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3. In words, define the random variable. 4. The distribution to use for the test is ________________. 5. Calculate the test statistic using your data. 6. Draw a graph and label it appropriately. Shade the actual level of significance. a. Graph: Figure 10.16 b. Calculate the p-value. 7. Do you reject or not reject the ... |
the sample standard deviations), will have these characteristics: ¯ • Random variable: X ¯ 1 − X 2 = the difference of the means • Distribution: normal distribution 10.3 Comparing Two Independent Population Proportions Test of two population proportions from independent samples. • Random variable: p^ A – p^ B = differ... |
and σ2 are the known population standard deviations. n1 and x¯ and n2 are the sample sizes. x¯ 2 means. μ1 and μ2 are the population means. are the sample 1 10.3 Comparing Two Independent Population Proportions Pooled Proportion: pc = xF + x M nF + n M Distribution for the differences: ⎡ ⎛ ⎣0, pc(1 − pc) ⎝ p′ A − p′B ... |
is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 10 | HYPOTHESIS TESTING WITH TWO SAMPLES 553 d. single mean e. two proportions f. single proportion 1. It is believed that 70% of males pass their drivers test in the first attempt, while 65% o... |
forests in the United States, 56 were coniferous or contained conifers. In a random sample of 80 forests in Mexico, 40 were coniferous or contained conifers. Is the proportion of conifers in the United States statistically more than the proportion of conifers in Mexico? 10. A new medicine is said to help improve sleep... |
distributions. 16. Are standard deviations known or unknown? 17. What is the random variable? 18. Is this a one-tailed or two-tailed test? 554 CHAPTER 10 | HYPOTHESIS TESTING WITH TWO SAMPLES Use the following information to answer the next 12 exercises: The U.S. Center for Disease Control reports that the mean life e... |
Wesley 86 Rodriguez 91 Table 10.18 3 7 31. What is the random variable? 32. State the null and alternative hypotheses. 33. What is the test statistic? 34. What is the p-value? 35. At the 1% significance level, what is your conclusion? Use the following information to answer the next five exercises. A researcher is tes... |
sample of 150 phones with OS2 had system failures within the first eight hours of operation. OS2 is believed to be more stable (have fewer crashes) than OS1. 46. Is this a test of means or proportions? 47. What is the random variable? 48. State the null and alternative hypotheses. 49. What is the p-value? 50. What can... |
software patch in reducing system failures over a six-month period. Results for randomly selected installations are shown in Table 10.21. The “before” value is matched to an “after” value, and the differences are calculated. The differences have a normal distribution. Test at the 1% significance level. Installation A ... |
in Appendix E. Please feel free to make copies of the solution sheets. For the online version of the book, it is suggested that you copy the.doc or the.pdf files. NOTE If you are using a Student's t-distribution for a homework problem in what follows, including for paired data, you may assume that the underlying popul... |
entry-level mechanical engineering salary is lower than the mean entry-level electrical engineering salary. 82. Marketing companies have collected data implying that teenage girls use more ring tones on their cellular phones than teenage boys do. In one particular study of 40 randomly chosen teenage girls and boys (20... |
16 CHAPTER 10 | HYPOTHESIS TESTING WITH TWO SAMPLES 559 c. There is insufficient evidence to conclude that the W teams score fewer goals, on average, than the E teams score. d. Unable to determine 89. Suppose a statistics instructor believes that there is no significant difference between the mean class scores of stati... |
deviation for 37 statistics night students were 75.41 and 19.73. The “day” subscript refers to the statistics day students. The “night” subscript refers to the statistics night students. An appropriate alternative hypothesis for the hypothesis test is: a. μday > μnight b. μday < μnight c. μday = μnight d. μday ≠ μnigh... |
year university. The sample means were $947 and $1,011, respectively. The population standard deviations are known to be $254 and $87, respectively. Conduct a hypothesis test to determine if the means are statistically the same. 96. Some manufacturers claim that non-hybrid sedan cars have a lower mean miles-per-gallon ... |
