CHAPTER NINE Testing the Difference Between Two Means

Testing the Difference Between Two Means, Two Proportions, and Two Variances CHAPTER 9 Outline 9 -1 Testing the Difference Between Two Means: Using the z Test 9 -2 Testing the Difference Between Two Means of Independent Samples: Using the t Test 9 -3 Testing the Difference Between Two Means: Dependent Samples 9 -4 Testing the Difference Between Proportions 9 -5 Testing the Difference Between Two Variances Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 2

Learning Objectives 1 2 3 4 5 Test the difference between sample means, using the z test. Test the difference between two means for independent samples, using the t test. Test the difference between two means for dependent samples. Test the difference between two proportions. Test the difference between two variances or standard deviations. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 3

9. 1 Testing the Difference Between Two Means: Using the z Test Assumptions: 1. Both samples are random samples. 2. The samples must be independent of each other. That is, there can be no relationship between the subjects in each sample. 3. The standard deviations of both populations must be known; and if the sample sizes are less than 30, the populations must be normally or approximately normally distributed. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 4

Testing the Difference Between Two Means: Large Samples Formula for the z test for comparing two means from independent populations Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 5

Hypothesis Testing Situations in the Comparison of Means Copyright © 2015 The Mc. Graw-Hill

Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9 -1 Example 9 -1 Page #491 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 8

Example 9 -1: Leisure Time A study using two random samples of 35 people each found that the average amount of time those in the age group of 26– 35 years spent per week on leisure activities was 39. 6 hours, and those in the age group of 46– 55 years spent 35. 4 hours. Assume that the population standard deviation for those in the first age group found by previous studies is 6. 3 hours, and the population standard deviation of those in the second group found by previous studies was 5. 8 hours. At α = 0. 05, can it be concluded that there is a significant difference in the average times each group spends on leisure activities? Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 9

Example 9 -1: Leisure Time A study using two random samples of 35 people each found that the average amount of time those in the age group of 26– 35 years spent per week on leisure activities was 39. 6 hours, and those in the age group of 46– 55 years spent 35. 4 hours. Assume that the population standard deviation for those in the first age group found by previous studies is 6. 3 hours, and the population standard deviation of those in the second group found by previous studies was 5. 8 hours. At α = 0. 05, can it be concluded that there is a significant difference in the average times each group spends on leisure activities? Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 10

Example 9 -1: Leisure Time Step 4: Make the decision. Reject the null hypothesis at α = 0. 05, since 2. 90 > 1. 96. Step 5: Summarize the results. There is enough evidence to support the claim that the means are not equal. That is, the average of the times spent on leisure activities is different for the groups. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 11

Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9 -1 Example 9 -2 Page #492 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 12

Example 9 -2: College Sports Offerings A researcher hypothesizes that the average number of sports that colleges offer for males is greater than the average number of sports that colleges offer for females. A sample of the number of sports offered by colleges is shown. At α = 0. 10, is there enough evidence to support the claim? Assume 1 and 2 = 3. 3. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 13

Example 9 -2: College Sports Offerings Step 1: State the hypotheses and identify the claim. H 0: μ 1 = μ 2 and H 1: μ 1 > μ 2 (claim) Step 2: Compute the test value. For the males: = 8. 6 and 1 = 3. 3 For the females: = 7. 9 and 2 = 3. 3 Substitute in the formula. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 14

Example 9 -2: College Sports Offerings Step 3: Find the P-value. For z = 1. 06, the area is 0. 8554. The P-value is 1. 0000 - 0. 8554 = 0. 1446. Step 4: Make the decision. Do not reject the null hypothesis. Step 5: Summarize the results. There is not enough evidence to support the claim that colleges offer more sports for males than they do for females. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 15

Confidence Intervals for the Difference Between Two Means Formula for the z confidence interval for the difference between two means from independent populations Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 16

Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9 -1 Example 9 -3 Page #478 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 17

Example 9 -3: Leisure Time Find the 95% confidence interval for the difference between the means for the data in Example 9– 1. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 18

9. 2 Testing the Difference Between Two Means: Using the t Test Formula for the t test for comparing two means from independent populations with unequal variances where the degrees of freedom are equal to the smaller of n 1 – 1 or n 2 – 1. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 19

9. 2 Testing the Difference Between Two Means of Independent Samples: Using the t Test Assumptions 1. The samples are random samples. 2. The sample data are independent of one another. 3. When the sample sizes are less than 30, the populations must be normally or approximately normally distributed. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 20

Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9 -2 Example 9 -4 Page #500 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 21

