 98.7 98.6 98.1 98.6 Table 13.30 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 723 FL FH ML MH FL FH ML MH |
97.2 97.8 97.1 97.1 98.7 98.6 98.2 98.7 97.2 97.9 97.2 97.1 98.7 98.7 98.2 98.8 97.4 98 97.3 97.4 98.7 98.7 98.2 98.8 97.6 98 97.4 97.5 98.8 98.8 98.2 98.8 97.7 98 97.4 97.6 98.8 98.8 98.3 98.9 97.8 98 97.4 97.7 98.8 98.8 98.4 99 97.8 98.1 97.5 97.8 98.8 98.9 98.4 99 97.9 98.3 97.6 97.9 99.2 99 98.5 99 97.9 98.3 97.6 98 99.3 99 98.5 99.2 98 98.3 97.8 98 99.1 98.6 99.5 98.2 98.4 97.8 98 99.1 98.6 98.2 98.4 97.8 98.3 99.2 98.7 98.2 98.4 97.9 98.4 99.4 99.1 98.2 98.4 98 98.2 98.5 98 98.2 98.6 98 98.4 98.6 98.6 99.9 99.3 100 99.4 100.8 Table 13.30 13.4 Test of Two Variances 76. Three students, Linda, Tuan, and Javier, are given five laboratory rats each for a nutritional experiment. Each ratโs weight is recorded in grams. Linda feeds her rats Formula A, Tuan feeds his rats Formula B, and Javier feeds his rats Formula C. At the end of a specified time period, each rat is weighed again and the net gain in grams is recorded. Linda's rats Tuan's rats Javier's rats 43.5 39.4 41.3 46.0 38.2 Table 13.31 47.0 40.5 38.9 46.3 44.2 51.2 40.9 37.9 45.0 48.6 Determine whether or not the variance in weight gain is statistically the same among Javierโs and Lindaโs rats. Test at a significance level of 10%. 77. A grassroots group opposed to a proposed increase in the gas tax claimed that |
the increase would hurt working-class people the most, since they commute the farthest to work. Suppose that the group randomly surveyed 24 individuals and asked them their daily one-way commuting mileage. The results are as follows. working-class professional (middle incomes) professional (wealthy) 17.8 26.7 Table 13.32 16.5 17.4 8.5 6.3 724 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA working-class professional (middle incomes) professional (wealthy) 49.4 9.4 65.4 47.1 19.5 51.2 Table 13.32 22.0 7.4 9.4 2.1 6.4 13.9 4.6 12.6 11.0 28.6 15.4 9.3 Determine whether or not the variance in mileage driven is statistically the same among the working class and professional (middle income) groups. Use a 5% significance level. 78. Refer to the data from Table 13.1. Examine practice laps 3 and 4. Determine whether or not the variance in lap time is statistically the same for those practice laps. Use the following information to answer the next two exercises. The following table lists the number of pages in four different types of magazines. home decorating news health computer 172 286 163 205 197 Table 13.33 87 94 123 106 101 82 153 87 103 96 104 136 98 207 146 79. Which two magazine types do you think have the same variance in length? 80. Which two magazine types do you think have different variances in length? 81. Is the variance for the amount of money, in dollars, that shoppers spend on Saturdays at the mall the same as the variance for the amount of money that shoppers spend on Sundays at the mall? Suppose that the Table 13.34 shows the results of a study. Saturday Sunday Saturday Sunday 75 18 150 94 62 73 Table 13.34 44 58 61 19 99 60 89 62 0 124 50 31 118 137 82 39 127 141 73 82. Are the variances for incomes on the East Coast and the West Coast the same? Suppose that Table 13.35 shows the results of a study. Income is shown in thousands of dollars. Assume that both distributions are normal. Use a level of significance of 0.05. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1 |
.16 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 725 East West 71 126 42 51 44 90 88 38 47 30 82 75 52 115 67 Table 13.35 83. Thirty men in college were taught a method of finger tapping. They were randomly assigned to three groups of ten, with each receiving one of three doses of caffeine: 0 mg, 100 mg, 200 mg. This is approximately the amount in no, one, or two cups of coffee. Two hours after ingesting the caffeine, the men had the rate of finger tapping per minute recorded. The experiment was double blind, so neither the recorders nor the students knew which group they were in. Does caffeine affect the rate of tapping, and if so how? Here are the data: 0 mg 100 mg 200 mg 0 mg 100 mg 200 mg 242 244 247 242 246 248 245 248 247 243 Table 13.36 246 250 248 246 245 245 248 248 244 242 246 247 250 246 244 248 252 250 248 250 84. King Manuel I, Komnenus ruled the Byzantine Empire from Constantinople (Istanbul) during the years 1145 to 1180 A.D. The empire was very powerful during his reign, but declined significantly afterwards. Coins minted during his era were found in Cyprus, an island in the eastern Mediterranean Sea. Nine coins were from his first coinage, seven from the second, four from the third, and seven from a fourth. These spanned most of his reign. We have data on the silver content of the coins: First Coinage Second Coinage Third Coinage Fourth Coinage 6.9 9.0 6.6 8.1 9.3 9.2 8.6 5.9 6.8 6.4 7.0 6.6 7.7 7.2 6.9 6.2 Table 13.37 4.9 5.5 4.6 4.5 5.3 5.6 5.5 5.1 6.2 5.8 5.8 726 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA Did the silver content of the coins change over the course of Manuelโs reign? Here are the means and variances of each coinage. The data are unbalanced. First Second Third Fourth Mean 6.7444 8.2429 4.875 5.6143 Variance 0.2953 1.2095 0.2025 0.1314 Table 13.38 85. The American League and the National League |
of Major League Baseball are each divided into three divisions: East, Central, and West. Many years, fans talk about some divisions being stronger (having better teams) than other divisions. This may have consequences for the postseason. For instance, in 2012 Tampa Bay won 90 games and did not play in the postseason, while Detroit won only 88 and did play in the postseason. This may have been an oddity, but is there good evidence that in the 2012 season, the American League divisions were significantly different in overall records? Use the following data to test whether the mean number of wins per team in the three American League divisions were the same or not. Note that the data are not balanced, as two divisions had five teams, while one had only four. Division Team Wins East East East East East NY Yankees 95 Baltimore 93 Tampa Bay 90 Toronto Boston 73 69 Table 13.39 Division Team Wins Central Detroit 88 Central Chicago Sox 85 Central Kansas City 72 Central Cleveland Central Minnesota 68 66 Table 13.40 Division Team Wins West West West West Oakland Texas 94 93 LA Angels 89 Seattle 75 Table 13.41 REFERENCES 13.2 The F Distribution and the F-Ratio Tomato Data, Marist College School of Science (unpublished student research) This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 727 13.3 Facts About the F Distribution Data from a fourth grade classroom in 1994 in a private K โ 12 school in San Jose, CA. Hand, D.J., F. Daly, A.D. Lunn, K.J. McConway, and E. Ostrowski. A Handbook of Small Datasets: Data for Fruitfly Fecundity. London: Chapman & Hall, 1994. Hand, D.J., F. Daly, A.D. Lunn, K.J. McConway, and E. Ostrowski. A Handbook of Small Datasets. London: Chapman & Hall, 1994, pg. 50. Hand, D.J., F. Daly, A.D. Lunn, K.J. McConway, and E. Ostrowski. A Handbook of Small Datasets. London: Chapman & Hall, 1994, pg. 118. โMLB Standings โ 2012.โ Available online at |
