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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... |
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, enlargin... |
189 198 – 189 = 9 67 153 150 153 – 150 = 3 70 163 164 163 – 164 = –1 71 159 169 159 – 169 = –10 Table 12.5 654 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION x y ŷ y – ŷ 69 151 160 151 – 160 = –9 69 159 160 159 – 160 = –1 Table 12.5 We are looking for all data points for which the residual is greater than 2s = 2(16.4)... |
The squares are 352; 172; 162; 62; 192; 92; 32; 12; 102; 92; 12 Then, add (sum) all the |y – ŷ| squared terms using the formula 11 ⎛ Σ i = 1 ⎝|yi − y^ 2 ⎠ i|⎞ = 11 Σ i = 1 εi 2 (Recall that yi – ŷi = εi.) = 352 + 172 + 162 + 62 + 192 + 92 + 32 + 12 + 102 + 92 + 12 = 2440 = SSE. The result, SSE is the Sum of Squared Er... |
, 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 st... |
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... |
, 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/s... |
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 Regressio... |
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 • The student will... |
Fuel Efficiency) Class Time: Names: Student Learning Outcomes • The student will calculate and construct the line of best fit between two variables. • 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 Co... |
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 ... |
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. Th... |
produced from a well-designed random sample or randomized experiment. The slope b and intercept a of the least-squares line estimate the slope β and intercept α of the population (true) regression line. To estimate the population standard deviation of y, σ, use the standard deviation of the residuals, s. s = SEE n − 2... |
Standard deviation of the residuals: s = SEE n − 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 REG... |
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 “year” the independent variable and “# AIDS cases diagnosed.” the dependent variab... |
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 ... |
hours and ŷ 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 ŷ = 0?) Table 12.14 contains real data for the first two decades of AI... |
the regression line. Figure 12.29 Obtain the graph on your calculator or computer. 43. Write the equation: ŷ= ____________ 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 d... |
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 th... |
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... |
. 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 ... |
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 women’s 100-meter freestyle (s... |
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 leas... |
? 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 mi... |
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 q... |
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... |
independent variable and “welfare family size” as the dependent variable, draw a scatter plot of the data. b. Calculate the least-squares line. Put the equation in the form of: ŷ = a + bx c. Find the correlation coefficient. Is it significant? d. Pick two years between 1969 and 1991 and find the estimated welfare fami... |
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 “size” as the indepen... |
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 t... |
� = 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 outlier... |
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 p... |
= 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 –1.5 (b = –1.5). 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 $1... |
correlation coefficient. When there is a linear association and it is appropriate to calculate a correlation, we cannot say that one variable “causes” the other variable. 49 We don’t know if the pre-1981 data was collected from a single year. So we don’t have an accurate x value for this figure. Regression equation: ŷ... |
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 = –0.57874 is not less than –0.811, so we do not reject the null ... |
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 student’s 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 retur... |
, 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 onl... |
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 = –0.08194; for each additional ounce in size, the cost per ounce decreases by 0.082 cents. 690 CHAPTER 12 | LIN... |
. 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. ŷ = −266.8863+0.1656x d. 0.9448; Yes e. 62.8233; 62.3265 f. ... |
b. Check student’s solution. c. There appears to be a linear relationship, with one outlier. 692 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION d. ŷ (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... |
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 AN... |
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/col... |
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 • k = the number of different groups • nj = the size of the jth group • sj = the sum of the values in the jth group • n = total number of all the values combined ... |
. 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 MSbetwee... |
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) ... |
(df) Mean Square (MS) F Factor (Between) SS(Factor) = SS(Between) = 2.2458 k – 1 = 3 groups – 1 = 2 SS(Error) = SS(Within) = 20.8542 SS(Total) = 2.2458 + 20.8542 = 23.1 n – k = 10 total data – 3 groups = 7 n – 1 = 10 total data – 1 = 9 Error (Within) Total Table 13.3 F = MS(Factor)/MS(Error) = 1.1229/2.9792 = 0.3769 M... |
(num) d f (denom) – 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 f... |
Make a decision: Since α > 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... |
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 μ1, μ2, μ3, μ4 be the population means of the sororities. Remembe... |
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... |
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 x¯ 2 Then MSbetween = ns x¯ 2 = (5)(0.413) where n... |
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... |
are the degrees of freedom for the numerator and n2 – 1 are the degrees of freedom for the denominator. If the null hypothesis is σ1 2 = σ2 2, then the F Ratio becomes F = ⎡ ⎢ ⎣ ⎡ ⎢ ⎣ (s1)2 (σ1)2 (s2)2 (σ2)2 ⎤ ⎥ ⎦ ⎤ ⎥ ⎦ = (s1)2 (s2)2. NOTE The F ratio could also be (s2)2 (s1)2. It depends on Ha and on which sample var... |
