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

2. Learning Objectives 1.Distinguish Independent and Related Populations 2.Solve Inference Problems for Two Populations Mean Proportion Variance 3.Determine Sample Size

3. 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?

4. Target Parameters Difference between Means   –   Difference between Proportions p  – p  Ratio of Variances

5. Possible Estimator 

6. Test Statistics What are the possible test statistics? Do we know the sampling distribution?

7. Not any two populations we can make inferences In some cases, we can. –Two independent populations –Two related population, but paired samples In many other cases, we cannot… –Two related population, but not paired samples

8. Independent & Related Populations 1.Different data sources Unrelated Independent Related 1.Same data source Paired or matched Repeated measures (before/after) 2.Use difference between each pair of observations d i = x 1i – x 2i 2.Use difference between the two sample means X 1 – X 2

9. Two Independent Populations Examples 1.An economist wishes to determine whether there is a difference in mean family income for households in two socioeconomic groups. 2.An admissions officer of a small liberal arts college wants to compare the mean SAT scores of applicants educated in rural high schools and in urban high schools.

10. Two Related Populations Examples 1.Nike wants to see if there is a difference in durability of two sole materials. One type is placed on one shoe, the other type on the other shoe of the same pair. 2.An analyst for Educational Testing Service wants to compare the mean GMAT scores of students before and after taking a GMAT review course.

11. Thinking Challenge 1.The life expectancy of light bulbs made in two different factories 2.Difference in hardness between two metals: one contains an alloy, one doesn’t 3.Tread life of two different motorcycle tires: one on the front, the other on the back Are they independent or related?

12. Two Population Inference Two Populations Z (Large sample) t (Paired sample) Z ProportionVariance F t (Small sample) Paired Indep. Mean

13. Comparing Two Means

14. Two Population Inference Two Populations Z (Large sample) t (Paired sample) Z ProportionVariance F t (Small sample) Paired Indep. Mean

15. Comparing Two Independent Means

16. Two Population Inference Two Populations Z (Large sample) t (Paired sample) Z ProportionVariance F t (Small sample) Paired Indep. Mean

17. 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

18. One Population Case

19. Two Population Case (independent) So far we do not know the sampling distribution of If these two populations are independent,

21. Large-Sample Inference for Two Independent Means

22. Two Population Inference Two Populations Z (Large sample) t (Paired sample) Z ProportionVariance F t (Small sample) Paired Indep. Mean

23. Conditions Required for Valid Large- Sample Inferences about μ 1 – μ 2 Assumptions Independent, random samples Can be approximated by the normal distribution when n 1  30 and n 2  30

24. Large-Sample Confidence Interval for μ 1 – μ 2 (Independent Samples) Confidence Interval

25. 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

26. Large-Sample Test for μ 1 – μ 2 (Independent Samples) Two Independent Sample Z-Test Statistic Hypothesized difference

27. 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?

28. Large-Sample Confidence Interval Solution

29. 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

30. Large-Sample Test for μ 1 – μ 2 (Independent Samples) Two Independent Sample Z-Test Statistic Hypothesized difference

31. 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)?

32. Large-Sample Test Solution H 0 : H a :   n 1 =, n 2 = Critical Value(s):.05 121 125 z 0 1.96-1.96 Reject H 0 0.025  1 -  2 = 0 (  1 =  2 )  1 -  2  0 (  1   2 )

33. Large-Sample Test Solution Test Statistic: Decision: Reject at  =.05 Conclusion: There is evidence of a difference in means

34. 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

35. Large-Sample Test Solution* H0: Ha:   n1 =, n2 = Critical Value(s): 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

36. Large-Sample Test Solution* Test Statistic: Decision: Do not reject at  =.10 Conclusion: There is no evidence of a difference in means

37. Small-Sample Inference for Two Independent Means

38. Two Population Inference Two Populations Z (Large sample) t (Paired sample) Z ProportionVariance F t (Small sample) Paired Indep. Mean

39. Conditions Required for Valid Small- Sample Inferences about μ 1 – μ 2 Assumptions Independent, random samples Populations are approximately normally distributed Population variances are equal

40. Small-Sample Confidence Interval for μ 1 – μ 2 (Independent Samples) Confidence Interval

41. 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.

42. Small-Sample Confidence Interval Solution df = n 1 + n 2 – 2 = 11 + 15 – 2 = 24 t.025 = 2.064

43. Two Independent Sample t–Test Statistic Small-Sample Test for μ 1 – μ 2 (Independent Samples) Hypothesized difference

44. 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, is there a difference in average yield (  =.05)? © 1984-1994 T/Maker Co.

45. H0: Ha:   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

47. Test Statistic: Decision: Do not reject at  =.05 Conclusion: There is no evidence of a difference in means Small-Sample Test Solution

48. 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

49. H0: Ha:   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*

51. Test Statistic: Decision: Do not reject at  =.05 Conclusion: There is no evidence of a difference in means Small-Sample Test Solution*

52. Paired Difference Experiments Small-Sample

53. Two Population Inference Two Populations Z (Large sample) t (Paired sample) Z ProportionVariance F t (Small sample) Paired Indep. Mean

54. Paired-Difference Experiments 1.Compares means of two related populations Paired or matched Repeated measures (before/after) 2.Eliminates variation among subjects

55. Conditions Required for Valid Small- Sample Paired-Difference Inferences Assumptions Random sample of differences Both population are approximately normally distributed

56. 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  ix 1i x 2i d i = x 1i – x 2i  nx 1n x 2n d n = x 1n – x 2n

57. 成對樣本 譬如說, 為了檢視投影片教學對於統計課是否有幫 助, 我們可以隨機選取 n 個學生, 並記錄其投影片 教學前與投影片教學後的成績 如果以 X 1i 與 X 2i 分別代表第 i 個同學在投影片教學 前後的成績, 則我們知道 X 1i 與 X 1j 相互獨立, 但是 X 1i 與 X 2i 則非獨立 諸如此類的樣本, 我們稱之為成對樣本

58. Paired-Difference Experiment Small-Sample Confidence Interval Sample MeanSample Standard Deviation d d n S (d i - d) 2 n i i n d i n     11 1 d d df = n d – 1

59. 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.