that the underlying population is normally distributed. (In general, you must first prove that assumption, however.) 99. A recent drug survey showed an increase in the use of drugs and alcohol among local high school seniors as compared to the national percent. Suppose that a survey of 100 local seniors and 100 nation... |
. In the United States in 2010 there were 629 reported cases of neuroinvasive West Nile virus out of a total of 1,021 reported cases and there were 486 neuroinvasive reported cases out of a total of 712 cases reported in 2011. Is the 2011 proportion of neuroinvasive West Nile virus cases more than the 2010 proportion o... |
index (BMI) is at least 30. The researchers wanted to determine if the proportion of women who are obese in the south is less than the proportion of southern men who are obese. The results are shown in Table 10.27. Test at the 1% level of significance. Number who are obese Sample size Men 42,769 Women 67,169 Table 10.... |
one pierced ear. Out of 92 females, 47 had at least one pierced ear. Do you believe that the proportion of males has reached the proportion of females? 113. Use the data sets found in Appendix C to answer this exercise. Is the proportion of race laps Terri completes slower than 130 seconds less than the proportion of ... |
5 10.4 Matched or Paired Samples DIRECTIONS: For each of the word problems, use a solution sheet to do the hypothesis test. The solution sheet is found in m47889 (http://cnx.org/content/m47889/latest/). Please feel free to make copies of the solution sheets. For the online version of the book, it is suggested that you... |
answer the next two exercises. An experiment is conducted to show that blood pressure can be consciously reduced in people trained in a “biofeedback exercise program.” Six subjects were randomly selected and blood pressure measurements were recorded before and after the training. The difference between blood pressures... |
http://cnx.org/content/col11562/1.16 CHAPTER 10 | HYPOTHESIS TESTING WITH TWO SAMPLES 565 Southern States 2012 2013 South Carolina 3,570 3,580 Tennessee 4,680 5,070 Texas Virginia Table 10.32 15,050 14,980 6,190 6,280 122. A traveler wanted to know if the prices of hotels are different in the ten cities that he visits... |
. 126. The mean number of English courses taken in a two–year time period by male and female college students is believed to be about the same. An experiment is conducted and data are collected from nine males and 16 females. 127. A football league reported that the mean number of touchdowns per game was five. A study ... |
41 right-handed preschool children were given several tests of motor skills to determine if there is evidence of a difference between the children based on this experiment. The experiment produced the means and standard deviations shown Table 10.36. Determine the appropriate test and best distribution to use for that ... |
17, 2013). “World Series History.” Baseball-Almanac, 2013. Available online at http://www.baseball-almanac.com/ws/wsmenu.shtml (accessed June 17, 2013). 10.2 Two Population Means with Known Standard Deviations Data from the United States Census Bureau. Available online at http://www.census.gov/prod/cen2010/briefs/c201... |
. “State States.aspx?ref=interactive (accessed June 17, 2013). States.” Gallup, the of 2013. Available online at http://www.gallup.com/poll/125066/State- “West Nile Virus.” Centers for Disease Control and Prevention. Available online at http://www.cdc.gov/ncidod/dvbid/ westnile/index.htm (accessed June 17, 2013). SOLUT... |
first eight hours of operation with OS1 and OS2. 49 0.1018 51 proportions 53 right-tailed 55 The random variable is the difference in proportions (percents) of the populations that are of two or more races in Nevada and North Dakota. 570 CHAPTER 10 | HYPOTHESIS TESTING WITH TWO SAMPLES 57 Our sample sizes are much gre... |
Ha: µ1 < µ2 ¯ c. X ¯ 1 − X 2 is the difference between the mean entry level salaries of mechanical engineers and electrical engineers. d. e. t108 test statistic: t = –0.82 f. p-value: 0.2061 g. Check student’s solution. h. i. Alpha: 0.05 ii. Decision: Do not reject the null hypothesis. iii. Reason for Decision: p-valu... |
reject H0. Conclusion: At the 1% level of significance, from the sample data, there is not sufficient evidence to conclude that the mean age of entering prostitution in Canada is lower than the mean age in the United States. 92 d 94 Subscripts: 1 = boys, 2 = girls a. H0: µ1 ≤ µ2 b. Ha: µ1 > µ2 c. The random variable i... |