http://espn.go.com/mlb/standings/_/year/2012. Mackowiak, P. A., Wasserman, S. S., and Levine, M. M. (1992), "A Critical Appraisal of 98.6 Degrees F, the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold August Wunderlich," Journal of the American Medical Association, 268, 1578-1580. 13.4 Test of Two Variances โMLB Vs. Division Standings โ 2012.โ Available online at http://espn.go.com/mlb/standings/_/year/2012/type/vs-division/ order/true. SOLUTIONS 1 Each population from which a sample is taken is assumed to be normal. 3 The populations are assumed to have equal standard deviations (or variances). 5 The response is a numerical value. 7 Ha: At least two of the group means ฮผ1, ฮผ2, ฮผ3 are not equal. 9 4,939.2 11 2 13 2,469.6 15 3.7416 17 3 19 13.2 21 0.825 23 Because a one-way ANOVA test is always right-tailed, a high F statistic corresponds to a low p-value, so it is likely that we will reject the null hypothesis. 25 The curves approximate the normal distribution. 27 ten 29 SS = 237.33; MS = 23.73 31 0.1614 33 two 35 SS = 5,700.4; MS = 2,850.2 37 3.6101 39 Yes, there is enough evidence to show that the scores among the groups are statistically significant at the 10% level. 728 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 43 The populations from which the two samples are drawn are normally distributed. 45 H0: ฯ1 = ฯ2 Ha: ฯ1 < ฯ2 or H0: ฯ1 2 = ฯ2 2 Ha: ฯ1 2 2 < ฯ2 47 4.11 49 0.7159 51 No, at the 10% level of significance, we do not reject the null hypothesis and state that the data do not show that the variation in drive times for the first worker is less than the variation in drive times for the second worker. 53 2.8674 55 Reject the null hypothesis. There is enough evidence to say that the variance of |
the grades for the first student is higher than the variance in the grades for the second student. 57 0.7414 59 SSbetween = 26 SSwithin = 441 F = 0.2653 62 df(denom) = 15 64 a. H0: ยตL = ยตT = ยตJ b. at least any two of the means are different c. df(num) = 2; df(denom) = 12 d. F distribution e. 0.67 f. 0.5305 g. Check studentโs solution. h. Decision: Do not reject null hypothesis; Conclusion: There is insufficient evidence to conclude that the means are different. 66 a. H0: ยต1 = ยต2 = ยต3 = ยต4 = ยต5 = ยต6 = ยต7 b. At least two mean lap times are different. c. df(num) = 6; df(denom) = 98 d. F distribution e. 1.69 f. 0.1319 g. Check studentโs solution. h. Decision: Do not reject null hypothesis; Conclusion: There is insufficient evidence to conclude that the mean lap times are different. 68 a. Ha: ยตd = ยตn = ยตh b. At least any two of the magazines have different mean lengths. c. df(num) = 2, df(denom) = 12 d. F distribtuion e. F = 15.28 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 729 f. p-value = 0.001 g. Check studentโs solution. h. i. Alpha: 0.05 ii. Decision: Reject the Null Hypothesis. iii. Reason for decision: p-value < alpha iv. Conclusion: There is sufficient evidence to conclude that the mean lengths of the magazines are different. 70 a. H0: ฮผo = ฮผh = ฮผf b. At least two of the means are different. c. df(n) = 2, df(d) = 13 d. F2,13 e. 0.64 f. 0.5437 g. Check studentโs solution. h. i. Alpha: 0.05 ii. Decision: Do not reject the null hypothesis. iii. |
Reason for decision: p-value > alpha iv. Conclusion: The mean scores of different class delivery are not different. 72 a. H0: ฮผp = ฮผm = ฮผh b. At least any two of the means are different. c. df(n) = 2, df(d) = 12 d. F2,12 e. 3.13 f. 0.0807 g. Check studentโs solution. h. i. Alpha: 0.05 ii. Decision: Do not reject the null hypothesis. iii. Reason for decision: p-value > alpha iv. Conclusion: There is not sufficient evidence to conclude that the mean numbers of daily visitors are different. 74 The data appear normally distributed from the chart and of similar spread. There do not appear to be any serious outliers, so we may proceed with our ANOVA calculations, to see if we have good evidence of a difference between the three groups. H0: ฮผ1 = ฮผ2 = ฮผ3; Ha: ฮผi โ ฮผj some i โ j. Define ฮผ1, ฮผ2, ฮผ3, as the population mean number of eggs laid by the three groups of fruit flies. F statistic = 8.6657; p-value = 0.0004 730 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA Figure 13.10 Decision: Since the p-value is less than the level of significance of 0.01, we reject the null hypothesis. Conclusion: We have good evidence that the average number of eggs laid during the first 14 days of life for these three strains of fruitflies are different. Interestingly, if you perform a two sample t-test to compare the RS and NS groups they are significantly different (p = 0.0013). Similarly, SS and NS are significantly different (p = 0.0006). However, the two selected groups, RS and SS are not significantly different (p = 0.5176). Thus we appear to have good evidence that selection either for resistance or for susceptibility involves a reduced rate of egg production (for these specific strains) as compared to flies that were not selected for resistance or susceptibility to DDT. Here, genetic selection has apparently involved a loss of fecundity. 76 a. H0 : ฯ1 2 2 = ฯ2 b. Ha : ฯ1 2 2 โ ฯ1 c. df(num) = 4; df(denom) = 4 d. F |
4, 4 e. 3.00 f. 2(0.1563) = 0.3126. Using the TI-83+/84+ function 2-SampFtest, you get the test statistic as 2.9986 and p-value directly as 0.3127. If you input the lists in a different order, you get a test statistic of 0.3335 but the p-value is the same because this is a two-tailed test. g. Check student't solution. h. Decision: Do not reject the null hypothesis; Conclusion: There is insufficient evidence to conclude that the variances are different. 78 a. H0: ฯ1 2 2 = ฯ2 b. Ha: ฯ1 2 2 โ ฯ1 c. df(n) = 19, df(d) = 19 d. F19,19 e. 1.13 f. 0.786 g. Check studentโs solution. h. i. Alpha:0.05 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA 731 ii. Decision: Do not reject the null hypothesis. iii. Reason for decision: p-value > alpha iv. Conclusion: There is not sufficient evidence to conclude that the variances are different. 80 The answers may vary. Sample answer: Home decorating magazines and news magazines have different variances. 82 a. H0: = ฯ1 2 2 = ฯ2 b. Ha: ฯ1 2 2 โ ฯ1 c. df(n) = 7, df(d) = 6 d. F7,6 e. 0.8117 f. 0.7825 g. Check studentโs solution. h. i. Alpha: 0.05 ii. Decision: Do not reject the null hypothesis. iii. Reason for decision: p-value > alpha iv. Conclusion: There is not sufficient evidence to conclude that the variances are different. 84 Here is a strip chart of the silver content of the coins: Figure 13.11 While there are differences in spread, it is not unreasonable to use ANOVA techniques. Here is the completed ANOVA table: Source of Variation Sum of Squares (SS) Degrees of Freedom (df) Mean Square ( |