60. Computation Table ObservationBeforeAfterDifference Sam8594-9 Tamika9487 7 Brian7879 Mike8788 Total- 4 d = –1S d = 6.53

61. Paired-Difference Experiment Confidence Interval Solution df = n d – 1 = 4 – 1 = 3 t.05 = 2.353

62. 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

63. Paired-Difference Experiment Small- Sample Test Statistic t d S n df = n – 1 0 d   D d Sample MeanSample Standard Deviation d didi n S (d i - d) 2 n i n d i n     11 1 d d

64. 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?

65. 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.

66. Computation Table ObservationBeforeAfterDifference Sam8594-9 Tamika9487 7 Brian7879 Mike8788 Total- 4 d = –1S d = 6.53

67. Paired-Difference Experiment Small- Sample Test Solution H0: Ha:  = 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

68. Test Statistic: Decision: Do not reject at  =.10 Conclusion: There is no evidence training was effective Paired-Difference Experiment Small- Sample Test Solution

69. 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

70. Paired-Difference Experiment Small- Sample Test Solution* H0: Ha:  = df = Critical Value(s): Test Statistic: Decision: Conclusion: t 0-2.998.01 Reject H 0  d = 0 (  d =  1 -  2 )  d < 0.01 8 - 1 = 7

71. Test Statistic: Decision: Reject at  =.01 Conclusion: There is evidence client’s brand (1) sells for less Paired-Difference Experiment Small- Sample Test Solution*

72. Comparing Two Population Proportions

73. Two Population Inference Two Populations Z (Large sample) t (Paired sample) Z ProportionVariance F t (Small sample) Paired Indep. Mean

74. Conditions Required for Valid Large- Sample Inference about p 1 – p 2 Assumptions Independent, random samples Normal approximation can be used if

75. Two Population Case (independent) A natural candidate of estimator would be but we do not know its sampling distribution

76. If two populations are independent

77. Large-Sample Confidence Interval for p 1 – p 2 Confidence Interval

78. 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.

79. Confidence Interval for p 1 – p 2 Solution

80. 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

81. Large-Sample Test for p 1 – p 2 Z-Test Statistic for Two Proportions

82. 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?

83. H0: Ha:  = 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

85. Test Statistic: Z = +2.90 Decision: Reject at  =.01 Conclusion: There is evidence of a difference in proportions

86. 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

87. Test Statistic: Decision: Conclusion: H0: Ha:  = 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*

89. Test Statistic: Z = –4.00 Decision: Reject at  =.05 Conclusion: There is evidence MA is less than CA

90. Determining Sample Size

91. Sample size for estimating μ 1 – μ 2 Sample size for estimating p 1 – p 2 ME = Margin of Error

92. 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.

93. Sample Size Example What sample size is needed to estimate p 1 – p 2 with 90% confidence and a width of.05?

94. Comparing Two Population Variances

95. Two Population Inference Two Populations Z (Large sample) t (Paired sample) Z ProportionVariance F t (Small sample) Paired Indep. Mean

96. F Distribution

98. F-Test for Equal Variances Critical Values 0 Reject H 0 Do Not Reject H 0 F 0 Note!  /2

100. Sampling Distribution Population 1  1  1 Select simple random sample, size n 1. Compute S 1 2 Population 2  2  2 Select simple random sample, size n 2. Compute S 2 2 Sampling Distributions for Different Sample Sizes Astronomical number of S 1 2 /S 2 2 values Compute F = S 1 2 /S 2 2 for every pair of n 1 & n 2 size samples

101. Conditions Required for a Valid F-Test for Equal Variances Assumptions Both populations are normally distributed —Test is not robust to violations Independent, random samples

102. F-Test for Equal Variances Hypotheses Hypotheses H 0 :  1 2 =  2 2 OR H 0 :  1 2   2 2 (or  ) H a :  1 2   2 2 H a :  1 2  2 2 (or >) Test Statistic F = s 1 2 /s 2 2 Two sets of degrees of freedom —  1 = n 1 – 1; 2 = n 2 – 1 Follows F distribution

103. F-Test for Equal Variances Critical Values 0 Reject H 0 Do Not Reject H 0 F 0 Note!  /2

104. 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.

105. F-Test for Equal Variances Solution H0:  1 2 =  2 2 Ha:  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

106. F-Test for Equal Variances Solution Test Statistic: Decision: Do not reject at  =.05 Conclusion: There is no evidence of a difference in variances

107. F-Test for Equal Variances Solution 0 Reject H 0 Do Not Reject H 0 F 0  /2 =.025

108. 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?

109. F-Test for Equal Variances Solution* H0:  1 2 =  2 2 Ha:  1 2   2 2  .05 1  24 2  20 Critical Value(s): Test Statistic: Decision: Conclusion: 0 F 2.41.429.025 Reject H 0.025

110. Critical Values Solution* 0 Reject H 0 Do Not Reject H 0 F 0  /2 =.025

111. F-Test for Equal Variances Solution* Test Statistic: Decision: Reject at  =.05 Conclusion: There is evidence of a difference in variances

112. Conclusion 1.Distinguished Independent and Related Populations 2.Solved Inference Problems for Two Populations Mean Proportion Variance 3.Determined Sample Size