2 (σ1)2 (s2)2 (σ2)2 ⎤ ⎥ ⎦ ⎤ ⎥ ⎦ = (s1)2 (s2)2 = 52.3 89.9 = 0.5818 Distribution for the test: F29,29 where n1 – 1 = 29 and n2 – 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 <... |
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 • The student will conduct a simple one-way ANOVA test i... |
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: • all populations of interest are norma... |
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, th... |
between MSwithin = SSwithin d fwithin F = MSbetween MSwithin F ratio when the groups are the same size: F = Mean of the F distribution: µ = d f (num) d f (denom) − 1 PRACTICE 13.1 One-Way ANOVA where: • k = the number of groups • nj = the size of the jth group • sj = the sum of the values in the jth group • n = the tot... |
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 M... |
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 lice... |
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 student’s grades have a standard dev... |
4 17.1 17.2 16.6 16.8 x¯ = ________ 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 inter... |
.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 com... |
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 show... |
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 var... |
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... |
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 ... |
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... |
.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 ... |
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, ... |
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 ... |
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: µL = µT = µJ 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 st... |
Reason for decision: p-value > alpha iv. Conclusion: The mean scores of different class delivery are not different. 72 a. H0: μp = μm = μh 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 student’s solution. h. i. Alpha: 0.05 ii. Decision: Do not reject the n... |
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. ... |
MS) F Factor (Between) Error (Within) Total Table 13.42 37.748 11.015 48.763 4 – 1 = 3 27 – 4 = 23 27 – 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 ... |
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... |
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 inferentia... |
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 outlie... |
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... |
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 mat... |
ax text for free at http://cnx.org/content/col11759/latest. SECTION 1.1 REAL NUMBERS: ALGEBRA ESSENTIALS 7 How To… 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... |
(15) − 2[(3) − 4 2 ] + 1 = 7(15) − 2(3 − 16) + 1 = 7(15) − 2(−13) + 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 e... |
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 + (−9)] + 23 = 29 and 15 + [(−9) + 23] = 29 Are subtraction and division associative? Review these examples. 8 − (3 − 1... |
5 + 3) Now, distribute −1 and simplify the result. 12 − (5 + 3) = 12 + (−1) ∙ (5 + 3) = 12 + [(−1) ∙ 5 + (−1) ∙ 3] = 12 + (−8) = 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... |
Property Distributive Property Identity Property Inverse Property Addition b + c) = (a + b) + c Multiplication a ∙ b = b ∙ a a(bc) = (ab)c a ∙ (b + c) = a ∙ b + a ∙ 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 invers... |
properties apply. a. ( − ) ] b. 5 · (6.2 + 0.4) c. 18 − (7 − 15) d. 23 ) · [ 11 · ( − 17 _ 18 17 _ 18 5 _ 23 ) ] e. 6 ċ (−3) + 6 ċ 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 e... |
. 3 5 4 _, π 3 c, n Try It #8 List the constants and variables for each algebraic expression. a. 2πr(r + h) b. 2(L + W) c. 4 y 3 + y Example 9 Evaluating an Algebraic Expression at Different Values Evaluate the expression 2x − 7 for each value for x. 1 _ c. x = 2 b. x = 1 a. x = 0 Solution a. Substitute 0 for x. b. Sub... |
= −85 e. Substitute 2 for m and 3 for n. √ Try It #10 Evaluate each expression for the given values. — 2m3n2 = √ = √ = √ = 12 — 2(2)3(3)2 — 2(8)(9) — 144 a. y + 3 _ y − 3 for y = 5 d. (p 2q)3 for p = −2, q = 3 b. 7 − 2t for t = −2 c. 1 _ πr 2 for r = 11 3 2 1 _ _ e. 4(m − n) − 5(n − m) for m =, n = 3 3 Download the Op... |
9)] = 2π(6)(15) = 180π The surface area is 180π 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) − L ∙ W. See Figure 5. Find the area of a matte for a photograph wi... |
(1 − p) d. 9r − (s + 2r) + (6 − 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... |
11) · 2 19. 6 + 2 · 2 − 1 20. 64 ÷ (8 + 4 · 2) 21. 9 + 4(22) 22. (12 ÷ 3 · 3)2 23. 25 ÷ 52 − 7 24. (15 − 7) · (3 − 7) 25. 2 · 4 − 9(−1) 1 _ 26. 42 − 25 · 5 27. 12(3 − 1) ÷ 6 ALGEBRAIC For the following exercises, solve for the variable. 28. 8(x + 3) = 64 29. 4y + 8 = 2y 30. (11a + 3) − 18a = −4 31. 4z − 2z(1 + 4) = 36... |
washing his neighbor’s 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 circumf... |
a rational number is also rational. 65. Determine whether the simplified expression is rational or irrational: √ — −18 − 4(5)(−1). 66. Determine whether the simplified expression is 67. The division of two whole numbers will always result rational or irrational: √ — −16 + 4(5) + 5. in what type of number? 68. What pro... |
, 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... |
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 ... |
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) · (x 2) · (x 2) 3 factors x · x ) · ( 2 factors factors = ( 2 factors = The exponent of the answer is the produc... |
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 − 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 ... |
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 ... |
4 = z−1 = z 4 z 4 (−5t 3)4 1 _ _ (−5t3)8 = (−5t 3)4 − 8 = (−5t 3)−4 = (−5t 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... |
(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. (−2w3)3 e. (e−2f 2)7 b. (2t)15 d. 1 ______ (−7z)4 Solution Use the product and quotient rules and the new definit... |
as a power of a quotient rule. = f 14 _ e14 (e −2f 2)7 = ( 7 f 2 _ e2 ) = = = (f 2)7 _ (e 2)7 f 2 · 7 _ e2 · 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... |
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