MS) F Factor (Between) Error (Within) Total Table 13.42 37.748 11.015 48.763 4 โ 1 = 3 27 โ 4 = 23 27 โ 1 = 26 12.5825 0.4789 26.272 732 CHAPTER 13 | F DISTRIBUTION AND ONE-WAY ANOVA P(F > 26.272) = 0; Reject the null hypothesis for any alpha. There is sufficient evidence to conclude that the mean silver content among the four coinages are different. From the strip chart, it appears that the first and second coinages had higher silver contents than the third and fourth. 85 Here is a stripchart of the number of wins for the 14 teams in the AL for the 2012 season. Figure 13.12 While the spread seems similar, there may be some question about the normality of the data, given the wide gaps in the middle near the 0.500 mark of 82 games (teams play 162 games each season in MLB). However, one-way ANOVA is robust. Here is the ANOVA table for the data: Source of Variation Sum of Squares (SS) Degrees of Freedom (df) Mean Square (MS) F Factor (Between) 344.16 Error (Within) Total Table 13.43 1,219.55 1,563.71 3 โ 1 = 2 14 โ 3 = 11 14 โ 1 = 13 172.08 110.87 26.272 1.5521 P(F > 1.5521) = 0.2548 Since the p-value is so large, there is not good evidence against the null hypothesis of equal means. We decline to reject the null hypothesis. Thus, for 2012, there is not any have any good evidence of a significant difference in mean number of wins between the divisions of the American League. Appendices are in Volume 2 846 INDEX INDEX Symbols ฮฑ, 477 A absolute value of a residual, 641 alternative hypothesis, 470 Analysis of Variance, 712 Area to the left, 343 Area to the right, 343 assumption, 475 average, 11 Average, 46, 394 B balanced design, 702 bar graph, 17 Bernoulli Trial, 235 Bernoulli Trials, 258 binomial distribution, 385, 427, 475 Binomial Distribution, 441, 497 Binomial Experiment, 258 binomial probability distribution, 235 Binomial Probability Distribution, 258 bivariate, 634 Blinding, 37, 46 Box plot, 121 Box |
plots, 94 box-and-whisker plots, 94 box-whisker plots, 94 C Categorical Variable, 46 Categorical variables, 11 central limit theorem, 369, 371, 378, 481 Central Limit Theorem, 394, 497 central limit theorem for means, 373 central limit theorem for sums, 375 chi-square distribution, 578 Cluster Sampling, 46 Coefficient of Correlation, 665 coefficient of determination, 645 Cohen's d, 532 complement, 165 conditional probability, 165, 298 Conditional Probability, 198, 316 confidence interval, 412, 422 Confidence Interval (CI), 441, 497 confidence intervals, 427 Confidence intervals, 469 confidence level, 413, 427 Confidence Level (CL), 441 contingency table, 180, 198, 588 Contingency Table, 605 continuity correction factor, 385 continuous, 14 Continuous Random Variable, 46 continuous random variable, 301 control group, 37 Control Group, 46 Convenience Sampling, 46 critical value, 345 cumulative distribution function (CDF), 302 Cumulative relative frequency, 30 Cumulative Relative Frequency, 46 D data, 9 Data, 11, 46 decay parameter, 316 degrees of freedom, 423 Degrees of Freedom (df), 441, 551 degrees of freedom (df), 527 Dependent Events, 198 descriptive statistics, 10 discrete, 14 Discrete Random Variable, 46 double-blind experiment, 37 Double-blinding, 46 E Empirical Rule, 340 empirical rule, 412 equal standard deviations, 694 Equally likely, 164 Equally Likely, 198 equally likely, 293 error bound, 427 error bound for a population mean, 413, 423 Error Bound for a Population Mean (EBM), 441 Error Bound for a Population Proportion (EBP), 441 event, 164 Event, 198 expected value, 228 Expected Value, 258 expected values, 579 experiment, 164 Experiment, 198 experimental unit, 37 Experimental Unit, 46 explanatory variable, 37 Explanatory Variable, 46 exponential distribution, 301, 381 Exponential Distribution, 316, 394 extrapolation, 651 F F distribution, 695 F ratio, 695 fair, 164 first quartile, 87 First Quartile, 121 frequency, 30, 76 Frequency, 46, 121 Frequency Polygon, 121 Frequency Table, 121 G geometric distribution, 242 Geometric Distribution, 258 Geometric Experiment, 258 goodness-of-fit test, 579 H histogram, 76 Histogram, 121 |
hypergeometric experiment, 260 Hypergeometric Experiment, 258 hypergeometric probability, 244 Hypergeometric Probability, 258 hypotheses, 470 Hypothesis, 497 hypothesis test, 475, 478, 498 hypothesis testing, 470 Hypothesis Testing, 497 I independent, 168, 176 Independent Events, 198 Independent groups, 526 inferential statistics, 10, 412 Inferential Statistics, 441 influential points, 651 informed consent, 39 Informed Consent, 46 Institutional Review Board, 46 Institutional Review Boards (IRB), 39 interpolation, 651 interquartile range, 87 Interquartile Range, 121 Interval, 121 interval scale, 29 L law of large numbers, 164, 378 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 INDEX 847 Least-Squares Line, 639 least-squares regression line, 640 level of measurement, 29 Level of Significance of the Test, 497 Line of Best Fit, 639 linear regression, 639 long-term relative frequency, 164 Lurking Variable, 46 lurking variables, 37 M margin of error, 412 matched pairs, 526 mean, 11, 98, 228, 370, 373, 379 Mean, 121, 258, 394 Mean of a Probability Distribution, 258 mean square, 695 median, 86, 98 Median, 121 memoryless property, 316 Midpoint, 121 mode, 100 Mode, 121 multivariate, 634 mutually exclusive, 170, 176 Mutually Exclusive, 198 N nominal scale, 29 Nonsampling Error, 46 normal approximation to the binomial, 385 Normal Distribution, 355, 394, 441, 497 normal distribution, 423, 427, 474 Normal distribution, 534 normally distributed, 370, 375, 475 null hypothesis, 470, 475, 475, 477 Numerical Variable, 46 Numerical variables, 11 O observed values, 579 One-Way ANOVA, 712 ordinal scale, 29 outcome, 164 Outcome, 198 outlier, 69, 87 Outlier, 121, 665 outliers, 651 P p-value, 475, 477, 478, 499, 497 paired data set, 84 Paired Data Set, 121 parameter, 11, 412 Parameter, 46, 441 Pareto chart, 17 Pearson, 10 Percentile, |
121 percentile, 377 percentiles, 86 pie chart, 17 placebo, 37 Placebo, 46 point estimate, 412 Point Estimate, 441 Poisson distribution, 316 Poisson probability distribution, 247, 260 Poisson Probability Distribution, 259 Pooled Proportion, 551 population, 11, 27 Population, 46 population variance, 596 potential outlier, 654 Probability, 10, 46, 198 probability, 164 probability density function, 288 probability distribution function, 226 Probability Distribution Function (PDF), 259 proportion, 11 Proportion, 46 Q Qualitative data, 13 Qualitative Data, 47 quantitative continuous data, 14 Quantitative data, 14 Quantitative Data, 47 quantitative discrete data, 14 quartiles, 86 Quartiles, 87, 121 R random assignment, 37 Random Assignment, 47 Random Sampling, 47 random variable, 226 Random variable, 528 Random Variable, 534 Random Variable (RV), 259 ratio scale, 29 relative frequency, 30, 76 Relative Frequency, 47, 121 replacement, 168 Representative Sample, 47 residual, 641 response variable, 37 Response Variable, 47 S sample, 11 Sample, 47 sample mean, 371 sample size, 371, 375 sample space, 164, 175, 186 Sample Space, 198 samples, 27 sampling, 11 Sampling Bias, 47 sampling distribution, 101 Sampling Distribution, 394 Sampling Error, 47 sampling variability of a statistic, 109 Sampling with Replacement, 47, 198 Sampling without Replacement, 47, 198 simple random sample, 475 Simple Random Sampling, 47 single population mean, 474 single population proportion, 474 Skewed, 121 standard deviation, 108, 422, 474, 475, 476, 480 Standard Deviation, 121, 441, 497, 551 Standard Deviation of a Probability Distribution, 259 standard error, 526 Standard Error of the Mean, 394 standard error of the mean., 371 standard normal distribution, 338 Standard Normal Distribution, 355 statistic, 11 Statistic, 47 statistics, 9 Stratified Sampling, 47 Student's t-distribution, 423, 474, 475 Student's t-Distribution, 441, 497 Sum of Squared Errors (SSE), 641 sum of squares, 695 Systematic Sampling, 47 T test for homogeneity, 592 test of a single variance, 596 test of independence, 588 848 INDEX test statistic, 534 The AND Event, 198 The Complement Event, 198 The Conditional Probability of A |
