2011 Pearson Education, Inc

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

3 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 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

5 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 7.1 Identifying the Target Parameter

7 Thinking Challenge How would you try to answer these questions? Who gets higher grades: males or females? Which program is faster to learn: Word or Excel?

8 Determining the Target Parameter Parameter Key Words or Phrases Mean difference; differences in averages Type of Data Quantitative p p ( 1) ( ) Differences between proportions, percentages, fractions, or rates; compare proportions Ration of variances; differences in variability or spread; compare variation Qualitative Quantitative

9 7.2 Comparing Two Population Means: Independent Sampling

10 Sampling Distribution Population Population Select simple random sample, n 1. Compute X 1 Compute X 1 X 2 for every pair of samples Select simple random sample, n 2. Compute X 2 Astronomical number of X 1 X 2 values Sampling Distribution 1-2

11 Large-Sample Confidence Interval for (μ 1 μ 2 ) 2 x 1 x 2 z 2 x1 x 2 x x 1 2 z n 1 n 2

12 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 [or 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: z x 1 x 2 D 0 x1 x 2 Rejection region: z < z [or z > z when H a : (µ 1 µ 2 ) > D 0 ] n 1 x1 x 2 n 2 s 2 1 s 2 2 n 1 n 2

13 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: z x 1 x 2 D 0 x1 x 2 Rejection region: z > z n 1 x1 x 2 n 2 s 2 1 s 2 2 n 1 n 2

14 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 x 1 x 2 will be approximately normal regardless of the shapes of the underlying 2 probability distributions of the populations. Also, s and s 2 will provide good approximations to 1 and when the samples are both large.] 2 2

15 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 Mean Std Dev What is the 95% confidence interval for the difference between the mean dividend yields? T/Maker Co.

16 Large-Sample Confidence Interval Solution 2 x 1 x 2 z n 1 n 2 ( ) 1.96 (1.3)2 121 (1.16)

17 Hypotheses for Means of Two Independent Populations Hypothesis H a No Difference Any Difference Research Questions Pop 1 Pop 2 Pop 1 < Pop 2 H Pop 1 Pop 2 Pop 1 > Pop

18 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 Mean Std Dev Is there a difference in average yield ( =.05)? T/Maker Co.

19 Large-Sample Test Solution H 0 : 1-2 = 0 ( 1 = 2 ) H a : ( 1 2 ).05 n 1 = 121, n 2 = 125 Critical Value(s): Reject H 0 Reject H z z Test Statistic: ( ) Decision: Reject at =.05 Conclusion: There is evidence of a difference in means 4.71

20 Large-Sample Test Thinking Challenge 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 Number Mean $ 6,012 $ 5,832 Std Dev $ 602 $ 497 Is there any difference in population means ( =.10)?

21 Large-Sample Test Solution* H 0 : 1-2 = 0 ( 1 = 2 ) H a : ( 1 2 ).10 n 1 = 35, n 2 = 35 Critical Value(s): Reject H 0 Reject H z z Test Statistic: ( ) Decision: Do not reject at = Conclusion: There is no evidence of a difference in means

22 Small-Sample Confidence Interval for (μ 1 μ 2 ) (Independent Samples) x 1 x 2 t 2 s 2 p 1 n 1 1 n 2 where s p 2 n 1 1 s 1 2 n 2 1 s 2 2 n 1 n 2 2 and t a/2 is based on (n 1 + n 2 2) degrees of freedom.

23 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 [or H a : (µ 1 µ 2 ) > D 0 ] Test statistic: t x x 1 2 D s p 1 n 1 n 2 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 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: t x x 1 2 D s p 1 n 1 n 2 Rejection region: t > t /2 where t a/2 is based on (n 1 + n 2 2) degrees of freedom.

25 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 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 Mean Std Dev Assuming normal populations, what is the 95% confidence interval for the difference between the mean dividend yields? T/Maker Co.

27 Small-Sample Confidence Interval Solution df = n 1 + n 2 2 = = 24 t.025 = s 2 p n 1 1 s n 2 1s 2 n 1 n

28 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 Mean Std Dev Assuming normal populations, And population variances are equal is there a difference in average yield ( =.05)? T/Maker Co.