. Reason for Decision: p-value < alpha iv. Conclusion: This is a weak test because alpha and the p-value are close. However, there is insufficient evidence to conclude that the mean difference is negative. 100 a. H0: PW = PB b. Ha: PW ≠ PB c. The random variable is the difference in the proportions of white and black s... |
Distribution: H0: p1 = p2 Ha: p1 > p2 A higher proportion of tablet owners are aged 16 to 29 years old than are 30 years old and older. Graph: righttailed Figure 10.22 p-value: 0.2354 Decision: Do not reject the H0. Conclusion: At the 1% level of significance, from the sample data, there is not sufficient evidence to ... |
statistic: 4.8963 d. e. f. p-value: 0.0004 g. Check student’s solution. h. i. Alpha: 0.05 ii. Decision: Reject the null hypothesis. iii. Reason for Decision: p-value < alpha iv. Conclusion: At the 5% level of significance, there is sufficient evidence to conclude that the mean difference in work times on days when eat... |
135 a This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 11 | THE CHI-SQUARE DISTRIBUTION 577 11 | THE CHI-SQUARE DISTRIBUTION Figure 11.1 The chi-square distribution can be used to find relationships between two things, like groce... |
notation for the chi-square distribution is: 2 χ ∼ χd f where df = degrees of freedom which depends on how chi-square is being used. (If you want to practice calculating chisquare probabilities then use df = n - 1. The degrees of freedom for the three major uses are each calculated differently.) For the χ2 distributio... |
of different data cells or categories The observed values are the data values and the expected values are the values you would expect to get if the null hypothesis were true. There are n terms of the form (O − E)2. E The number of degrees of freedom is df = (number of categories – 1). The goodness-of-fit test is almos... |
per term Actual number of students 0–2 35 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 11 | THE CHI-SQUARE DISTRIBUTION 581 Number of absences per term Actual number of students 3–5 6–8 9+ Table 11.4 40 20 5 b. What is the nu... |
+ 9 + 15 = 60), there would be 12 absences on Monday, 12 on Tuesday, 12 on Wednesday, 12 on Thursday, and 12 on Friday. These numbers are the expected (E) values. The values in the table are the observed (O) values or data. This time, calculate the χ2 test statistic by hand. Make a chart with the following headings an... |
2 GOF. Enter the list names for the Observed list and the Expected list. Enter the degrees of freedom and press calculate or draw. Make sure you clear any lists before you start. To Clear Lists in the calculators: Go into STAT EDIT and arrow up to the list name area of the particular list. Press CLEAR and then arrow do... |
96 (0.55)(600) = 330 (0.11)(600) = 66 (0.08)(600) = 48 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 11 | THE CHI-SQUARE DISTRIBUTION 585 Therefore, the expected frequencies are 60, 96, 330, 66, and 48. In the TI calculators, ... |
to the name area "L3". Enter (L1-L2)^2/L2 and ENTER. Press 2nd QUIT. Press 2nd LIST and arrow over to MATH. Press 5. You should see "sum" (Enter L3). Rounded to 2 decimal places, you should see 29.65. Press 2nd DISTR. Press 7 or Arrow down to 7:χ2cdf and press ENTER. Enter (29.65,1E99,4). Rounded to four places, you s... |
flipping two fair coins is {HH, HT, TH, TT}. Out of 100 flips, you would expect 25 HH, 25 HT, 25 TH, and 25 TT. This is the expected distribution. The question, "Are the coins fair?" is the same as saying, "Does the distribution of the coins (20 HH, 27 HT, 30 TH, 23 TT) fit the expected distribution?" Random Variable:... |
should see.3430, which is the p-value. The newer TI-84 calculators have in STAT TESTS the test Chi2 GOF. To run the test, put the observed values (the data) into a first list and the expected values (the values you expect if the null hypothesis is true) 588 CHAPTER 11 | THE CHI-SQUARE DISTRIBUTION into a second list. ... |
(A AND B) = P(A)P(B). A AND B is the event that a driver received a speeding violation last year and also used a cell phone while driving. Suppose, in a study of drivers who received speeding violations in the last year, and who used cell phone while driving, that 755 people were surveyed. Out of the 755, 70 had a spee... |
In a volunteer group, adults 21 and older volunteer from one to nine hours each week to spend time with a disabled senior citizen. The program recruits among community college students, four-year college students, and nonstudents. In Table 11.15 is a sample of the adult volunteers and the number of hours they voluntee... |
