1. © 2011 Pearson Education, Inc

2. Statistics for Business and Economics Chapter 7 Inferences Based on Two Samples: Confidence Intervals & Tests of Hypotheses

3. © 2011 Pearson Education, Inc Content 1.Identifying the Target Parameter 2.Comparing Two Population Means: Independent Sampling 3.Comparing Two Population Means: Paired Difference Experiments 4.Comparing Two Population Proportions: Independent Sampling 5.Determining the Sample Size 6.Comparing Two Population Variances: Independent Sampling

4. © 2011 Pearson Education, Inc Learning Objectives 1.Learn how to compare two populations using confidence intervals and tests of hypotheses 2.Apply these inferential methods to problems where we want to compare Two population means Two population proportions Two population variances

5. © 2011 Pearson Education, Inc Learning Objectives 3.Determine the sizes of the samples necessary to estimate the difference between two population parameters with a specified margin of error

6. © 2011 Pearson Education, Inc 7.1 Identifying the Target Parameter

7. © 2011 Pearson Education, Inc Thinking Challenge Who gets higher grades: males or females? Which program is faster to learn: Word or Excel? How would you try to answer these questions?

8. © 2011 Pearson Education, Inc Determining the Target Parameter ParameterKey Words or PhrasesType of Data   –   Mean difference; differences in averages Quantitative p  – p  Differences between proportions, percentages, fractions, or rates; compare proportions Qualitative Ration of variances; differences in variability or spread; compare variation Quantitative

9. © 2011 Pearson Education, Inc 7.2 Comparing Two Population Means: Independent Sampling

10. © 2011 Pearson Education, Inc Sampling Distribution Population 1  1  1 Select simple random sample, n 1. Compute X 1 Compute X 1 – X 2 for every pair of samples Population 2  2  2 Select simple random sample, n 2. Compute X 2 Astronomical number of X 1 – X 2 values  1 -  2 Sampling Distribution

11. © 2011 Pearson Education, Inc Large-Sample Confidence Interval for (μ 1 – μ 2 )

12. © 2011 Pearson Education, Inc Large-Sample Test of Hypothesis for (µ 1 – µ 2 ) One-Tailed Test H 0 : (µ 1 – µ 2 ) = D 0 H a : (µ 1 – µ 2 ) D 0 ] where D 0 = Hypothesized difference between the means (the difference is often hypothesized to be equal to 0) Test statistic: Rejection region: z < –z  [or z > z  when H a : (µ 1 – µ 2 ) > D 0 ]

13. © 2011 Pearson Education, Inc Large-Sample Test of Hypothesis for (µ 1 – µ 2 ) Two-Tailed Test H 0 : (µ 1 – µ 2 ) = D 0 H a : (µ 1 – µ 2 ) ≠ D 0 where D 0 = Hypothesized difference between the means (the difference is often hypothesized to be equal to 0) Test statistic: Rejection region: |z| > z 

14. © 2011 Pearson Education, Inc Conditions Required for Valid Large-Sample Inferences about (μ 1 – μ 2 ) 1.The two samples are randomly selected in an independent manner from the two target populations. 2.The sample sizes, n 1 and n 2, are both large (i.e., n 1 ≥ 30 and n 2 ≥ 30). [Due to the Central Limit Theorem, this condition guarantees that the sampling distribution of will be approximately normal regardless of the shapes of the underlying probability distributions of the populations. Also, and will provide good approximations to and when the samples are both large.]

15. © 2011 Pearson Education, Inc Large-Sample Confidence Interval Example You’re a financial analyst for Charles Schwab. You want to estimate the difference in dividend yield between stocks listed on NYSE and NASDAQ. You collect the following data: NYSE NASDAQ Number 121125 Mean3.272.53 Std Dev1.301.16 What is the 95% confidence interval for the difference between the mean dividend yields?