Example 9 -4: Weights of Newborn Infants A researcher wishes to see if the average weights of newborn male infants are different from the average weights of newborn female infants. She selects a random sample of 10 male infants and finds the mean weight is 7 pounds 11 ounces and the standard deviation of the sample is 8 ounces. She selects a random sample of 8 female infants and finds that the mean weight is 7 pounds 4 ounces and the standard deviation of the sample is 5 ounces. Can it be concluded at α = 0. 05 that the mean weight of the males is different from the mean weight of the females? Assume that the variables are normally distributed. Step 1: State the hypotheses and identify the claim. H 0: μ 1 = μ 2 and H 1: μ 1 μ 2 (claim) Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 22

Example 9 -4: Weights of Newborn Infants Step 2: Find the critical values. Since the test is two-tailed and α = 0. 05, the degrees of freedom are the smaller of n 1 – 1 or n 2 – 1. In this case, n 1 – 1 = 10 – 1 = 9 and n 2 – 1 = 8 – 1 = 7. From Table F, the critical values are +2. 365 and -2. 365. Step 3: Find the Test Value. Change the means to ounces. (1 lb = 16 oz ) Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 23

Example 9 -4: Farm Sizes Step 4: Make the decision. Do not reject the null hypothesis. Step 5: Summarize the results. There is not enough evidence to support the claim that the mean of the weights of the male infants is different from the mean of the weights of the female infants. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 24

Confidence Intervals for the Difference Between Two Means Formula for the t confidence interval for the difference between two means from independent populations with unequal variances d. f. = smaller value of n 1 – 1 or n 2 – 1. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 25

Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9 -2 Example 9 -5 Page #501 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 26

Example 9 -5: Confidence Intervals Find the 95% confidence interval for the difference between the means for the data in Example 9– 4. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 27

9. 3 Testing the Difference Between Two Means: Dependent Samples When the values are dependent, do a t test on the differences. Denote the differences with the symbol D, the mean of the population differences with μD, and the sample standard deviation of the differences with s. D. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 28

Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9 -3 Example 9 -6 Page #510 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 29

Example 9 -6: Bank Deposits A sample of nine local banks shows their deposits (in billions of dollars) 3 years ago and their deposits (in billions of dollars) today. At α = 0. 05, can it be concluded that the average in deposits for the banks is greater today than it was 3 years ago? Use α = 0. 05. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 30

Example 9 -6: Bank Deposits Step 1: State the hypotheses and identify the claim. H 0: μD = 0 and H 1: μD < 0 (claim) Step 2: Find the critical value. The degrees of freedom are n – 1 = 9 – 1 = 8. The critical value for a left-tailed test with α = 0. 05 is t = – 1. 860. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 31

Example 9 -6: Bank Deposits Step 3: Compute the test value. Copyright © 2015

Example 9 -6: Bank Deposits Step 4: Make the decision. Do not reject the null hypothesis since the test value, – 1. 674, is greater than the critical value, – 1. 860. Step 5: Summarize the results. There is not enough evidence to show that the deposits have increased over the last 3 years. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 36

Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9 -3 Example 9 -7 Page #512 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 37

Example 9 -7: Cholesterol Levels A dietitian wishes to see if a person’s cholesterol level will change if the diet is supplemented by a certain mineral. Six subjects were pretested, and then they took the mineral supplement for a 6 -week period. The results are shown in the table. (Cholesterol level is measured in milligrams per deciliter. ) Can it be concluded that the cholesterol level has been changed at α = 0. 10? Assume the variable is approximately normally distributed. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 38

Example 9 -7: Cholesterol Levels Step 1: State the hypotheses and identify the claim. H 0: μD = 0 and H 1: μD 0 (claim) Step 2: Find the critical value. The degrees of freedom are 5. At α = 0. 10, the critical values are ± 2. 015. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 39

Example 9 -7: Cholesterol Levels Step 3: Compute the test value. Before (X 1) After (X 2) 210 235 208 190 172 244 D = X 1 – X 2 190 170 20 65 – 2 2 – 1 210 188 173 228 16 D 2 400 4225 4 4 1 256 Σ D = 100 Σ D 2 = 4890 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 40

Example 9 -7: Cholesterol Levels Step 3: Compute the test value. Step 4: Make the decision. Do not reject the null. Step 5: Summarize the results. There is not enough evidence to support the claim that the mineral changes a person’s cholesterol level. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 41

Confidence Interval for the Mean Difference Formula for the t confidence interval for the mean difference Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 42

Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9 -3 Example 9 -8 Page #514 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 43

Example 9 -8: Confidence Intervals Find the 90% confidence interval for the difference between the means for the data in Example 9– 7. Since 0 is contained in the interval, the decision is to not reject the null hypothesis H 0: μD = 0. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 44

9. 4 Testing the Difference Between Proportions Copyright © 2015 The Mc. Graw-Hill Companies,

Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9 -4 Example 9 -9 Page #521 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 46

Example 9 -9: Vaccination Rates In the nursing home study mentioned in the chapteropening Statistics Today, the researchers found that 12 out of 34 small nursing homes had a resident vaccination rate of less than 80%, while 17 out of 24 large nursing homes had a vaccination rate of less than 80%. At α = 0. 05, test the claim that there is no difference in the proportions of the small and large nursing homes with a resident vaccination rate of less than 80%. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 47

Example 9 -9: Vaccination Rates Step 1: State the hypotheses and identify the claim. H 0: p 1 = p 2 (claim) and H 1: p 1 p 2 Step 2: Find the critical value. Since α = 0. 05, the critical values are – 1. 96 and +1. 96. Step 3: Compute the test value. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 48

Example 9 -9: Vaccination Rates Step 4: Make the decision. Reject the null hypothesis. Step 5: Summarize the results. There is enough evidence to reject the claim that there is no difference in the proportions of small and large nursing homes with a resident vaccination rate of less than 80%. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 49

Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9 -4 Example 9 -10 Page #522 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 50

Example 9 -10: Male and Female Workers A survey of 200 randomly selected male and female workers (100 in each group) found that 7% of the male workers said that they worked more than 5 days per week while 11% of the female workers said that they worked more than 5 days per week. At α = 0. 01, can it be concluded that the percentage of males who work more than 5 days per week is less than the percentage of female workers who work more than 5 days per week? Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 51

Example 9 -10: Male and Female Workers Step 1: State the hypotheses and identify the claim. H 0: p 1 = p 2 and H 1: p 1 < p 2 (claim) Step 2: Find the critical value. Since α = 0. 01, the critical value is -2. 33. Step 3: Compute the test value. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 52

Example 9 -10: Texting While Driving Step 4: Make the decision. Do not reject the null hypothesis since -0. 99 > -2. 33 Step 5: Summarize the results. There is not enough evidence to support the claim that the proportion of men who say that they work more than 5 days a week is less than the proportion of women who say that they work more than 5 days a week. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 53

Confidence Interval for the Difference Between Proportions Formula for the confidence interval for the difference between proportions Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 54

Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9 -4 Example 9 -11 Page #523 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 55

Example 9 -11: Confidence Intervals Find the 95% confidence interval for the difference of the proportions for the data in Example 9– 9. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 56

Example 9 -11: Confidence Intervals Find the 95% confidence interval for the difference of the proportions for the data in Example 9– 9. Since 0 is not contained in the interval, the decision is to reject the null hypothesis H 0: p 1 = p 2. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 57

9. 5 Testing the Difference Between Two Variances n In addition to comparing two means, statisticians are interested in comparing two variances or standard deviations. n For the comparison of two variances or standard deviations, an F test is used. n The F test should not be confused with the chi-square test, which compares a single sample variance to a specific population variance, as shown in Chapter 8. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 58

Characteristics of the F Distribution 1. The values of F cannot be negative, because variances are always positive or zero. 2. The distribution is positively skewed. 3. The mean value of F is approximately equal to 1. 4. The F distribution is a family of curves based on the degrees of freedom of the variance of the numerator and the degrees of freedom of the variance of the denominator. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 59

Shapes of the F Distribution Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission

Testing the Difference Between Two Variances where the larger of the two variances is placed in the numerator regardless of the subscripts. (See note on page 519. ) The F test has two terms for the degrees of freedom: that of the numerator, n 1 – 1, and that of the denominator, n 2 – 1, where n 1 is the sample size from which the larger variance was obtained. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 61

Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9 -5 Example 9 -12 Page #529 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 62

Example 9 -12: Table H Find the critical value for a right-tailed F test when α = 0. 05, the degrees of freedom for the numerator (abbreviated d. f. N. ) are 15, and the degrees of freedom for the denominator (d. f. D. ) are 21. Since this test is right-tailed with a 0. 05, use the 0. 05 table. The d. f. N. is listed across the top, and the d. f. D. is listed in the left column. The critical value is found where the row and column intersect in the table. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 63