/1.16 CHAPTER 11 | THE CHI-SQUARE DISTRIBUTION 591 Compare α and the p-value: Since no α is given, assume α = 0.05. p-value = 0.0113. α > p-value. Make a decision: Since α > p-value, reject H0. This means that the factors are not independent. Conclusion: At a 5% level of significance, from the data, there is sufficient... |
level and the need to succeed in school. A random sample of 400 students took a test that measured anxiety level and need to succeed in school. Table 11.18 shows the results. De Anza College wants to know if anxiety level and need to succeed in school are independent events. 592 CHAPTER 11 | THE CHI-SQUARE DISTRIBUTIO... |
test for homogeneity, follow the same procedure as with the test of independence. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 NOTE The expected value for each cell needs to be at least five in order for you to use this test. CHAPTER... |
each. Press 2nd QUIT. Press STAT and arrow over to TESTS. Arrow down to C:χ2-TEST. Press ENTER. You should see Observed:[A] and Expected:[B]. Arrow down to Calculate. Press ENTER. The test statistic is 10.1287 and the p-value = 0.0175. Do the procedure a second time but arrow down to Draw instead of calculate. Compare... |
voter preferences was not the same before and after the earthquake. Degrees of Freedom (df): df = number of columns – 1 = 3 – 1 = 2 2 Distribution for the test: χ2 Calculate the test statistic: χ2 = 3.2603 (calculator or computer) Probability statement: p-value=P(χ2 > 3.2603) = 0.1959 Press the MATRX key and arrow ove... |
"fits" a known distribution. In this case there will be a single qualitative survey question or a single outcome of an experiment 596 CHAPTER 11 | THE CHI-SQUARE DISTRIBUTION from a single population. Goodness-of-Fit is typically used to see if the population is uniform (all outcomes occur with equal frequency), the p... |
points. One of his best students thinks otherwise. The student claims that the standard deviation is more than five points. If the student were to conduct a hypothesis test, what would the null and alternative hypotheses be? Solution 11.10 Even though we are given the population standard deviation, we can set up the t... |
5, and σ = 7.2. Graph: Figure 11.8 Probability statement: p-value = P ( χ2 < 5.67) = 0.000042 Compare α and the p-value: α = 0.05; p-value = 0.000042; α > p-value Make a decision: Since α > p-value, reject H0. This means that you reject σ2 = 7.22. In other words, you do not think the variation in waiting times is 7.2 m... |
last ten amounts. Be sure to include the express lane, if it is open.) NOTE You may need to combine two categories so that each cell has an expected value of at least five. 1. Record the values. __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ __________ ___... |
. Exponential Distribution Test to see if grocery receipts follow the exponential distribution with decay parameter 1 x¯. 1. Using 1 x¯ as the decay parameter, X ~ Exp(_________). 2. Calculate the following: a. b. lowest value = ________ first quartile = ________ c. 37th percentile = ________ d. median = ________ e. 63... |
of snack food? Why or why not? Hypothesis Test Conduct a hypothesis test to determine if the factors are independent: 1. H0: ________ 2. Ha: ________ 3. What distribution should you use for a hypothesis test? 4. Why did you choose this distribution? 5. Calculate the test statistic. 6. Find the p-value. 7. Sketch a gra... |
test that uses the chi-square distribution. The null hypothesis for this test states that the data come from the assumed distribution. The test compares observed values against the values you would expect to have if your data followed the assumed distribution. The test is almost always right-tailed. Each observation o... |
of different data cells or categories df = k − 1 degrees of freedom 11.3 Test of Independence Test of Independence • The number of degrees of freedom is equal to (number of columns - 1)(number of rows - 1). • The test statistic is Σ (i ⋅ j) (O – E)2 E where O = observed values, E = expected values, i = the number of r... |
Test Determine the appropriate test to be used in the next three exercises. 6. An archeologist is calculating the distribution of the frequency of the number of artifacts she finds in a dig site. Based on previous digs, the archeologist creates an expected distribution broken down by grid sections in the dig site. Onc... |