16. © 2011 Pearson Education, Inc Large-Sample Confidence Interval Solution

17. © 2011 Pearson Education, Inc Hypotheses for Means of Two Independent Populations HaHa Hypothesis Research Questions No Difference Any Difference Pop 1  Pop 2 Pop 1 < Pop 2 Pop 1  Pop 2 Pop 1 > Pop 2 H0H0

18. © 2011 Pearson Education, Inc Large-Sample Test Example You’re a financial analyst for Charles Schwab. You want to find out if there is a difference in dividend yield between stocks listed on NYSE and NASDAQ. You collect the following data: NYSE NASDAQ Number 121125 Mean3.272.53 Std Dev1.301.16 Is there a difference in average yield (  =.05)?

19. © 2011 Pearson Education, Inc Large-Sample Test Solution H 0 : H a :   n 1 =, n 2 = Critical Value(s): Test Statistic: Decision: Conclusion:.05 121 125 Reject at  =.05 There is evidence of a difference in means z 0 1.96-1.96 Reject H 0 0.025  1 -  2 = 0 (  1 =  2 )  1 -  2  0 (  1   2 )

20. © 2011 Pearson Education, Inc You’re an economist for the Department of Education. You want to find out if there is a difference in spending per pupil between urban and rural high schools. You collect the following: Urban Rural Number3535 Mean$ 6,012 $ 5,832 Std Dev$ 602$ 497 Is there any difference in population means (  =.10)? Large-Sample Test Thinking Challenge

21. © 2011 Pearson Education, Inc Large-Sample Test Solution* H 0 : H a :   n 1 =, n 2 = Critical Value(s): Test Statistic: Decision: Conclusion: Do not reject at  =.10 There is no evidence of a difference in means z 0 1.645-1.645.05 Reject H 0 0.05  1 -  2 = 0 (  1 =  2 )  1 -  2  0 (  1   2 ).10 35 35

22. © 2011 Pearson Education, Inc Small-Sample Confidence Interval for ( μ 1 – μ 2 ) (Independent Samples) where and t a/2 is based on (n 1 + n 2 – 2) degrees of freedom.

23. © 2011 Pearson Education, Inc Small-Sample Test of Hypothesis for (µ 1 – µ 2 ) One-Tailed Test H 0 : (µ 1 – µ 2 ) = D 0 H a : (µ 1 – µ 2 ) D 0 ] Test statistic: Rejection region: t < –t  [or t > t  when H a : (µ 1 – µ 2 ) > D 0 ] where t  is based on (n 1 + n 2 – 2) degrees of freedom.

24. © 2011 Pearson Education, Inc Small-Sample Test of Hypothesis for (µ 1 – µ 2 ) Two-Tailed Test H 0 : (µ 1 – µ 2 ) = D 0 H a : (µ 1 – µ 2 ) ≠ D 0 Test statistic: Rejection region: |t| > t  /2 where t a/2 is based on (n 1 + n 2 – 2) degrees of freedom.

25. © 2011 Pearson Education, Inc Conditions Required for Valid Small-Sample Inferences about (μ 1 – μ 2 ) 1.The two samples are randomly selected in an independent manner from the two target populations. 2.Both sampled populations have distributions that are approximately equal. 3.The populations variances are equal (i.e., ).

26. © 2011 Pearson Education, Inc Small-Sample Confidence Interval Example You’re a financial analyst for Charles Schwab. You want to estimate the difference in dividend yield between stocks listed on the NYSE and NASDAQ. You collect the following data: NYSE NASDAQ Number 1115 Mean3.272.53 Std Dev1.301.16 Assuming normal populations, what is the 95% confidence interval for the difference between the mean dividend yields? © 1984-1994 T/Maker Co.

27. © 2011 Pearson Education, Inc Small-Sample Confidence Interval Solution df = n 1 + n 2 – 2 = 11 + 15 – 2 = 24 t.025 = 2.064

28. © 2011 Pearson Education, Inc Small-Sample Test Example You’re a financial analyst for Charles Schwab. Is there a difference in dividend yield between stocks listed on the NYSE and NASDAQ? You collect the following data: NYSE NASDAQ Number 1115 Mean3.272.53 Std Dev1.301.16 Assuming normal populations, And population variances are equal is there a difference in average yield (  =.05)? © 1984-1994 T/Maker Co.

