Chapter. Hypothesis Testing with Two Samples. Copyright 2015, 2012, and 2009 Pearson Education, Inc. 1

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2 Two Sample Hypothesis Test Compares two parameters from two populations Sampling methods: Independent Samples The sample selected from one population is not related to the sample selected from the second population Dependent Samples (paired or matched samples) Each member of one sample corresponds to a member of the other sample Copyright 2015, 2012, and 2009 Pearson Education, Inc 2

4 Example: Independent and Dependent Samples Classify the pair of samples as independent or dependent Sample 1: Weights of 65 college students before their freshman year begins Sample 2: Weights of the same 65 college students after their freshmen year Solution: Dependent Samples (The samples can be paired with respect to each student) Copyright 2015, 2012, and 2009 Pearson Education, Inc 4

5 Example: Independent and Dependent Samples Classify the pair of samples as independent or dependent Sample 1: Scores for 38 adults males on a psycho- logical screening test for attention-deficit hyperactivity disorder Sample 2: Scores for 50 adult females on a psycho- logical screening test for attention-deficit hyperactivity disorder Solution: Independent Samples (Not possible to form a pairing between the members of the samples; the sample sizes are different, and the data represent scores for different individuals) Copyright 2015, 2012, and 2009 Pearson Education, Inc 5

6 Two Sample Hypothesis Test with Independent Samples 1 Null hypothesis H 0 A statistical hypothesis that usually states there is no difference between the parameters of two populations Always contains the symbol, =, or 2 Alternative hypothesis H a A statistical hypothesis that is true when H 0 is false Always contains the symbol >,, or < Copyright 2015, 2012, and 2009 Pearson Education, Inc 6

7 Two Sample Hypothesis Test with Independent Samples H 0 : μ 1 = μ 2 H 0 : μ 1 μ 2 H 0 : μ 1 μ 2 H a : μ 1 μ 2 H a : μ 1 > μ 2 H a : μ 1 < μ 2 Regardless of which hypotheses you use, you always assume there is no difference between the population means, or μ 1 = μ 2 Copyright 2015, 2012, and 2009 Pearson Education, Inc 7

8 Two Sample z-test for the Difference Between Means Three conditions are necessary to perform a z-test for the difference between two population means μ 1 and μ 2 1 The population standard deviations are known 2 The samples are randomly selected 3 The samples are independent 4The populations are normally distributed or each sample size is at least 30 Copyright 2015, 2012, and 2009 Pearson Education, Inc 8

9 Two Sample z-test for the Difference Between Means If these requirements are met, the sampling distribution for x1 x2 (the difference of the sample means) is a normal distribution with x x x x 1 Sampling distribution Mean: for x1 x2: When no difference is assumed, the mean is 0 Standard error: x x x x n n 1 2 σ xx σ xx 1 2 x x 1 2 Copyright 2015, 2012, and 2009 Pearson Education, Inc 9

10 Two Sample z-test for the Difference Between Means Test statistic is The standardized test statistic is z x x where 1 2 x x 1 2 xx n1 n2 1 2 Can be used when these conditions are met: 1 Both σ 1 and σ 2 are known 2 The samples are random 3 The samples are independent x x The populations are normally distributed or both n 1 30 and n 2 30 Copyright 2015, 2012, and 2009 Pearson Education, Inc 10

11 Using a Two-Sample z-test for the Difference Between Means (Large Independent Samples) In Words 1 Verify that σ 1 and σ 2 are known, the samples are random and independent, and either the pops are normally dist or both n 1 30 and n 2 30 In Symbols 2 State the claim mathematically Identify the null and alternative hypotheses 3 Specify the level of significance State H 0 and H a Identify Copyright 2015, 2012, and 2009 Pearson Education, Inc 11

12 Using a Two-Sample z-test for the Difference Between Means (Large Independent Samples) In Words In Symbols 4 Determine the critical value(s) 5 Determine the rejection region(s) Use Table 4 in Appendix B 6 Find the standardized test statistic and sketch the sampling distribution z x x x x 1 2 Copyright 2015, 2012, and 2009 Pearson Education, Inc 12

13 Using a Two-Sample z-test for the Difference Between Means (Large Independent Samples) In Words In Symbols 7 Make a decision to reject or fail to reject the null hypothesis 8 Interpret the decision in the context of the original claim If z is in the rejection region, reject H 0 Otherwise, fail to reject H 0 Copyright 2015, 2012, and 2009 Pearson Education, Inc 13

