8-1 Copyright © 2014, 2011, and 2008 Pearson Education, Inc.

8-3 Copyright © 2014, 2011, and 2008 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

8-4 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Learning Objectives 1.Learn how to identify the target parameter for comparing two populations. 2.Learn how to compare two population means using confidence intervals and tests of hypotheses 3.Apply these inferential methods to problems where we want to compare Two population proportions Two population variances

8-6 Copyright © 2014, 2011, and 2008 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?

8-7 Copyright © 2014, 2011, and 2008 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 Ratio of variances; differences in variability or spread; compare variation Quantitative

8-9 Copyright © 2014, 2011, and 2008 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

8-11 Copyright © 2014, 2011, and 2008 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 ]

8-12 Copyright © 2014, 2011, and 2008 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 

8-13 Copyright © 2014, 2011, and 2008 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.]

8-14 Copyright © 2014, 2011, and 2008 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?

8-16 Copyright © 2014, 2011, and 2008 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

8-17 Copyright © 2014, 2011, and 2008 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)?

8-18 Copyright © 2014, 2011, and 2008 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 )

8-19 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 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 Number3535 Mean$ 6,012 $ 5,832 Std Dev$ 602$ 497 Is there any difference in population means (  =.10)?

8-20 Copyright © 2014, 2011, and 2008 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

8-22 Copyright © 2014, 2011, and 2008 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.

8-23 Copyright © 2014, 2011, and 2008 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  /2 is based on (n 1 + n 2 – 2) degrees of freedom.

8-24 Copyright © 2014, 2011, and 2008 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., ).

8-25 Copyright © 2014, 2011, and 2008 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.

8-27 Copyright © 2014, 2011, and 2008 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 equal population variances, is there a difference in average yield (  =.05)? © 1984-1994 T/Maker Co.

8-28 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Small-Sample Test Solution 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

8-30 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Small-Sample Test Solution 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 Do not reject at  =.05 There is no evidence of a difference in means

8-31 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. 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 Number1511 Mean22.0020.27 Std Dev4.77 3.64

8-32 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Small-Sample Test Solution* 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

8-34 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Small-Sample Test Solution* 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 Do not reject at  =.05 There is no evidence of a difference in means

8-35 Copyright © 2014, 2011, and 2008 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.

8-36 Copyright © 2014, 2011, and 2008 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...

8-37 Copyright © 2014, 2011, and 2008 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.

8-38 Copyright © 2014, 2011, and 2008 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.

8-40 Copyright © 2014, 2011, and 2008 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 ]

8-41 Copyright © 2014, 2011, and 2008 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.

8-42 Copyright © 2014, 2011, and 2008 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 

8-43 Copyright © 2014, 2011, and 2008 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.

8-44 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Paired-Difference Confidence Interval for µ d = (µ 1 – µ 2 ) Large Sample Small Sample where t  /2 is based on (n d – 1) degrees of freedom.

8-45 Copyright © 2014, 2011, and 2008 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.

8-46 Copyright © 2014, 2011, and 2008 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.

8-47 Copyright © 2014, 2011, and 2008 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

8-48 Copyright © 2014, 2011, and 2008 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.

8-51 Copyright © 2014, 2011, and 2008 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

8-52 Copyright © 2014, 2011, and 2008 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?

8-53 Copyright © 2014, 2011, and 2008 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.

8-54 Copyright © 2014, 2011, and 2008 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

8-56 Copyright © 2014, 2011, and 2008 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

8-57 Copyright © 2014, 2011, and 2008 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

8-58 Copyright © 2014, 2011, and 2008 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

8-60 Copyright © 2014, 2011, and 2008 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.

8-62 Copyright © 2014, 2011, and 2008 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, and

8-63 Copyright © 2014, 2011, and 2008 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:

8-64 Copyright © 2014, 2011, and 2008 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:

8-65 Copyright © 2014, 2011, and 2008 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.

8-67 Copyright © 2014, 2011, and 2008 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

8-68 Copyright © 2014, 2011, and 2008 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?

8-69 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Test for Two Proportions Solution 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

8-71 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Test for Two Proportions Solution 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 Reject at  =.01 There is evidence of a difference in proportions z = +2.90

8-72 Copyright © 2014, 2011, and 2008 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

8-73 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Test for Two Proportions Solution* 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

8-75 Copyright © 2014, 2011, and 2008 Pearson Education, Inc. Test Statistic: Decision: Conclusion: Test for Two Proportions Solution* 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 z = –4.00 Reject at  =.05 There is evidence MA is less than CA

8-76 Copyright © 2014, 2011, and 2008 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

8-78 Copyright © 2014, 2011, and 2008 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.

8-80 Copyright © 2014, 2011, and 2008 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

8-81 Copyright © 2014, 2011, and 2008 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

8-82 Copyright © 2014, 2011, and 2008 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 of a difference

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