29. © 2011 Pearson Education, Inc H 0 : H a :   df  Critical Value(s):  1 –  2 = 0 (  1 =  2 )  1 –  2  0 (  1   2 ).05 11 + 15 – 2 = 24 t 02.064-2.064.025 Reject H 0 0.025 Small-Sample Test Solution

30. © 2011 Pearson Education, Inc Small-Sample Test Solution

31. © 2011 Pearson Education, Inc H 0 : H a :   df  Critical Value(s): Test Statistic: Decision: Conclusion:  1 –  2 = 0 (  1 =  2 )  1 –  2  0 (  1   2 ).05 11 + 15 – 2 = 24 t 02.064-2.064.025 Reject H 0 0.025 Small-Sample Test Solution Do not reject at  =.05 There is no evidence of a difference in means

32. © 2011 Pearson Education, Inc You’re a research analyst for General Motors. Assuming equal variances, is there a difference in the average miles per gallon (mpg) of two car models (  =.05)? You collect the following: Sedan Van Number1511 Mean22.0020.27 Std Dev4.77 3.64 Small-Sample Test Thinking Challenge

33. © 2011 Pearson Education, Inc H 0 : H a :   df  Critical Value(s): t 02.064-2.064.025 Reject H 0 0.025  1 –  2 = 0 (  1 =  2 )  1 –  2  0 (  1   2 ).05 15 + 11 – 2 = 24 Small-Sample Test Solution*

34. © 2011 Pearson Education, Inc Small-Sample Test Solution*

35. © 2011 Pearson Education, Inc H 0 : H a :   df  Critical Value(s): Test Statistic: Decision: Conclusion: t 02.064-2.064.025 Reject H 0 0.025  1 –  2 = 0 (  1 =  2 )  1 –  2  0 (  1   2 ).05 15 + 11 – 2 = 24 Small-Sample Test Solution* Do not reject at  =.05 There is no evidence of a difference in means

36. © 2011 Pearson Education, Inc Approximate Small-Sample Procedures when 1. Equal sample sizes (n 1 = n 2 = n) Confidence interval: Test statistic H 0 : where t is based on v = n 1 + n 2 – 2 = 2(n – 1) degrees of freedom.

37. © 2011 Pearson Education, Inc Approximate Small-Sample Procedures when 2. Unequal sample sizes (n 1 ≠ n 2 ) Confidence interval: Test statistic H 0 : where t is based on degrees of freedom equal to...

38. © 2011 Pearson Education, Inc Approximate Small-Sample Procedures when Note: The value of v will generally not be an integer. Round v down to the nearest integer to use the t-table.

39. © 2011 Pearson Education, Inc What Should You Do if the Assumptions Are Not Satisfied? If you are concerned that the assumptions are not satisfied, use the Wilcoxon rank sum test for independent samples to test for a shift in population distributions. See Chapter 14.

40. © 2011 Pearson Education, Inc 7.3 Comparing Two Population Means: Paired Difference Experiments

41. © 2011 Pearson Education, Inc Paired-Difference Test of Hypothesis for µ d = (µ 1 – µ 2 ) One-Tailed Test H 0 : µ d = D 0 H a : µ d D 0 ] Large Sample Test statistic: Rejection region: z < –z  [or z > z  when H a : (µ d > D 0 ]

42. © 2011 Pearson Education, Inc Paired-Difference Test of Hypothesis for µ d = (µ 1 – µ 2 ) One-Tailed Test H 0 : µ d = D 0 H a : µ d D 0 ] Small Sample Test statistic: Rejection region: t < –t  [or t > t  when H a : (µ d > D 0 ] where t  is based on (n d – 1) degrees of freedom.