29 H 0 : H a : Small-Sample Test 1 2 = 0 ( 1 = 2 ) ( 1 2 ).05 df = 24 Critical Value(s): Reject H 0 Reject H Solution t

30 Small-Sample Test Solution s 2 p n 1 1 s n 2 1s 2 n 1 n t x x s p 1 n 1 n

31 H 0 : H a : 1 2 = 0 ( 1 = 2 ) ( 1 2 ).05 df = 24 Critical Value(s): Small-Sample Test Reject H 0 Reject H Solution t Test Statistic: t 1.53 Decision: Do not reject at =.05 Conclusion: There is no evidence of a difference in means

32 Small-Sample Test Thinking Challenge 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 Number Mean Std Dev

33 H 0 : H a : Small-Sample Test 1 2 = 0 ( 1 = 2 ) ( 1 2 ).05 df = 24 Critical Value(s): Reject H 0 Reject H Solution* t

34 Small-Sample Test Solution* s 2 p n 1 1 s n 2 1 s 2 n 1 n t x x s 2 p 1 n 1 n

35 H 0 : H a :.05 df = 24 Critical Value(s): Small-Sample Test 1 2 = 0 ( 1 = 2 ) ( 1 2 ) Reject H 0 Reject H Solution* t Test Statistic: t 1.00 Decision: Do not reject at =.05 Conclusion: There is no evidence of a difference in means

36 Approximate Small-Sample Procedures when Equal sample sizes (n 1 = n 2 = n) Confidence interval: x 1 x 2 t 2 s s 2 n Test statistic H 0 : t x 1 x 2 s s 2 n where t is based on v = n 1 + n 2 2 = 2(n 1) degrees of freedom.

37 Approximate Small-Sample Procedures when Unequal sample sizes (n 1 n 2 ) Confidence interval: x 1 x 2 t 2 s 2 1 n 1 s 2 2 n 2 Test statistic H 0 : t x 1 x 2 s 2 1 n 1 s 2 2 n 2 where t is based on degrees of freedom equal to...

38 Approximate Small-Sample Procedures when v 2 s n 1 s 2 2 n 2 2 s 2 1 n 1 n 1 1 s 2 2 n 2 n 2 1 Note: The value of v will generally not be an integer. Round v down to the nearest integer to use the t-table.

39 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 7.3 Comparing Two Population Means: Paired Difference Experiments

41 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 [or H a : µ d > D 0 ] Large Sample Test statistic: z d D 0 d D 0 d n d s d n d Rejection region: z < z [or z > z when H a : (µ d > D 0 ]

42 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 [or H a : µ d > D 0 ] Small Sample Test statistic: t d D 0 s d n d 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 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: z d D 0 d D 0 d n d s d n d Rejection region: z < z

44 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: t d D 0 s d n d Rejection region: t > t where t is based on (n d 1) degrees of freedom.

45 Paired-Difference Confidence Interval for µ d = (µ 1 µ 2 ) Large Sample d z 2 d Small Sample d t n d d z 2 s d n d 2 where t a/2 is based on (n d 1) degrees of freedom. s d n d

46 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 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 Paired-Difference Experiment Data Collection Table Observation Group 1 Group 2 Difference 1 x 11 x 21 d 1 = x 11 x 21 2 x 12 x 22 d 2 = x 12 x 22 i x 1i x 2i d i = x 1i x 2i n x 1n x 2n d n = x 1n x 2n

49 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: Name Before (1) After (2) Sam Tamika Brian Mike Find a 90% confidence interval for the mean difference in test scores.

50 Computation Table Observation Before After Difference Sam Tamika Brian Mike Total 4 d = 1 s d = 6.53

51 Paired-Difference Experiment Confidence Interval Solution df = n d 1 = 4 1 = 3 t.05 = d t d S d n d

52 Hypotheses for Paired- Difference Experiment Hypothesis H a No Difference Any Difference Research Questions Pop 1 Pop 2 Pop 1 < Pop 2 H d d d 0 Pop 1 Pop 2 Pop 1 > Pop 2 0 d Note: d i = x 1i x 2i for i th observation 0 d d 0

53 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: Name Before After Sam Tamika Brian Mike At the.10 level of significance, was the training effective?

54 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 < Defining d = B A and substituting into (4) gives d. 6. The alternative hypothesis is H a : d 0.

55 Paired-Difference Experiment Small-Sample Test Solution H 0 : d = 0 ( d = B A ) H a : d < 0 =.10 df = 4 1 = 3 Critical Value(s): Reject H t

56 Computation Table Observation Before After Difference Sam Tamika Brian Mike Total 4 d = 1 s d = 6.53

57 Paired-Difference Experiment Small-Sample Test Solution H 0 : d = 0 ( d = B A ) Test Statistic: H a : d < 0 d D0 10 =.10 t.306 Sd 6.53 df = 4 1 = 3 nd 4 Critical Value(s): Decision: Reject H 0 Do not reject at = t Conclusion: There is no evidence training was effective

58 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 $

59 Paired-Difference Experiment Small-Sample Test Solution* H 0 : d = 0 ( d = 1 2 ) H a : d < 0 =.01 df = 8 1 = 7 Critical Value(s): Reject H t Test Statistic: t d d s d 1.16 n d Decision: Reject at =.01 Conclusion: There is evidence client s brand (1) sells for less

60 7.4 Comparing Two Population Proportions: Independent Sampling

61 Properties of the Sampling Distribution of (p 1 p 2 ) 1. The mean of the sampling distribution of ˆp 1 ˆp 2 is (p 1 p 2 ); that is, Eˆp 1 ˆp 2 p 1 p 2 2. The standard deviation of the sampling distribution of is ˆp 1 ˆp 2 ˆp 1 ˆp 2 p 1 q 1 p 2q 2 n 1 n 2 3. If the sample sizes n 1 and n 2 are large, the sampling distribution of ˆp 1 ˆp 2 is approximately normal.