GIVEN B, 198 The Conditional Probability of One Event Given Another Event, 198 The Law of Large Numbers, 259 The Or Event, 198 The OR of Two Events, 198 The standard deviation, 534 third quartile, 87 treatments, 37 Treatments, 47 tree diagram, 185 Tree Diagram, 198 Type 1 Error, 497 Type 2 Error, 498 Type I error, 472, 477 Type II error, 472 U unfair, 165 Uniform Distribution, 316, 394 uniform distribution, 379 V variable, 11 Variable, 47 Variable (Random Variable), 551 variance, 109 Variance, 121, 712 Variance between samples, 695 Variance within samples, 695 variances, 694 Variation, 27 Venn diagram, 191 Venn Diagram, 199 Z z-score, 355, 423 z-scores, 338 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16onrepeating, and are nonterminating: {h | h is not a rational number}. _ n โฃ m and n are integers and n โ 0 } m Example 5 Differentiating the Sets of Numbers Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q'). a. โ โ 36 8 _ b. 3 c. โ โ 73 d. โ6 e. 3.2121121112 โฆ Solution โ a. โ 36 = 6 _ 8 _ #6 = 2. b. 3 c. โ โ 73 d. โ6 e. 3.2121121112...'ร ร Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 6 CHAPTER 1 PREREQUISITES Try It #5 Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q'). a. โ 35 _ 7 b. 0 c. โ โ 169 d. โ โ 24 e. 4.763763763 โฆ Performing Calculations Using the Order of Operations When we multiply a number by itself, we square it or raise it to a power of 2. For example, 4 2 = 4 โ 4 = |
16. We can raise any number to any power. In general, the exponential notation a n means that the number or variable a is used as a factor n times. n factors In this notation, a n is read as the nth power of a, where a is called the base and n is called the exponent. A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. 2 _ For example, 24 + 6 โ โ 4 2 is a mathematical expression. 3 To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations. This is a sequence of rules for evaluating such expressions. Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols. The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right. Letโs take a look at the expression provided. 2 _ 24 + 6 โ โ 4 2 3 There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify 4 2 as 16. 2 _ โ 4 2 24 + 6 โ 3 2 _ โ 16 24 + 6 โ 3 Next, perform multiplication or division, left to right. 2 _ 24 + 6 โ โ 16 3 24 + 4 โ 16 Lastly, perform addition or subtraction, left to right. 2 _ Therefore, 24 + 6 โ โ 4 2 = 12. 3 24 + 4 โ 16 28 โ 16 12 For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result. order of operations Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS: P(arentheses) E(xponents) M(ultiplication) and D(ivision) A(ddition) and S(ubtraction) Download the OpenSt |
ax text for free at http://cnx.org/content/col11759/latest. SECTION 1.1 REAL NUMBERS: ALGEBRA ESSENTIALS 7 How Toโฆ Given a mathematical expression, simplify it using the order of operations. 1. Simplify any expressions within grouping symbols. 2. Simplify any expressions containing exponents or radicals. 3. Perform any multiplication and division in order, from left to right. 4. Perform any addition and subtraction in order, from left to right. Example 6 Using the Order of Operations Use the order of operations to evaluate each of the following expressions. b. 5 2 โ 4 ______ 7 โ โ โ 11 โ 2 c(4 โ 1) e. 7(5 โ 3) โ 2[(6 โ 3) โ 4 2 ] + 1 a. (3 โ 2) 2 โ 4(6 + 2) d. 14 โ Solution a. (3 โ 2) 2 โ 4(6 + 2) = (6) 2 โ 4(8) = 36 โ 4(8) = 36 โ 32 = 4 Simplify parentheses. Simplify exponent. Simplify multiplication. Simplify subtraction. 5 2 __ b. 7 โ โ โ 11 โ ______ 7 5 2 โ 4 ______ 7 25 โ 4 ______ 7 21 ___ Simplify grouping symbols (radical). Simplify radical. Simplify exponent. Simplify subtraction in numerator. Simplify division. Simplify subtraction. Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped. c. 6 โ |5 โ 8| + 3(4 โ 1) = 6 โ |โ3| + 3(3) = 6 โ 3 + 3(3 = 12 Simplify inside grouping symbols. Simplify absolute value. Simplify multiplication. Simplify subtraction. Simplify addition. d. 14 โ = Simplify exponent. = 14 โ 3 โ 2 _ 2 โ 5 โ 9 14 โ 6 _ 10 โ 9 8 _ = 1 = 8 Simplify products. Simplify differences. Simplify quotient. In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step. e 7(5 โ 3) โ 2[(6 โ 3) โ 4 2 ] + 1 = 7 |
(15) โ 2[(3) โ 4 2 ] + 1 = 7(15) โ 2(3 โ 16) + 1 = 7(15) โ 2(โ13) + 1 = 105 + 26 + 1 = 132 Simplify inside parentheses. Simplify exponent. Subtract. Multiply. Add. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 8 CHAPTER 1 PREREQUISITES Try It #6 Use the order of operations to evaluate each of the following expressions5 โ 4) 2 a. โ 1 1 __ __ ยท 9 2 [. 3 2 b __________ 9 โ 6 e. [ (3 โ 8) 2 โ 4] โ (3 โ 8) c. |1.8 โ 4.3| + 0.4 โ โ 15 + 10 Using Properties of Real Numbers For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics. Commutative Properties The commutative property of addition states that numbers may be added in any order without affecting the sum. a + b = b + a We can better see this relationship when using real numbers. and (โ2) + 7 = 5 7 + (โ2) = 5 Similarly, the commutative property of multiplication states that numbers may be multiplied in any order without affecting the product. a โ b = b โ a Again, consider an example with real numbers. (โ11) โ (โ4) = 44 and (โ4) โ (โ11) = 44 It is important to note that neither subtraction nor division is commutative. For example, 17 โ 5 is not the same as 5 โ 17. Similarly, 20 รท 5 โ 5 รท 20. Associative Properties The associative property of multiplication tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same. Consider this example. a(bc) = (ab)c (3 โ 4) โ 5 = 60 and 3 โ (4 โ 5) = 60 The associative property of addition tells |
us that numbers may be grouped differently without affecting the sum. a + (b + c) = (a + b) + c This property can be especially helpful when dealing with negative integers. Consider this example. [15 + (โ9)] + 23 = 29 and 15 + [(โ9) + 23] = 29 Are subtraction and division associative? Review these examples. 8 โ (3 โ 15) $#(8 โ 3) โ 15 8 โ ( โ 12) $ 5 โ 15 20 โ โ10 64 รท (8 รท 4) $ (64 รท 8) รท 4 64 รท 2 $ 8 รท 4 32 โ 2 As we can see, neither subtraction nor division is associative. Distributive Property The distributive property states that the product of a factor times a sum is the sum of the factor times each term in the sum. a โ (b + c) = a โ b + a โ c Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 1.1 REAL NUMBERS: ALGEBRA ESSENTIALS 9 This property combines both addition and multiplication (and is the only property to do so). Let us consider an example. 4 โ [12 + (โ7)] = 4 โ 12 + 4 โ (โ7) = 48 + (โ28) = 20 Note that 4 is outside the grouping symbols, so we distribute the 4 by multiplying it by 12, multiplying it by โ7, and adding the products. To be more precise when describing this property, we say that multiplication distributes over addition. The reverse is not true, as we can see in this example. 6 + (3 โ 5) $ (6 + 3) โ (6 + 5) 6 + (15) $ (9) โ (11) 21 โ 99 Multiplication does not distribute over subtraction, and division distributes over neither addition nor subtraction. A special case of the distributive property occurs when a sum of terms is subtracted. a โ b = a + (โb) For example, consider the difference 12 โ (5 + 3). We can rewrite the difference of the two terms 12 and (5 + 3) by turning the subtraction expression into addition of the opposite. So instead of subtracting (5 + 3), we add the opposite. 12 + (โ1) โ ( |