.8% Total = 100% 14. If the ethnicities of AIDS victims followed the ethnicities of the total county population, fill in the expected number of cases per ethnic group. Perform a goodness-of-fit test to determine whether the occurrence of AIDS cases follows the ethnicities of the general population of Santa Clara County... |
11.31 21 18 16 12 6 73 14 16 17 14 6 67 6 8 15 21 10 60 41 42 48 47 22 200 26. State the hypotheses. H0: _______ Ha: _______ 27. df = _______ 28. How many passengers are expected to travel between 201 and 300 miles and purchase second-class tickets? 29. How many passengers are expected to travel between 401 and 500 mi... |
the region corresponding to the p-value. Figure 11.15 State the decision and conclusion (in a complete sentence) for the following preconceived levels of α. This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 11 | THE CHI-SQUARE DIS... |
of independence similar to tests for homogeneity? 58. How are tests of independence different from tests for homogeneity? 11.6 Test of a Single Variance Use the following information to answer the next three exercises: An archer’s standard deviation for his hits is six (data is measured in distance from the center of ... |
to determine if the die is fair. The data in Table 11.34 are the result of the 120 rolls. Face Value Frequency Expected Frequency 1 2 3 4 5 6 Table 11.34 15 29 16 15 30 15 73. The marital status distribution of the U.S. male population, ages 15 and older, is as shown in Table 11.35. Marital Status Percent Expected Fre... |
e population, based on ethnicity. 76. The City of South Lake Tahoe, CA, has an Asian population of 1,419 people, out of a total population of 23,609. Suppose that a survey of 1,419 self-reported Asians in the Manhattan, NY, area yielded the data in Table 11.38. Conduct a goodness-of-fit test to determine if the self-re... |
at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 11 | THE CHI-SQUARE DISTRIBUTION 615 Major Other Undecided Table 11.40 Men - Expected Major Men - Actual Major 8.2% 6.6% 400 340 Read the statement and decide whether it is true or false. 79. In a goodness-of-fit te... |
hypothesis test at the 5% significance level to determine whether the survey participants are a representative sample of the USA obese population. Age Class (Years) Obese (Percentage) Expected USA average (Percentage) 20–30 31–40 41–50 51–60 61–70 Table 11.42 75.0 26.5 13.6 21.9 21.0 32.6 32.6 36.6 36.6 39.7 616 CHAPT... |
over Niagara Falls 15 Poconos Europe 15 10 Table 11.46 25 25 25 25 25 15 20 10 5 This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 11 | THE CHI-SQUARE DISTRIBUTION 617 Location 20–29 30–39 40–49 50 and over Virgin Islands 20 25 15... |
000 20 Table 11.50 25 40 10 70 5 30 618 CHAPTER 11 | THE CHI-SQUARE DISTRIBUTION Annual Salary Not a high school graduate High school graduate College graduate Masters or doctorate $40,000–$50,000 10 $50,000–$60,000 5 $60,000+ 0 Table 11.50 20 10 5 40 20 10 55 60 150 Read the statement and decide whether it is true or ... |
okequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 11 | THE CHI-SQUARE DISTRIBUTION 619 AsianAmerican White/NonHispanic AfricanAmerican 433 234 43 710 41 24 16 71 Opinion/ Ethnicity Against tax In Favor of tax No opinion 48 54 16 Column Total 118 Table 11.53 11.4 Test for Homogeneit... |
result you obtained? Reasons for Homeschooling Applicable Reason (in thousands of respondents) Most Important Reason (in thousands of respondents) Concern about the environment of other schools 1,321 309 Row Total 1,630 Table 11.56 620 CHAPTER 11 | THE CHI-SQUARE DISTRIBUTION Reasons for Homeschooling Dissatisfaction ... |
5 Comparison of the Chi-Square Tests This content is available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 11 | THE CHI-SQUARE DISTRIBUTION 621 For each word problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E for the ... |
, which distribution would you use? 120. If an additional test were done on the claim of the average delay, but 45 flights were surveyed, which distribution would you use? For each word problem, use a solution sheet to solve the hypothesis test problem. Go to Appendix E for the chi-square solution sheet. Round expected... |
622 CHAPTER 11 | THE CHI-SQUARE DISTRIBUTION # of births Frequency 0 1 2 3 Table 11.59 5 30 10 5 126. According to an avid aquarist, the average number of fish in a 20-gallon tank is 10, with a standard deviation of two. His friend, also an aquarist, does not believe that the standard deviation is two. She counts the ... |