62 Large-Sample (1 )% Confidence Interval for (p 1 p 2 ) ˆp 1 ˆp 2 z 2 ˆp 1 ˆp 2 p ˆp 1 ˆp 2 z 1 q 1 2 p 2q 2 n 1 n 2 ˆp 1 ˆp 2 z 2 ˆp 1 ˆq 1 n 1 ˆp 2 ˆq 2 n 2

63 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 ˆp 1 ˆp 2 will be approximately normal. (This condition will be satisfied if both n 1 ˆp 1 15, n 1 ˆq 1 15, n 2 ˆp 2 15, and n 2 ˆq 2 15.)

64 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 [or H a : (p 1 p 2 ) > 0 ] Test statistic: z ˆp 1 ˆp 2 ˆp 1 ˆp 2 Rejection region: z < z [or z > z when H a : (p 1 p 2 ) > 0 ] ˆp 1 ˆp 2 p 1 q 1 p 2q 2 Note: ˆp ˆq 1 1 n 1 n 2 n 1 n 2, ˆp x 1 x 2. n 1 n 2

65 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: z ˆp 1 ˆp 2 ˆp 1 ˆp 2 Rejection region: z > z ˆp 1 ˆp 2 p 1 q 1 p 2q 2 Note: ˆp ˆq 1 1 n 1 n 2 n 1 n 2, ˆp x 1 x 2. n 1 n 2

66 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 Confidence Interval for p 1 p 2 Solution pˆ pˆ qˆ qˆ p 1 p

68 Hypotheses for Two Proportions Hypothesis H a No Difference Any Difference Research Questions Pop 1 Pop 2 Pop 1 < Pop 2 H 0 p p p 1 p 2 0 p p2 0 p 1 p2 0 Pop 1 Pop 2 Pop 1 > Pop 2 p p 1 p2 0 1 p2 0

69 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 H0: Ha: Test for Two Proportions p 1 p 2 = 0 p 1 p 2 0 =.01 n 1 = 78 n 2 = 82 Critical Value(s): Reject H 0 Reject H Solution z

71 Test for Two Proportions Solution öp 1 x 1 n öp 2 x 2 n öp x 1 x 2 n 1 n z öp 1 öp 2 p 1 p n 1 n 1 2 öp 1 öp

72 Test for Two Proportions Solution H0: Ha: p 1 p 2 = 0 p 1 p 2 0 =.01 n 1 = 78 n 2 = 82 Critical Value(s): Reject H 0 Reject H z Test Statistic: z = Decision: Reject at =.01 Conclusion: There is evidence of a difference in proportions

73 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? CA MA

74 H0: p MA p CA = 0 Ha: p MA p CA < 0 =.05 n MA = 1500 n CA = 1500 Critical Value(s):.05 Test for Two Proportions Reject H 0 Solution* z

75 Test for Two Proportions Solution* öp MA x MA n MA öp CA x CA n CA öp x MA x CA n MA n CA z

76 Test for Two Proportions Solution* H0: p MA p CA = 0 Ha: p MA p CA < 0 =.05 n MA = 1500 n CA = 1500 Critical Value(s): Reject H z Test Statistic: z = 4.00 Decision: Reject at =.05 Conclusion: There is evidence MA is less than CA

77 7.5 Determining Sample Size

78 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: n 1 n 2 z (ME) 2 You will need to substitute estimates for the values of 2 and 2 before solving for the sample size. These 2 2 estimates might be sample variances s 1 and s 2 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. 1 2

79 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: n 1 n 2 z 2 2 p 1 q 1 p 2 q 2 (ME) 2 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 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. n n (5.8) 1 2 2

81 Sample Size Example What sample size is needed to estimate p 1 p 2 with 90% confidence and a width of.05? n ME width n (.025)

82 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

83 Key Ideas Determining the Sample Size Estimating : n 1 n 2 z (ME) 2 Estimating p p : n 1 n 2 z 2 2 p 1 q 1 p 2 q 2 (ME) 2

84 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

85 Key Ideas Conditions Required for Inferences about 1 2 / 2 2 Large or small Samples: 1. Independent random samples 2. Both populations normal

86 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

87 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 n 2 p 2 15, n 2 q 2 15

88 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