43. © 2011 Pearson Education, Inc Paired-Difference Test of Hypothesis for µ d = (µ 1 – µ 2 ) Two-Tailed Test H 0 : µ d = D 0 H a : µ d ≠ D 0 Large Sample Test statistic: Rejection region: |z| < z 

44. © 2011 Pearson Education, Inc Paired-Difference Test of Hypothesis for µ d = (µ 1 – µ 2 ) Two-Tailed Test H 0 : µ d = D 0 H a : µ d ≠ D 0 Small Sample Test statistic: Rejection region: |t| > t  where t  is based on (n d – 1) degrees of freedom.

45. © 2011 Pearson Education, Inc Paired-Difference Confidence Interval for µ d = (µ 1 – µ 2 ) Large Sample Small Sample where t a/2 is based on (n d – 1) degrees of freedom.

46. © 2011 Pearson Education, Inc Conditions Required for Valid Large - Sample Inferences about µ d 1.A random sample of differences is selected from the target population of differences. 2.The sample size n d is large (i.e., n d ≥ 30); due to the Central Limit Theorem, this condition guarantees that the test statistic will be approximately normal regardless of the shape of the underlying probability distribution of the population.

47. © 2011 Pearson Education, Inc Conditions Required for Valid Small - Sample Inferences about µ d 1.A random sample of differences is selected from the target population of differences. 2.The population of differences has a distribution that is approximately normal.

48. © 2011 Pearson Education, Inc Paired-Difference Experiment Data Collection Table ObservationGroup 1Group 2Difference 1x 11 x 21 d 1 = x 11 – x 21 2x 12 x 22 d 2 = x 12 – x 22  ix1ix1i x2ix2i d i = x 1i – x 2i  nx1nx1n x2nx2n d n = x 1n – x 2n

49. © 2011 Pearson Education, Inc Paired-Difference Experiment Confidence Interval Example You work in Human Resources. You want to see if there is a difference in test scores after a training program. You collect the following test score data: NameBefore (1)After (2) Sam8594 Tamika9487 Brian7879 Mike8788 Find a 90% confidence interval for the mean difference in test scores.

50. © 2011 Pearson Education, Inc Computation Table ObservationBeforeAfterDifference Sam8594–9 Tamika9487 7 Brian7879–1 Mike8788–1 Total– 4 d = –1s d = 6.53

51. © 2011 Pearson Education, Inc Paired-Difference Experiment Confidence Interval Solution df = n d – 1 = 4 – 1 = 3 t.05 = 2.353

52. © 2011 Pearson Education, Inc Hypotheses for Paired- Difference Experiment HaHa Hypothesis Research Questions No Difference Any Difference Pop 1  Pop 2 Pop 1 < Pop 2 Pop 1  Pop 2 Pop 1 > Pop 2 H0H0 Note: d i = x 1i – x 2i for i th observation

53. © 2011 Pearson Education, Inc Paired-Difference Experiment Small-Sample Test Example You work in Human Resources. You want to see if a training program is effective. You collect the following test score data: NameBefore After Sam8594 Tamika9487 Brian7879 Mike8788 At the.10 level of significance, was the training effective?

54. © 2011 Pearson Education, Inc Null Hypothesis Solution 1.Was the training effective? 2.Effective means ‘Before’ < ‘After’. 3.Statistically, this means  B <  A. 4.Rearranging terms gives  B –  A < 0. 5.Defining  d =  B –  A and substituting into (4) gives  d . 6.The alternative hypothesis is H a :  d  0.