Example 9 -12: Table H Find the critical value for a right-tailed F test when α = 0. 05, the degrees of freedom for the numerator (abbreviated d. f. N. ) are 15, and the degrees of freedom for the denominator (d. f. D. ) are 21. F = 2. 18 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 64

Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9 -5 Example 9 -13 Page #530 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 65

Example 9 -13: Table H Find the critical value for a two-tailed F test with α = 0. 05 when the sample size from which the variance for the numerator was obtained was 21 and the sample size from which the variance for the denominator was obtained was 12. When you are conducting a two-tailed test, α is split; and only the right tail is used. The reason is that F 1. Since this is a two-tailed test with α = 0. 05, the 0. 05/2 = 0. 025 table must be used. Here, d. f. N. = 21 – 1 = 20, and d. f. D. = 12 – 1 = 11. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 66

Example 9 -13: Table H Find the critical value for a two-tailed F test with α = 0. 05 when the sample size from which the variance for the numerator was obtained was 21 and the sample size from which the variance for the denominator was obtained was 12. F = 3. 23 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 67

Notes for the Use of the F Test 1. The larger variance should always be placed in the numerator of the formula regardless of the subscripts. (See note on page 534. ) 2. For a two-tailed test, the α value must be divided by 2 and the critical value placed on the right side of the F curve. 3. If the standard deviations instead of the variances are given in the problem, they must be squared for the formula for the F test. 4. When the degrees of freedom cannot be found in Table H, the closest value on the smaller side should be used. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 68

Assumptions for Using the F Test 1. The samples must be random samples. 2. The populations from which the samples were obtained must be normally distributed. (Note: The test should not be used when the distributions depart from normality. ) 3. The samples must be independent of each other. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 69

Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9 -5 Example 9 -14 Page #532 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 70

Example 9 -14: Heart Rates of Smokers A medical researcher wishes to see whether the variance of the heart rates (in beats per minute) of smokers is different from the variance of heart rates of people who do not smoke. Two samples are selected, and the data are as shown. Using α = 0. 05, is there enough evidence to support the claim? Step 1: State the hypotheses and identify the claim. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 71

Example 9 -14: Heart Rates of Smokers Step 2: Find the critical value. Use the 0. 025 table in Table H since α = 0. 05 and this is a two-tailed test. Here, d. f. N. = 25, and d. f. D. = 17. The critical value is 2. 56 (d. f. N. 24 was used). Step 3: Compute the test value. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 72

Example 9 -14: Heart Rates of Smokers Step 4: Make the decision. Reject the null hypothesis, since 3. 6 > 2. 56. Step 5: Summarize the results. There is enough evidence to support the claim that the variance of the heart rates of smokers and nonsmokers is different. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 73

Chapter 9 Testing the Difference Between Two Means, Two Proportions, and Two Variances Section 9 -5 Example 9 -15 Page #532 Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 74

Example 9 -15: Noise Levels of Power Mowers The mean noise level of a random sample of 16 riding power mowers is 93. 2 decibels, and the standard deviation is 4. 3 decibels, while the mean noise level of a random sample of 12 push power mowers is 89. 5 decibels and the standard deviation is 3. 6 decibels. Is there enough evidence at α = 0. 01 to conclude that the variance of the noise levels of the riding power mowers is greater than the variance of the noise levels of the push power mowers? Assume the noise levels of both types of power mowers are normally distributed. Step 1: State the hypotheses and identify the claim. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 75

Example 9 -15: Noise Levels of Power Mowers The mean noise level of a random sample of 16 riding power mowers is 93. 2 decibels, and the standard deviation is 4. 3 decibels, while the mean noise level of a random sample of 12 push power mowers is 89. 5 decibels and the standard deviation is 3. 6 decibels. Is there enough evidence at α = 0. 01 to conclude that the variance of the noise levels of the riding power mowers is greater than the variance of the noise levels of the push power mowers? Assume the noise levels of both types of power mowers are normally distributed. Step 2: Find the critical value. Here, d. f. N. = 16, d. f. D. = 11, and α = 0. 01. The critical value is F = 4. 25. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 76

Example 9 -15: Noise Levels of Power Mowers Step 3: Compute the test value. Step 4: Make the decision. Do not reject the null hypothesis since 1. 43 < 4. 25. Step 5: Summarize the results. There is not enough evidence to support the claim that the variance of the noise levels of the riding power mowers is greater than the variance of the noise levels of the push power mowers. Copyright © 2015 The Mc. Graw-Hill Companies, Inc. Permission required for reproduction or display. Bluman Chapter 9 77