Chapter 8. Inferences Based on a Two Samples Confidence Intervals and Tests of Hypothesis
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1 Chapter 8 Inferences Based on a Two Samples Confidence Intervals and Tests of Hypothesis Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 1
2 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 2
3 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 3
4 8.1 Identifying the Target Parameter Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 4
5 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? Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 5
6 Determining the Target Parameter Parameter Key Words or Phrases Type of Data Mean difference; differences in averages Quantitative p p Differences between proportions, percentages, fractions, or rates; compare proportions Qualitative Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 6
7 8.2 Comparing Two Population Means: Independent Sampling Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 7
8 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 8
9 Large-Sample Confidence Interval 2 2 1, 2 known: for (μ 1 μ 2 ) 2 x 1 x 2 z 2 x1 x 2 x 1 x 2 z n 1 n 2, unknown: s x1 x2 z 2 x1 x x2 1 x2 z 2 n s n Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 9
10 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) z x 1 x 2 D 0 x1 x 2 Test statistic: Rejection region: z < z [or z > z when H a : (µ 1 µ 2 ) > D 0 ] 2 1 x1 x 2 2 n 1 2 n 2 s 2 1 s 2 2 n 1 n 2 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 10
11 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 2 1 x1 x 2 2 n 1 2 n 2 s 2 1 s 2 2 n 1 n 2 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 11
12 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 x 1 x 2 of will be approximately normal regardless of the shapes of the underlying probability distributions 2 2 of the populations. Also, s and s will provide good approximations to 1 and 2 when the samples are both large.] Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 12
13 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. Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 13
14 Large-Sample Confidence Interval Solution s x 1 x2 z 2 n s n (1.3) (1.16) ( ) Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 14
15 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 15
16 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. Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 16
17 H 0 : H a : Large-Sample Test 1-2 = 0 ( 1 = 2 ) ( 1 2 ).05 n 1 = 121, n 2 = 125 Critical Value(s): Reject H 0 Reject H Solution z z Test Statistic: ( ) Decision: Reject at =.05 Conclusion: There is evidence of a difference in means 4.71 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 17
18 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)? Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 18
19 Large-Sample Test Solution* H 0 : H a : 1-2 = 0 ( 1 = 2 ) ( 1 2 ) n 1 =.10 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 19
20 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 /2 is based on (n 1 + n 2 2) degrees of freedom. Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 20
21 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 1 x 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. Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 21
22 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 1 x 2 D s p 1 n 1 n 2 Rejection region: t > t /2 where t /2 is based on (n 1 + n 2 2) degrees of freedom. Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 22
23 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., ) Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 23
24 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. Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 24
25 Small-Sample Confidence Interval Solution df = n 1 + n 2 2 = = 24 t.025 = n1 1 s1 n2 1 s2 sp n n Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 25
26 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 equal population variances, is there a difference in average yield ( =.05)? T/Maker Co. Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 26
27 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 27
28 Small-Sample Test Solution s 2 p 1 1 n s n s n n t x x s 2 p n1 n Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 28
29 Small-Sample Test Solution H 0 : H a : 1 2 = 0 ( 1 = 2 ) ( 1 2 ).05 df = 24 Critical Value(s): Reject H 0 Reject H t Test Statistic: t 1.53 Decision: Do not reject at =.05 Conclusion: There is no evidence of a difference in means Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 29
30 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 30
31 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 31
32 Small-Sample Test Solution* s 2 p n 1 s n 1 s n n t x x s 2 p n1 n Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 32
33 Small-Sample Test Solution* H 0 : H a :.05 df = 24 Critical Value(s): = 0 ( 1 = 2 ) ( 1 2 ) Reject H 0 Reject H t Test Statistic: t 1.00 Decision: Do not reject at =.05 Conclusion: There is no evidence of a difference in means Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 33
34 8.3 Comparing Two Population Means: Paired Difference Experiments Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 34
35 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 ] Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 35