55. © 2011 Pearson Education, Inc Paired-Difference Experiment Small-Sample Test Solution H 0 : H a :  = df = Critical Value(s):  d = 0 (  d =  B –  A )  d < 0.10 4 – 1 = 3 t 0-1.638.10 Reject H 0

56. © 2011 Pearson Education, Inc Computation Table ObservationBeforeAfterDifference Sam8594–9 Tamika9487 7 Brian7879–1 Mike8788–1 Total– 4 d = –1s d = 6.53

57. © 2011 Pearson Education, Inc Paired-Difference Experiment Small-Sample Test Solution H 0 : H a :  = df = Critical Value(s): Test Statistic: Decision: Conclusion:  d = 0 (  d =  B –  A )  d < 0.10 4 – 1 = 3 t 0-1.638.10 Reject H 0 Do not reject at  =.10 There is no evidence training was effective

58. © 2011 Pearson Education, Inc Paired-Difference Experiment Small-Sample Test Thinking Challenge You’re a marketing research analyst. You want to compare a client’s calculator to a competitor’s. You sample 8 retail stores. At the.01 level of significance, does your client’s calculator sell for less than their competitor’s? (1)(2) Store Client Competitor 1$ 10$ 11 2811 3710 4912 51111 61013 7912 8810

59. © 2011 Pearson Education, Inc Paired-Difference Experiment Small-Sample Test Solution* H 0 : H a :  = df = Critical Value(s): Test Statistic: Decision: Conclusion: Reject at  =.01 There is evidence client’s brand (1) sells for less t 0-2.998.01 Reject H 0  d = 0 (  d =  1 –  2 )  d < 0.01 8 – 1 = 7

60. © 2011 Pearson Education, Inc 7.4 Comparing Two Population Proportions: Independent Sampling

61. © 2011 Pearson Education, Inc Properties of the Sampling Distribution of (p 1 – p 2 ) 1.The mean of the sampling distribution of is (p 1 – p 2 ); that is, 2.The standard deviation of the sampling distribution of is 3.If the sample sizes n 1 and n 2 are large, the sampling distribution of is approximately normal.

62. © 2011 Pearson Education, Inc Large-Sample (1 –  )% Confidence Interval for (p 1 – p 2 )

63. © 2011 Pearson Education, Inc Conditions Required for Valid Large-Sample Inferences about (p 1 – p 2 ) 1.The two samples are randomly selected in an independent manner from the two target populations. 2.The sample sizes, n 1 and n 2, are both large so that the sampling distribution of will be approximately normal. (This condition will be satisfied if both

64. © 2011 Pearson Education, Inc Large-Sample Test of Hypothesis about (p 1 – p 2 ) One-Tailed Test H 0 : (p 1 – p 2 ) = 0 H a : (p 1 – p 2 ) 0 ] Test statistic: Rejection region: z < –z  [or z > z  when H a : (p 1 – p 2 ) > 0 ] Note:

65. © 2011 Pearson Education, Inc Large-Sample Test of Hypothesis about (p 1 – p 2 ) Two-Tailed Test H 0 : (p 1 – p 2 ) = 0 H a : (p 1 – p 2 ) ≠ 0 Test statistic: Rejection region: | z | > z  Note:

66. © 2011 Pearson Education, Inc Confidence Interval for p 1 – p 2 Example As personnel director, you want to test the perception of fairness of two methods of performance evaluation. 63 of 78 employees rated Method 1 as fair. 49 of 82 rated Method 2 as fair. Find a 99% confidence interval for the difference in perceptions.

67. © 2011 Pearson Education, Inc Confidence Interval for p 1 – p 2 Solution

68. © 2011 Pearson Education, Inc Hypotheses for Two Proportions HaHa Hypothesis Research Questions No Difference Any Difference Pop 1  Pop 2 Pop 1 < Pop 2 Pop 1  Pop 2 Pop 1 > Pop 2 H0H0

69. © 2011 Pearson Education, Inc Test for Two Proportions Example As personnel director, you want to test the perception of fairness of two methods of performance evaluation. 63 of 78 employees rated Method 1 as fair. 49 of 82 rated Method 2 as fair. At the.01 level of significance, is there a difference in perceptions?