5 + 3) Now, distribute โ1 and simplify the result. 12 โ (5 + 3) = 12 + (โ1) โ (5 + 3) = 12 + [(โ1) โ 5 + (โ1) โ 3] = 12 + (โ8) = 4 This seems like a lot of trouble for a simple sum, but it illustrates a powerful result that will be useful once we introduce algebraic terms. To subtract a sum of terms, change the sign of each term and add the results. With this in mind, we can rewrite the last example. 12 โ (5 + 3) = 12 + (โ5 โ 3) = 12 + (โ8) = 4 Identity Properties The identity property of addition states that there is a unique number, called the additive identity (0) that, when added to a number, results in the original number. a + 0 = a The identity property of multiplication states that there is a unique number, called the multiplicative identity (1) that, when multiplied by a number, results in the original number. For example, we have (โ6) + 0 = โ6 and 23 โ 1 = 23. There are no exceptions for these properties; they work for every real number, including 0 and 1. a โ 1 = a Inverse Properties The inverse property of addition states that, for every real number a, there is a unique number, called the additive inverse (or opposite), denotedโa, that, when added to the original number, results in the additive identity, 0. For example, if a = โ8, the additive inverse is 8, since (โ8) + 8 = 0. a + (โa) = 0 The inverse property of multiplication holds for all real numbers except 0 because the reciprocal of 0 is not defined. The property states that, for every real number a, there is a unique number, called the multiplicative inverse (or 1 _ a, that, when multiplied by the original number, results in the multiplicative identity, 1. reciprocal), denoted 1 _ a = 1 a โ Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 10 CHAPTER 1 PREREQUISITES 3 2 1 _ _ a, is โ _ For example, if a = โ because, the reciprocal, denoted properties of real numbers The following properties hold for real numbers a, b, and c. Commutative Property Associative |
Property Distributive Property Identity Property Inverse Property Addition b + c) = (a + b) + c Multiplication a โ b = b โ a a(bc) = (ab)c a โ (b + c) = a โ b + a โ c There exists a unique real number called the additive identity, 0, such that, for any real number a a + 0 = a Every real number a has an additive inverse, or opposite, denoted โa, such that a + (โa) = 0 There exists a unique real number called the multiplicative identity, 1, such that, for any real number a a โ 1 = a Every nonzero real number a has a multiplicative inverse, or reciprocal, 1 _ a, such that denoted 1 a โ ( _ a ) = 1 Example 7 Using Properties of Real Numbers Use the properties of real numbers to rewrite and simplify each expression. State which properties apply. a. 3 โ 6 + 3 โ 4 b. (5 + 8) + (โ8) c. 6 โ (15 + 9) Solution a. b. c. d6 + 4) = 3 โ 10 = 30 (5 + 8) + (โ8) = 5 + [8 + (โ8)] = 5 + 0 = 5 6 โ (15 + 9) = 6 + [(โ15) + (โ9)] = 6 + (โ24) = โ18 2 7 4 7 2 4 ) โ ( ) = โ ( __ __ __ __ __ __ โ โ 7 4 7 3 4 3 2 7 4 ) โ = ( __ __ __ โ 4 3 7 2 __ = 1 โ 3 2 __ = 3 7 2 4 ) โ ( __ __ __ โ d. 4 3 7 e. 100 โ [0.75 + (โ2.38)] Distributive property Simplify. Simplify. Associative property of addition Inverse property of addition Identity property of addition Distributive property Simplify. Simplify. Commutative property of multiplication Associative property of multiplication Inverse property of multiplication Identity property of multiplication e. 100 โ [0.75 + (โ2.38)] = 100 โ 0.75 + 100 โ (โ2.38) = 75 + (โ238) = โ163 Distributive property Simplify. Simplify. Try It #7 Use the properties of real numbers to rewrite and simplify each expression. State which |
properties apply. a. ( โ ) ] b. 5 ยท (6.2 + 0.4) c. 18 โ (7 โ 15) d. 23 ) ยท [ 11 ยท ( โ 17 _ 18 17 _ 18 5 _ 23 ) ] e. 6 ฤ (โ3) + 6 ฤ 3 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 1.1 REAL NUMBERS: ALGEBRA ESSENTIALS 11 Evaluating Algebraic Expressions So far, the mathematical expressions we have seen have involved real numbers only. In mathematics, we may see 4 2 m 3 n 2. In the expression x + 5, 5 is called a constant because it does not vary ฯ r 3, or โ _ expressions such as x + 5, 3 โ and x is called a variable because it does. (In naming the variable, ignore any exponents or radicals containing the variable.) An algebraic expression is a collection of constants and variables joined together by the algebraic operations of addition, subtraction, multiplication, and division. We have already seen some real number examples of exponential notation, a shorthand method of writing products of the same factor. When variables are used, the constants and variables are treated the same way. (โ3) 5 = (โ3) โ (โ3) โ (โ3) โ (โ3) โ (โ3) (2 โ 7) 3 = (2 โ 7) โ (2 โ 7) โ (2 โ 7yz) 3 = (yz) โ (yz) โ (yz) In each case, the exponent tells us how many factors of the base to use, whether the base consists of constants or variables. Any variable in an algebraic expression may take on or be assigned different values. When that happens, the value of the algebraic expression changes. To evaluate an algebraic expression means to determine the value of the expression for a given value of each variable in the expression. Replace each variable in the expression with the given value, then simplify the resulting expression using the order of operations. If the algebraic expression contains more than one variable, replace each variable with its assigned value and simplify the expression as before. Example 8 Describing Algebraic Expressions List the constants and variables for each algebraic expression. a. 3 c. โ โ 2 m 3 n 2 Solution Constants Variables a |
. 3 5 4 _, ฯ 3 c, n Try It #8 List the constants and variables for each algebraic expression. a. 2ฯr(r + h) b. 2(L + W) c. 4 y 3 + y Example 9 Evaluating an Algebraic Expression at Different Values Evaluate the expression 2x โ 7 for each value for x. 1 _ c. x = 2 b. x = 1 a. x = 0 Solution a. Substitute 0 for x. b. Substitute 1 for x. d. x = โ4 2x โ 7 = 2(0) โ 7 = 0 โ 7 = โ7 2x โ 7 = 2(1) โ 7 = 2 โ 7 = โ5 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 12 CHAPTER 1 PREREQUISITES 1 __ for x. c. Substitute 2 d. Substitute โ4 for x. 1 ) โ 7 2x โ 7 = 2 ( __ 2 = 1 โ 7 = โ6 2x โ 7 = 2(โ4) โ 7 = โ8 โ 7 = โ15 Try It #9 Evaluate the expression 11 โ 3y for each value for y. a. y = 2 b. y = 0 2 _ c. y =# 3 d. y = โ5 Example 10 Evaluating Algebraic Expressions Evaluate each expression for the given values. a. x + 5 for x = โ5 b. t _ 2tโ1 for t = 10 4 _ ฯ r 3 for r = 5 c. 3 d. a + ab + b for a = 11, b = โ8 e. โ โ 2 m 3 n 2 for m = 2, n = 3 Solution a. Substitute โ5 for x. b. Substitute 10 for t. c. Substitute 5 for r. x + 5 = (โ5) + 5 = 0 t _ 2t โ 1 = (10) _ 2(10) โ 1 = 10 _ 20 โ 1 = 10 _ 19 4 4 _ _ ฯr 3 = ฯ(5)3 3 3 4 _ = ฯ(125) 3 = 500 _ ฯ 3 d. Substitute 11 for a and โ8 for b. a + ab + b = (11) + (11)(โ8) + (โ8) = 11 โ 88 โ 8 |