149; 169; 164; 139; 154; 150; 157; 171; 152; 161; 141; 166; 172; BRINGING IT TOGETHER: HOMEWORK 130. a. Explain why a goodness-of-fit test and a test of independence are generally right-tailed tests. b. If you did a left-tailed test, what would you be testing? REFERENCES 11.1 Facts About the Chi-Square Distribution Da... |
ing Sugar-Sweetened Beverages If Proceeds are Tied to Improving School Nutrition and Physical Activity Programs.” The Field Poll, released Feb. 14, 2013. Available online at http://field.com/fieldpollonline/subscribers/ Rls2436.pdf (accessed May 24, 2013). Harris Interactive, “Favorite Flavor of Ice Cream.” Available o... |
from the expected test scores. 15 H0: the distribution of AIDS cases follows the ethnicities of the general population of Santa Clara County. 17 right-tailed 19 88,621 21 Graph: Check student’s solution. Decision: Reject the null hypothesis. Reason for the Decision: p-value < alpha Conclusion (write out in complete se... |
vary. Sample answer: Tests of independence and tests for homogeneity both calculate the test statistic the same way ∑ (i j) (O - E)2 E. In addition, all values must be greater than or equal to five. 59 a test of a single variance 61 a left-tailed test 63 H0: σ2 = 0.812; Ha: σ2 > 0.812 65 a test of a single variance 67... |
solution. h. i. Alpha = 0.05 ii. Decision: Do not reject null when a = 0.05 and a = 0.01 iii. Reason for decision: p-value > alpha iv. Conclusion: There is insufficient evidence to conclude that the distribution of actual college majors of graduating females fits the distribution of their expected majors. 79 true 81 t... |
there is insufficient evidence to conclude that honeymoon location and bride age are dependent. 91 a. H0: The types of fries sold are independent of the location. b. Ha: The types of fries sold are dependent on the location. c. df = 6 d. chi-square distribution with df = 6 e. test statistic =18.8369 f. p-value = 0.004... |
iii. Reason for decision: p-value > alpha iv. Conclusion: There is insufficient evidence to conclude that the distribution of personality types is different for business and social science majors. 103 a. H0: The distribution for fish caught is the same in Green Valley Lake and in Echo Lake. b. Ha: The distribution for... |
to conclude that the distribution of technology use for statistics homework is not the same for statistics students at community colleges and at universities. 110 225 112 H0: σ2 ≤ 150 114 36 116 Check student’s solution. 118 The claim is that the variance is no more than 150 minutes. 120 a Student's t- or normal distr... |
⎠ = 0.0681 = p-value: P⎛ ⎝x7 = 13.169 ; 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 11 | THE CHI-SQUARE DISTRIBUTION 631 Decision: Do not reject the null hypothesis. Reason for decision: p-value > alpha Conclusio... |
this chapter, you will be studying the simplest form of regression, "linear regression" with one independent variable (x). This involves data that fits a line in two dimensions. You will also study correlation which measures how strong the relationship is. 12.1 | Linear Equations Linear regression for two variables is... |
. Find the equation that expresses the total cost in terms of the number of students in a class. Slope and Y-Intercept of a Linear Equation For the linear equation y = a + bx, b = slope and a = y-intercept. From algebra recall that the slope is a number that describes the steepness of a line, and the y-intercept is the... |
a scatter plot. The following example illustrates a scatter plot. Example 12.5 In Europe and Asia, m-commerce is popular. M-commerce users have special mobile phones that work like electronic wallets as well as provide phone and Internet services. Users can do everything from paying for parking to buying a TV set or s... |
her jump shot each week. She records the following data: X (hours practicing jump shot) Y (points scored in a game) 5 7 9 10 11 12 Table 12.2 15 22 28 31 33 36 Construct a scatter plot and state if what Amelia thinks appears to be true. A scatter plot shows the direction of a relationship between the variables. A clea... |
) finger length, do you think you could predict that person's height? Collect data from your class (pinky finger length, in inches). The independent variable, x, is pinky finger length and the dependent variable, y, is height. For each set of data, plot the points on graph paper. Make your graph big enough and use a ru... |