70. © 2011 Pearson Education, Inc H 0 : H a :  = n 1 = n 2 = Critical Value(s): p 1 – p 2 = 0 p 1 – p 2  0.01 7882 z 0 2.58-2.58 Reject H 0 0.005 Test for Two Proportions Solution

71. © 2011 Pearson Education, Inc Test for Two Proportions Solution

72. © 2011 Pearson Education, Inc H 0 : H a :  = n 1 = n 2 = Critical Value(s): Test Statistic: Decision: Conclusion: p 1 – p 2 = 0 p 1 – p 2  0.01 7882 z 0 2.58–2.58 Reject H 0 0.005 Test for Two Proportions Solution Reject at  =.01 There is evidence of a difference in proportions z = +2.90

73. © 2011 Pearson Education, Inc Test for Two Proportions Thinking Challenge You’re an economist for the Department of Labor. You’re studying unemployment rates. In MA, 74 of 1500 people surveyed were unemployed. In CA, 129 of 1500 were unemployed. At the.05 level of significance, does MA have a lower unemployment rate than CA? MA CA

74. © 2011 Pearson Education, Inc H 0 : H a :  = n MA = n CA = Critical Value(s): p MA – p CA = 0 p MA – p CA < 0.05 1500 z 0-1.645.05 Reject H 0 Test for Two Proportions Solution*

75. © 2011 Pearson Education, Inc Test for Two Proportions Solution*

76. © 2011 Pearson Education, Inc Test Statistic: Decision: Conclusion: H 0 : H a :  = n MA = n CA = Critical Value(s): p MA – p CA = 0 p MA – p CA < 0.05 1500 z 0-1.645.05 Reject H 0 Test for Two Proportions Solution* z = –4.00 Reject at  =.05 There is evidence MA is less than CA

77. © 2011 Pearson Education, Inc 7.5 Determining Sample Size

78. © 2011 Pearson Education, Inc Determination of Sample Size for Estimating µ 1 – µ 2 To estimate (μ 1 – μ 2 ) with a given margin of error ME and with confidence level (1 –  ), use the following formula to solve for equal sample sizes that will achieve the desired reliability: You will need to substitute estimates for the values of and before solving for the sample size. These estimates might be sample variances and from prior sampling (e.g., a pilot sample) or from an educated (and conservatively large) guess based on the range–that is, s ≈ R/4.

79. © 2011 Pearson Education, Inc Determination of Sample Size for Estimating p 1 – p 2 To estimate (p 1 – p 2 ) with a given margin of error ME and with confidence level (1 –  ), use the following formula to solve for equal sample sizes that will achieve the desired reliability: You will need to substitute estimates for the values of p 1 and p 2 before solving for the sample size. These estimates might be based on prior samples, obtained from educated guesses, or, most conservatively, specified as p 1 = p 2 =.5.

80. © 2011 Pearson Education, Inc Sample Size Example What sample size is needed to estimate μ 1 – μ 2 with 95% confidence and a margin of error of 5.8? Assume prior experience tells us σ 1 =12 and σ 2 =18.

81. © 2011 Pearson Education, Inc Sample Size Example What sample size is needed to estimate p 1 – p 2 with 90% confidence and a width of.05?

82. © 2011 Pearson Education, Inc 7.6 Comparing Two Population Variances: Independent Sampling

83. © 2011 Pearson Education, Inc F-Test for Equal Population Variances One-Tailed Test H0:H0: H a : (or H a : ) Test statistic: Rejection region: F > F  where F  is based on v 1 = numerator degrees of freedom and v 2 = denominator degrees of freedom; v 1 and v 2 are the degrees of freedom for the numerator and denominator sample variances, respectively.

84. © 2011 Pearson Education, Inc F-Test for Equal Population Variances Two-Tailed Test H0:H0: Ha:Ha: Test statistic: Rejection region: F > F  where F  /2 is based on v 1 = numerator degrees of freedom and v 2 = denominator degrees of freedom; v 1 and v 2 are the degrees of freedom for the numerator and denominator sample variances, respectively.

85. © 2011 Pearson Education, Inc Conditions Required for a Valid F-Test for Equal Variances 1.Both sampled populations are normally distributed. 2. The samples are random and independent.