36 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. Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 36
37 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 37
38 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. Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 38
39 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 /2 is based on (n d 1) degrees of freedom. s d n d Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 39
40 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. Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 40
41 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. Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 41
42 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 42
43 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. Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 43
44 Computation Table Observation Before After Difference Sam Tamika Brian Mike Total 4 d = 1 s d = 6.53 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 44
45 Paired-Difference Experiment Confidence Interval Solution df = n d 1 = 4 1 = 3 t.05 = d t d S d n d Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 45
46 Hypotheses for Paired- Difference Experiment Hypothesis H a No Difference Any Difference Research Questions Pop 1 Pop 2 Pop 1 < Pop 2 Pop 1 Pop 2 Pop 1 > Pop 2 H d d d d Note: d i = x 1i x 2i for i th observation 0 0 d d 0 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 46
47 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? Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 47
48 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. Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 48
49 Paired-Difference Experiment H 0 : H a : Small-Sample Test Solution = df = d = 0 ( d = B A ) d < = 3 Critical Value(s): Reject H t Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 49
50 Computation Table Observation Before After Difference Sam Tamika Brian Mike Total 4 d = 1 s d = 6.53 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 50
51 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 51
52 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 $ Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 52
53 Paired-Difference Experiment Small-Sample Test Solution* H 0 : H a : =.01 df = 8 1 = 7 Critical Value(s): Reject H d = 0 ( d = 1 2 ) d < 0 0 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide
54 8.4 Comparing Two Population Proportions: Independent Sampling Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 54
55 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, Eˆp 1 ˆp 2 p 1 p 2 2. The standard deviation of the sampling distribution of ˆp 1 ˆp 2 is ˆp 1 ˆp 2 ˆp 1 ˆp 2 p 1q 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. Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 55
56 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 56
57 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 垐, and 1p1 15, n1q 1 15 n p垐 15, n q 15.) Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 57
58 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: Rejection region: z < z [or z > z when H a : (p 1 p 2 ) > 0 ] z ˆp 1 ˆp 2 ˆp 1 ˆp 2 ˆp 1 ˆp 2 p 1 q 1 p 2q 2 ˆp ˆq 1 1 Note: n 1 n 2 n 1 n 2, ˆp x 1 x 2. n 1 n 2 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 58
59 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 59
60 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. Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 60
61 Confidence Interval for p 1 p 2 Solution pˆ pˆ qˆ qˆ p p Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 61
62 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 62
63 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? Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 63
64 H0: Ha: Test for Two Proportions p 1 p 2 = 0 p 1 p 2 0 = n 1 = n 2 = 82 Critical Value(s): Reject H 0 Reject H Solution z Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 64
65 Test for Two Proportions Solution pˆ x 63 x pˆ n1 n2 x 1 2 ˆ.70 p n x n z p p p p 2.90 ˆ1 ˆ pˆ 1 pˆ n n2 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 65
66 Test for Two Proportions Solution H0: Ha: p 1 p 2 = 0 p 1 p 2 0 = n 1 = 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 66
67 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 67
68 Test for Two Proportions Solution* H0: Ha: = n MA = p MA p CA = 0 p MA p CA < n CA = 1500 Critical Value(s): Reject H z Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 68
69 Test for Two Proportions Solution* pˆ MA xma 74 xca pˆca.0860 n 1500 n 1500 MA CA xma xca pˆ.0677 n n z MA CA Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 69
70 Test for Two Proportions Solution* H0: Ha: = n MA = p MA p CA = 0 p MA p CA < 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 70
71 Key Ideas Key Words for Identifying the Target Parameter d p 1 p 2 Difference in means or averages Paired difference in means or averages Difference in proportions, fractions, percentages, rates Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 71
72 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 72
73 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 73
74 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 74
75 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 Copyright 2018, 2014, and 2011 Pearson Education, Inc. Slide - 75
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