= โ85 e. Substitute 2 for m and 3 for n. โ Try It #10 Evaluate each expression for the given values. โ 2m3n2 = โ = โ = โ = 12 โ 2(2)3(3)2 โ 2(8)(9) โ 144 a. y + 3 _ y โ 3 for y = 5 d. (p 2q)3 for p = โ2, q = 3 b. 7 โ 2t for t = โ2 c. 1 _ ฯr 2 for r = 11 3 2 1 _ _ e. 4(m โ n) โ 5(n โ m) for m =, n = 3 3 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 1.1 REAL NUMBERS: ALGEBRA ESSENTIALS 13 Formulas An equation is a mathematical statement indicating that two expressions are equal. The expressions can be numerical or algebraic. The equation is not inherently true or false, but only a proposition. The values that make the equation true, the solutions, are found using the properties of real numbers and other results. For example, the equation 2x + 1 = 7 has the unique solution x = 3 because when we substitute 3 for x in the equation, we obtain the true statement 2(3) + 1 = 7. A formula is an equation expressing a relationship between constant and variable quantities. Very often, the equation is a means of finding the value of one quantity (often a single variable) in terms of another or other quantities. One of the most common examples is the formula for finding the area A of a circle in terms of the radius r of the circle: A = ฯr 2. For any value of r, the area A can be found by evaluating the expression ฯr 2. Example 11 Using a Formula A right circular cylinder with radius r and height h has the surface area S (in square units) given by the formula S = 2ฯr(r + h). See Figure 4. Find the surface area of a cylinder with radius 6 in. and height 9 in. Leave the answer in terms of ฯ. r h Solution Evaluate the expression 2ฯr(r + h) for r = 6 and h = 9. Figure 4 Right circular cylinder S = 2ฯr(r + h) = 2ฯ(6)[(6) + ( |
9)] = 2ฯ(6)(15) = 180ฯ The surface area is 180ฯ square inches. Try It #11 A photograph with length L and width W is placed in a matte of width 8 centimeters (cm). The area of the matte (in square centimeters, or cm2) is found to be A = (L + 16)(W + 16) โ L โ W. See Figure 5. Find the area of a matte for a photograph with length 32 cm and width 24 cm. 8 cm 24 cm 8 cm 32 cm Figure 5 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 14 CHAPTER 1 PREREQUISITES Simplifying Algebraic Expressions Sometimes we can simplify an algebraic expression to make it easier to evaluate or to use in some other way. To do so, we use the properties of real numbers. We can use the same properties in formulas because they contain algebraic expressions. Example 12 Simplifying Algebraic Expressions Simplify each algebraic expression. a. 3x โ 2y + x โ 3y โ 7 b. 2r โ 5(3 โ r) + 4 5 2 t + 2s ) s ) โ ( c. ( 4t โ _ _ 4 3 d. 2mn โ 5m + 3mn + n Solution a. b. 3x โ 2y + x โ 3y โ 7 = 3x + x โ 2y โ 3y โ 7 = 4x โ 5y โ 7 2r โ 5(3 โ r) + 4 = 2r โ 15 + 5r + 4 = 2r + 5y โ 15 + 4 = 7r โ 11 5 5 2 2 t + 2s ) = 4t โ s ) โ ( c. 4t โ โ 2s 4 4 3 3 5 2 _ _ s โ 2s t โ = 4t โ 4 3 = t โ 10 _ 3 13 _ s 4 Commutative property of addition Simplify. Distributive property Commutative property of addition Simplify. Distributive property Commutative property of addition Simplify. d. mn โ 5m + 3mn + n = 2mn + 3mn โ 5m + n = 5mn โ 5m + n Commutative property of addition Simplify. Try It #12 Simplify each algebraic expression. t t Example 13 Simplifying a Formula c. 4p(q โ 1) + q |
(1 โ p) d. 9r โ (s + 2r) + (6 โ s) A rectangle with length L and width W has a perimeter P given by P = L + W + L + W. Simplify this expression. Solution = 2L + 2W P = 2(L + W) Commutative property of addition Simplify. Distributive property Try It #13 If the amount P is deposited into an account paying simple interest r for time t, the total value of the deposit A is given by A = P + Prt. Simplify the expression. (This formula will be explored in more detail later in the course.) Access these online resources for additional instruction and practice with real numbers. โข Simplify an Expression (http://openstaxcollege.org/l/simexpress) โข Evaluate an Expression1 (http://openstaxcollege.org/l/ordofoper1) โข Evaluate an Expression2 (http://openstaxcollege.org/l/ordofoper2) Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 1.1 SECTION EXERCISES 15 1.1 SECTION EXERCISES VERBAL โ 1. Is โ 2 an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category. 2. What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for? 3. What do the Associative Properties allow us to do when following the order of operations? Explain your answer. NUMERIC For the following exercises, simplify the given expression. 4. 10 + 2 ยท (5 โ 3) 5. 6 รท 2 โ (81 รท 32) 6. 18 + (6 โ 8)3 7. โ2 ยท [16 รท (8 โ 4)2] 2 8. 4 โ 6 + 2 ยท 7 9. 3(5 โ 8) 10. 4 + 6 โ 10 รท 2 11. 12 รท (36 รท 9) + 6 12. (4 + 5)2 รท 3 13. 3 โ 12 ยท 2 + 19 14. 2 + 8 ยท 7 รท 4 15. 5 + (6 + 4) โ 11 16. 9 โ 18 รท 32 17. 14 ยท 3 รท 7 โ 6 18. 9 โ (3 + |
11) ยท 2 19. 6 + 2 ยท 2 โ 1 20. 64 รท (8 + 4 ยท 2) 21. 9 + 4(22) 22. (12 รท 3 ยท 3)2 23. 25 รท 52 โ 7 24. (15 โ 7) ยท (3 โ 7) 25. 2 ยท 4 โ 9(โ1) 1 _ 26. 42 โ 25 ยท 5 27. 12(3 โ 1) รท 6 ALGEBRAIC For the following exercises, solve for the variable. 28. 8(x + 3) = 64 29. 4y + 8 = 2y 30. (11a + 3) โ 18a = โ4 31. 4z โ 2z(1 + 4) = 36 32. 4y(7 โ 2)2 = โ200 33. โ(2x)2 + 1 = โ3 34. 8(2 + 4) โ 15b = b 35. 2(11c โ 4) = 36 36. 4(3 โ 1)x = 4 1 _ (8w โ 42) = 0 37. 4 For the following exercises, simplify the expression. 38. 4x + x(13 โ 7) 39. 2y โ (4)2 y โ 11 a __ 40. 23 (64) โ 12a รท 6 41. 8b โ 4b(3) + 1 42. 5l รท 3l ยท (9 โ 6) 43. 7z โ 3 + z ยท 62 44. 4 ยท 3 + 18x รท 9 โ 12 45. 9(y + 8) โ 27 9 t โ 4 ) 2 46. ( _ 6 47. 6 + 12b โ 3 ยท 6b 48. 18y โ 2(1 + 7y) 50. 8(3 โ m) + 1(โ8) 51. 9x + 4x(2 + 3) โ 4(2x + 3x) 4 ) 49. ( _ 9 2 ยท 27x 52. 52 โ 4(3x) Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 16 CHAPTER 1 PREREQUISITES REAL-WORLD APPLICATIONS For the following exercises, consider this scenario: Fred earns $40 mowing lawns. He spends $10 on mp3s, puts half of what is left in a savings account, and gets another $5 for |
washing his neighborโs car. 53. Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations. For the following exercises, solve the given problem. 55. According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by ฯ. Is the circumference of a quarter a whole number, a rational number, or an irrational number? 54. How much money does Fred keep? 56. Jessica and her roommate, Adriana, have decided to share a change jar for joint expenses. Jessica put her loose change in the jar first, and then Adriana put her change in the jar. We know that it does not matter in which order the change was added to the jar. What property of addition describes this fact? For the following exercises, consider this scenario: There is a mound of g pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel. 57. Write the equation that describes the situation. 58. Solve for g. For the following exercise, solve the given problem. 59. Ramon runs the marketing department at his company. His department gets a budget every year, and every year, he must spend the entire budget without going over. If he spends less than the budget, then his department gets a smaller budget the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing budget. He must spend the budget such that 2,500,000 โ x = 0. What property of addition tells us what the value of x must be? TECHNOLOGY For the following exercises, use a graphing calculator to solve for x. Round the answers to the nearest hundredth. 3 _ 60. 0.5(12.3)2 โ 48x = 5 EXTENSIONS 61. (0.25 โ 0.75)2x โ 7.2 = 9.9 62. If a whole number is not a natural number, what must the number be? 64. Determine whether the statement is true or false: The product of a rational and irrational number is always irrational. 63. Determine whether the statement is true or false: The multiplicative inverse of |