86. © 2011 Pearson Education, Inc F-Test for Equal Variances Example You’re a financial analyst for Charles Schwab. You want to compare dividend yields between stocks listed on the NYSE & NASDAQ. You collect the following data: NYSE NASDAQ Number 2125 Mean3.272.53 Std Dev1.301.16 Is there a difference in variances between the NYSE & NASDAQ at the.05 level of significance? © 1984-1994 T/Maker Co.

87. © 2011 Pearson Education, Inc F-Test for Equal Variances Solution H 0 : H a :   1  2   1 2 =  2 2 12  2212  22.05 2024

88. © 2011 Pearson Education, Inc F-Test for Equal Variances Solution 0 Reject H 0 Do Not Reject H 0 F 0  /2 =.025

89. © 2011 Pearson Education, Inc F-Test for Equal Variances Solution H 0 :  1 2 =  2 2 H a :  1 2   2 2  .05 1  20 2  24 Critical Value(s): Test Statistic: Decision: Conclusion: 0 F 2.33.415.025 Reject H 0.025 Do not reject at  =.05 There is no evidence of a difference in variances

90. © 2011 Pearson Education, Inc F-Test for Equal Variances Thinking Challenge You’re an analyst for the Light & Power Company. You want to compare the electricity consumption of single-family homes in two towns. You compute the following from a sample of homes : Town 1Town 2 Number 25 21 Mean$ 85$ 68 Std Dev $ 30 $ 18 At the.05 level of significance, is there evidence of a difference in variances between the two towns?

91. © 2011 Pearson Education, Inc F-Test for Equal Variances Solution* H 0 : H a :   1  2   1 2 =  2 2 12  2212  22.05 2420

92. © 2011 Pearson Education, Inc Critical Values Solution* 0 Reject H 0 Do Not Reject H 0 F 0  /2 =.025

93. © 2011 Pearson Education, Inc F-Test for Equal Variances Solution* H 0 :  1 2 =  2 2 H a :  1 2   2 2  .05 1  24 2  20 Critical Value(s): Test Statistic: Decision: Conclusion: Reject at  =.05 There is evidence of a difference in variances 0 F 2.41.429.025 Reject H 0.025

94. © 2011 Pearson Education, Inc Key Ideas Key Words for Identifying the Target Parameter   –   Difference in means or averages  d Paired difference in means or averages p 1 – p 2 Difference in proportions, fractions, percentages, rates Ratio (or difference) in variances, spreads

95. © 2011 Pearson Education, Inc Key Ideas Determining the Sample Size Estimating   –   : Estimating p  – p  :

96. © 2011 Pearson Education, Inc Key Ideas Conditions Required for Inferences about µ 1 – µ 2 Large Samples: 1.Independent random samples 2.n 1 ≥ 30, n 2 ≥ 30 Small Samples: 1.Independent random samples 2.Both populations normal 3.

97. © 2011 Pearson Education, Inc Key Ideas Conditions Required for Inferences about Large or small Samples: 1.Independent random samples 2.Both populations normal

98. © 2011 Pearson Education, Inc Key Ideas Conditions Required for Inferences about µ d Large Samples: 1.Random sample of paired differences 2.n d ≥ 30 Small Samples: 1.Random sample of paired differences 2.Population of differences is normal

99. © 2011 Pearson Education, Inc Key Ideas Conditions Required for Inferences about p 1 – p 2 Large Samples: 1.Independent random samples 2.n 1 p 1 ≥ 15, n 1 q 1 ≥ 15 3.n 2 p 2 ≥ 15, n 2 q 2 ≥ 15

100. © 2011 Pearson Education, Inc Key Ideas Using a Confidence Interval for (µ 1 – µ 2 ) or (p 1 – p 2 ) to Determine whether a Difference Exists 1.If the confidence interval includes all positive numbers (+,+): Infer µ 1 > µ 2 or p 1 > p 2 2.If the confidence interval includes all negative numbers (–,–): Infer µ 1 < µ 2 or p 1 < p 2 3.If the confidence interval includes 0 (–, +): Infer no evidence or a difference