of the the form (x, y) and each point ofthe line of best fit using leastsquares linear regression has the form (x, ŷ). The ŷ is read "y hat" and is the estimated value of y. It is the value of y obtained using the regression line. It is not generally equal to y from data. This content is available for free at http://t... |
b = Σ(x − x¯ )(y − y¯ ) Σ(x − x¯ ) 2. y^ = a + bx The sample means of the x values and the y values are x¯ and y¯, respectively. The best fit line always passes through the point ( x¯, y¯ ). The slope b can be written as b = r⎛ ⎝ s y s x ⎞ ⎠ where sy = the standard deviation of the y values and sx = the standard devia... |
use the line to predict the final exam score for a student who earned a grade of 50 on the third exam, because 50 is not within the domain of the x-values in the sample data, which are between 65 and 75. UNDERSTANDING SLOPE The slope of the line, b, describes how changes in the variables are related. It is important t... |
to find the values a = –173.513, and b = 4.8273; the equation of the best fit line is ŷ = –173.51 + 4.83x The two items at the bottom are r2 = 0.43969 and r = 0.663. For now, just note where to find these values; we will discuss them in the next two sections. Graphing the Scatterplot and Regression Line 1. We are assu... |
between x and y. The correlation coefficient, r, developed by Karl Pearson in the early 1900s, is numerical and provides a measure of strength and direction of the linear association between the independent variable x and the dependent variable y. The correlation coefficient is calculated as r = nΣ(xy) − (Σx)(Σy) ⎡ ⎡ ... |
AR REGRESSION AND CORRELATION 645 The formula for r looks formidable. However, computer spreadsheets, statistical software, and many calculators can quickly calculate r. The correlation coefficient r is the bottom item in the output screens for the LinRegTTest on the TI-83, TI-83+, or TI-84+ calculator (see previous se... |
the sample data is strong enough to use to model the relationship in the population. The sample data are used to compute r, the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we have only have sample data, we cannot ca... |
Ha: ρ ≠ 0 WHAT THE HYPOTHESES MEAN IN WORDS: • Null Hypothesis H0: The population correlation coefficient IS NOT significantly different from zero. There IS NOT a significant linear relationship(correlation) between x and y in the population. • Alternate Hypothesis Ha: The population correlation coefficient IS signifi... |
available for free at http://textbookequity.org/introductory-statistics or at http://cnx.org/content/col11562/1.16 CHAPTER 12 | LINEAR REGRESSION AND CORRELATION 647 The formula for the test statistic is. The value of the test statistic, t, is shown in the computer or calculator output along with the p-value. The test... |
= n - 2 = 10 - 2 = 8. The critical values associated with df = 8 are -0.632 and + 0.632. If r < negative critical value or r > positive critical value, then r issignificant. Since r = 0.801 and 0.801 > 0.632, r is significant and the line may be usedfor prediction. If you view this example on a number line, it will he... |
line of best fit is: ŷ = –173.51+4.83x with r = 0.6631 and there are n = 11 data points. Can the regression line be used for prediction? Given a third-exam score (x value), can we use the line to predict the final exam score (predicted y value)? H0: ρ = 0 Ha: ρ ≠ 0 α = 0.05 • Use the "95% Critical Value" table for r w... |
between the two critical values so r is not significant. 12.10 For a given line of best fit, you compute that r = 0 using n = 100 data points. Can the line be used for prediction? Why or why not? Assumptions in Testing the Significance of the Correlation Coefficient Testing the significance of the correlation coeffici... |
More y values lie near the line than are scattered further away from the line. 12.5 | Prediction Recall the third exam/final exam example. We examined the scatterplot and showed that the correlation coefficient is significant. We found the equation of the best-fit line for the final exam grade as a function of the gra... |
predicting outside of the observed x values observed in the data is called extrapolation. 12.11 Data are collected on the relationship between the number of hours per week practicing a musical instrument and scores on a math test. The line of best fit is as follows: ŷ = 72.5 + 2.8x What would you predict the score on ... |
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