a rational number is also rational. 65. Determine whether the simplified expression is rational or irrational: โ โ โ18 โ 4(5)(โ1). 66. Determine whether the simplified expression is 67. The division of two whole numbers will always result rational or irrational: โ โ โ16 + 4(5) + 5. in what type of number? 68. What property of real numbers would simplify the following expression: 4 + 7(x โ 1)? Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 1.2 EXPONENTS AND SCIENTIFIC NOTATION 17 LEARNING OBJECTIVES In this section students will: โข Use the product rule of exponents. โข Use the quotient rule of exponents. โข Use the power rule of exponents. โข Use the zero exponent rule of exponents. โข Use the negative rule of exponents. โข Find the power of a product and a quotient. โข Simplify exponential expressions. โข Use scienti๏ฌc notation. 1. 2 EXPONENTS AND SCIENTIFIC NOTATION Mathematicians, scientists, and economists commonly encounter very large and very small numbers. But it may not be obvious how common such figures are in everyday life. For instance, a pixel is the smallest unit of light that can be perceived and recorded by a digital camera. A particular camera might record an image that is 2,048 pixels by 1,536 pixels, which is a very high resolution picture. It can also perceive a color depth (gradations in colors) of up to 48 bits per frame, and can shoot the equivalent of 24 frames per second. The maximum possible number of bits of information used to film a one-hour (3,600-second) digital film is then an extremely large number. Using a calculator, we enter 2,048 ยท 1,536 ยท 48 ยท 24 ยท 3,600 and press ENTER. The calculator displays 1.304596316E13. What does this mean? The โE13โ portion of the result represents the exponent 13 of ten, so there are a maximum of approximately 1.3 ยท 1013 bits of data in that one-hour film. In this section, we review rules of exponents first and then apply them to calculations involving very large or small numbers. Using the Product Rule of Exponents Consider the product x 3 โ x 4. Both terms have the same base, x |
, but they are raised to different exponents. Expand each expression, and then rewrite the resulting expression. 3 factors 4 factors factors = The result is that. Notice that the exponent of the product is the sum of the exponents of the terms. In other words, when multiplying exponential expressions with the same base, we write the result with the common base and add the exponents. This is the product rule of exponents. Now consider an example with real numbers. am ยท an = am + n 23 ยท 24 = 23 + 4 = 27 We can always check that this is true by simplifying each exponential expression. We find that 23 is 8, 24 is 16, and 27 is 128. The product 8 โ 16 equals 128, so the relationship is true. We can use the product rule of exponents to simplify expressions that are a product of two numbers or expressions with the same base but different exponents. the product rule of exponents For any real number a and natural numbers m and n, the product rule of exponents states that am ยท an = am + n Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 18 CHAPTER 1 PREREQUISITES Example 1 Using the Product Rule Write each of the following products with a single base. Do not simplify further. a. t 5 โ t 3 b. (โ3)5 โ (โ3) c. x 2 โ x 5 โ x 3 Solution Use the product rule to simplify each expression. a. t 5 โ t3 = t5 + 3 = t8 b. (โ3)5 โ (โ3) = (โ3)5 โ (โ3)1 = (โ3)5 + 1 = (โ3)6 c. x 2 โ x 5 โ x 3 At first, it may appear that we cannot simplify a product of three factors. However, using the associative property of multiplication, begin by simplifying the first twox 2 ยท x 5) ยท x 3 = (x 2 + 5)ยท = x10 Notice we get the same result by adding the three exponents in one step = x10 Try It #1 Write each of the following products with a single base. Do not simplify further. a. ( 2 _ y ) ยท ( c. t3 ยท t6 ยท t5 Using the Quotient Rule of Exponents The quotient rule of exponents allows us to simplify an expression that |
divides two numbers with the same base but ym ___ yn, where m > n. different exponents. In a similar way to the product rule, we can simplify an expression such as y 9 _ y 5. Perform the division by canceling common factors. Consider the example y9 ___ y5 = ___ ___ % __________ 1 = y 4 Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend. = am โ n am ___ an y9 __ y5 = y9 โ 5 = y4 In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents. For the time being, we must be aware of the condition m > n. Otherwise, the difference m โ n could be zero or negative. Those possibilities will be explored shortly. Also, instead of qualifying variables as nonzero each time, we will simplify matters and assume from here on that all variables represent nonzero real numbers. the quotient rule of exponents For any real number a and natural numbers m and n, such that m > n, the quotient rule of exponents states that am ___ an = am โ n Example 2 Using the Quotient Rule Write each of the following products with a single base. Do not simplify further. a. (โ2)14 ______ (โ2)9 b. t 23 __ t 15 c. โ 5 2 ) ( z โ ________ 2 z โ โ Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 1.2 EXPONENTS AND SCIENTIFIC NOTATION 19 Solution Use the quotient rule to simplify each expression. a. = (โ2)14 โ 9 = (โ2)5 b. = t 23 โ 15 = t 8 (โ2)14 ______ (โ2)9 t23 __ t15 2 ) ( z โ ________ 2 z โ โ โ 5 c ) Try It #2 Write each of the following products with a single base. Do not simplify further. a. s75 __ s68 b. (โ3)6 _____ โ3 c. (ef 2)5 _ (ef 2)3 Using the Power Rule of Exponents Suppose an exponential expression is raised to some power. Can we simplify the result? Yes. To do this, we use the power rule of exponents. Consider the expression (x 2)3. The |
expression inside the parentheses is multiplied twice because it has an exponent of 2. Then the result is multiplied three times because the entire expression has an exponent of 3. 3 factors (x 2) 3 = (x 2) ยท (x 2) ยท (x 2) 3 factors x ยท x ) ยท ( 2 factors factors = ( 2 factors = The exponent of the answer is the product of the exponents: (x 2. In other words, when raising an exponential expression to a power, we write the result with the common base and the product of the exponents. (a m)n = am โ n Be careful to distinguish between uses of the product rule and the power rule. When using the product rule, different terms with the same bases are raised to exponents. In this case, you add the exponents. When using the power rule, a term in exponential notation is raised to a power. In this case, you multiply the exponents. Product Rule Power Rule 53 โ 54 = 53 + 3a)7 โ (3a)10 = (3a)7 + 10 = 57 = x 7 = (3a)17 but but but (53)4 = 5 3 โ 4 (x 5)2 = x 5 โ 2 ((3a)7)10 = (3a)7 โ 10 = 512 = x 10 = (3a)70 the power rule of exponents For any real number a and positive integers m and n, the power rule of exponents states that (am)n = am โ n Example 3 Using the Power Rule Write each of the following products with a single base. Do not simplify further. a. (x 2) 7 b. ((2t)5) 3 c. ((โ3)5) 11 Solution Use the power rule to simplify each expression. a. (x 2) 7 = x 2 โ 7 = x 14 b. ((2t)5)3 = (2t)5 โ 3 = (2t)15 c. ((โ3)5)11 = (โ3)5 โ 11 = (โ3)55 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 20 CHAPTER 1 PREREQUISITES Try It #3 Write each of the following products with a single base. Do not simplify further. c. ((โg)4)4 a. ((3y) |
8)3 b. (t 5)7 Using the Zero Exponent Rule of Exponents Return to the quotient rule. We made the condition that m > n so that the difference m โ n would never be zero or negative. What would happen if m = n? In this case, we would use the zero exponent rule of exponents to simplify the expression to 1. To see how this is done, let us begin with an example. t 8 __ t 8 % t8 __ % t8 = = 1 If we were to simplify the original expression using the quotient rule, we would have t 8 __ t 8 If we equate the two answers, the result is t0 = 1. This is true for any nonzero real number, or any variable representing a real number. a0 = 1 The sole exception is the expression 00. This appears later in more advanced courses, but for now, we will consider the value to be undefined. = t 8 โ 8 = t 0 the zero exponent rule of exponents For any nonzero real number a, the zero exponent rule of exponents states that a0 = 1 Example 4 Using the Zero Exponent Rule Simplify each expression using the zero exponent rule of exponents. c 3 __ a. c 3 b. โ3x 5 _____ x 5 c. (j 2k)4 _ (j 2k) โ (j 2k)3 d. 5(rs2)2 _ (rs2)2 Solution Use the zero exponent and other rules to simplify each expression. c3 __ a. c3 = c3 โ 3 = c0 b. = โ3 โ โ3x5 _____ x5 x5 __ x5 = โ3 โ x5 โ 5 = โ3 โ x0 = โ3 โ 1 = โ3 c. (j2k)4 _ (j2k) โ (j2k)3 = = (j2k)4 _ (j2k)1 + 3 (j2k)4 _ (j2k)4 = (j2k)4 โ 4 = (j2k)0 = 1 d. 5(rs2)2 ______ (rs2)2 = 5(rs2)2 โ 2 = 5(rs2)0 = 5 โ 1 = 5 Use the product rule in the denominator. Simplify. Use the quotient rule. Simplify. Use the quotient rule. |
Simplify. Use the zero exponent rule. Simplify. Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 1.2 EXPONENTS AND SCIENTIFIC NOTATION 21 Try It #4 Simplify each expression using the zero exponent rule of exponents. t 7 __ a. t 7 b. (de2)11 _ 2(de2)11 c Using the Negative Rule of Exponents Another useful result occurs if we relax the condition that m > n in the quotient rule even further. For example, can we h3 _ h5? When m < n โthat is, where the difference m โ n is negativeโwe can use the negative rule of exponents simplify to simplify the expression to its reciprocal. Divide one exponential expression by another with a larger exponent. Use our example, h3 _ h5. h3 _ h5 = h ยท h ยท h __ __ % h2 If we were to simplify the original expression using the quotient rule, we would have h3 _ h5 = h3 โ 5 = hโ2 Putting the answers together, we have hโ2 = a nonzero real number. 1 _ h2. This is true for any nonzero real number, or any variable representing A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction barโfrom numerator to denominator or vice versa. aโn = 1 _ an and an = 1 _ aโn We have shown that the exponential expression an is defined when n is a natural number, 0, or the negative of a natural number. That means that an is defined for any integer n. Also, the product and quotient rules and all of the rules we will look at soon hold for any integer n. the negative rule of exponents For any nonzero real number a and natural number n, the negative rule of exponents states that aโn = 1 _ an Example 5 Using the Negative Exponent Rule Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents. a. ฮธ 3 _ ฮธ 10 Solution b. z 2 ฤ z _ z 4 c. (โ5t 3)4 _ (โ5t 3)8 a. b. c. ฮธ 3 1 _ _ ฮธ 10 = ฮธ 3 โ 10 = ฮธโ7 = ฮธ7 = z3 โ |
4 = zโ1 = z 4 z 4 (โ5t 3)4 1 _ _ (โ5t3)8 = (โ5t 3)4 โ 8 = (โ5t 3)โ4 = (โ5t 3)4 = = Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 22 CHAPTER 1 PREREQUISITES Try It #5 Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents. a. (โ3t)2 ______ (โ3t)8 b. f 47 _____ f 49 โ
f c. 2k 4 _ 5k 7 Example 6 Using the Product and Quotient Rules Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents. โ7z_ (โ7z)5 b. (โx)5 โ (โx)โ5 a. b 2 โ bโ8 c. Solution 1 __ a. b 2 โ bโ8 = b 2 โ 8 = b โ6 = b 6 b. (โx)5 โ (โx)โ5 = (โx)5 โ 5 = (โx)0 = 1 c. โ7z _ (โ7z)5 = (โ7z)1 _ (โ7z)5 = (โ7z)1 โ 5 = (โ7z)โ4 = 1 _ (โ7z)4 Try It #6 Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents. a. tโ11 โ
t 6 b. 2512 ____ 2513 Finding the Power of a Product To simplify the power of a product of two exponential expressions, we can use the power of a product rule of exponents, which breaks up the power of a product of factors into the product of the powers of the factors. For instance, consider (pq)3. We begin by using the associative and commutative properties of multiplication to regroup the factors. 3 factors (pq)3 = (pq) ยท (pq) ยท (pq factors = factors = p3 ยท q3 In other words, (pq)3 = p3 ยท q3. the power of a product rule of exponents For any nonzero real number a and natural number n, the negative rule of exponents states that |
(ab)n = an bn Example 7 Using the Power of a Product Rule Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents. a. (ab 2)3 c. (โ2w3)3 e. (eโ2f 2)7 b. (2t)15 d. 1 ______ (โ7z)4 Solution Use the product and quotient rules and the new definitions to simplify each expression. a. (ab 2)3 = (a)3 โ (b2)3 = a1 โ 3 โ b2 โ 3 = a3b6 b. (2t)15 = (2)15 โ (t)15 = 215t15 = 32,768t15 c. (โ2w3)3 = (โ2)3 โ (w3)3 = โ8 โ w 3 โ 3 = โ8w9 d. 1 _ (โ7z)4 = 1 _ (โ7)4 โ (z)4 = 1 _ 2,401z 4 e. (eโ2f 2)7 = (eโ2)7 โ (f 2)7 = eโโ14f 14 = f 14 _ e14 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. SECTION 1.2 EXPONENTS AND SCIENTIFIC NOTATION 23 Try It #7 Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents. a. (g 2h3)5 b. (5t)3 c. (โ3y5)3 d. 1 _______ (a6b7)3 e. (r 3sโ2)4 Finding the Power of a Quotient To simplify the power of a quotient of two expressions, we can use the power of a quotient rule, which states that the power of a quotient of factors is the quotient of the powers of the factors. For example, letโs look at the following example. (e โ2f 2)7 = f 14 _ e14 Letโs rewrite the original problem differently and look at the result. 7 (e โ2f 2)7 = ( f 2 _ e2 ) It appears from the last two steps that we can use the power of a product rule |
as a power of a quotient rule. = f 14 _ e14 (e โ2f 2)7 = ( 7 f 2 _ e2 ) = = = (f 2)7 _ (e 2)7 f 2 ยท 7 _ e2 ยท 7 f 14 _ e14 the power of a quotient rule of exponents For any real numbers a and b and any integer n, the power of a quotient rule of exponents states that n = a _ ) ( b an __ bn Example 8 Using the Power of a Quotient Rule Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents. a. ( 3 z11 ) 4 _ 6 b. ( p _ q3 ) c. ( 27 โ1 t2 ) _ d. ( j3kโ2)4 e. (mโ2nโ2)3 Solution 27 3 = 6 = 43 z11 ) a. ( 64 64 4 _ _ _ _ z11 โ 3 = (z11)3 = z33 p6 p1 โ 6 p6 q3 ) b. ( p _ _ _ _ q3 โ 6 = (q3)6 = q18 (โ1)27 โ1 โ1 โ1 โ1 c. ( t2 ) _ _ _ _ _ (t2)27 = t54 = t2 โ 27 = t54 d. ( j3kโ2)4 = ( j 12 j 3 โ 4 (j 3)4 j 3 k2 ) k2)4 = k8 = ( 13 e. (mโ2nโ2)3 = ( 1 1 (m2n2)3 ) = m2n2 ) _ _ _ (m2)3(n2)3 = 4 = = 3 1 _ m2 โ 3 โ n2 โ 3 = 1 _ m6n6 Download the OpenStax text for free at http://cnx.org/content/col11759/latest. 24 CHAPTER 1 PREREQUISITES Try It #8 Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents. a. ( _ c ) b5 e. (cโ5dโ3)4 d. (pโ4q3)8 c. ( 35 3 4 u8 |
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