14 Example: Two-Sample z-test for the Difference Between Means A credit card watchdog group claims that there is a difference in the mean credit card debts of households in California and Illinois The results of a random survey of 250 households from each state are shown at the left The two samples are independent Assume that σ 1 = $1045 for California and σ 2 = $1350 for Illinois Do the results support the group s claim? Use α = 005 (Source: PlasticEconomycom) Copyright 2015, 2012, and 2009 Pearson Education, Inc 14

Solution: Two-Sample z-test for the Difference Between Means H 0 : μ 1 = μ 2 H a : μ 1 μ 2 005 n 1 = 250, n 2 = 250 Rejection Region: Test Statistic: (4777-4866) - 0 z =» -082 1045 2 250 + 13502 250

15 Solution: Two-Sample z-test for the Difference Between Means H 0 : μ 1 = μ 2 H a : μ 1 μ n 1 = 250, n 2 = 250 Rejection Region: Test Statistic: ( ) - 0 z =» Decision: Fail to Reject H 0 At the 5% level of significance, there is not enough evidence to support the group s claim that there is a difference in the mean credit card debts of households in New York and Texas Copyright 2015, 2012, and 2009 Pearson Education, Inc 15

Example: Using Technology to Perform a Two-Sample z-test A travel agency claims that the average daily cost of meals and lodging for vacationing in Texas is less than the average daily cost in

16 Example: Using Technology to Perform a Two-Sample z-test A travel agency claims that the average daily cost of meals and lodging for vacationing in Texas is less than the average daily cost in Virginia The table at the left shows the results of a random survey of vacationers in each state The two samples are independent Assume that σ 1 = $19 for Texas and σ 2 = $24 for Virginia, and that both populations are normally distributed At α = 001, is there enough evidence to support the claim? (Adapted from American Automobile Association) Copyright 2015, 2012, and 2009 Pearson Education, Inc 16

18 Solution: Using Technology to Perform a Two-Sample z-test Rejection Region: z Decision: Fail to Reject H 0 At the 1% level of significance, there is not enough evidence to support the travel agency s claim Copyright 2015, 2012, and 2009 Pearson Education, Inc 18

19 Two Sample t-test for the Difference Between Means A two-sample t-test is used to test the difference between two population means μ 1 and μ 2 when 1 σ 2 and σ 2 are unknown, 2 the samples are random, 3 the samples are independent, and 4 the populations are normally distributed or both n 1 30 and n 2 30 Copyright 2015, 2012, and 2009 Pearson Education, Inc 19

20 The Two Sample t-test for the Difference Between Means x 1 - x 2 The standardized test statistic is ( ) - ( m 1 - m ) 2 t = x 1 - x 2 s x1 -x 2 The standard error and the degrees of freedom of the sampling distribution depend on whether the 2 2 population variances and are equal 1 2 Copyright 2015, 2012, and 2009 Pearson Education, Inc 20

21 Two Sample t-test for the Difference Between Means Variances are equal Information from the two samples is combined to calculate a pooled estimate of the standard deviation ˆ n s n s n n The standard error for the sampling distribution of x x is 1 2 s x1 -x 2 df= n 1 + n 2 2 = ˆ s 1 n n 2 ˆ s Copyright 2015, 2012, and 2009 Pearson Education, Inc 21

22 Two Sample t-test for the Difference Between Means Variances are not equal If the population variances are not equal, then the standard error is s x1 = s s 2 2 -x 2 n 1 n 2 df = smaller of n 1 1 or n 2 1 Copyright 2015, 2012, and 2009 Pearson Education, Inc 22

24 Two-Sample t-test for the Difference Between Means (Small Independent Samples) In Words 1 Verify that 1 and 2 are unknown, the samples are random and independent, and either the populations are normally distributed or both n 1 30 and n 2 30 In Symbols 2 State the claim mathematically Identify the null and alternative hypotheses 3 Specify the level of significance State H 0 and H a Identify Copyright 2015, 2012, and 2009 Pearson Education, Inc 24

25 Two-Sample t-test for the Difference Between Means (Small Independent Samples) In Words In Symbols 4 Determine the degrees of freedom 5 Determine the critical value(s) 6 Determine the rejection region(s) df = n 1 + n 2 2 or df = smaller of n 1 1 or n 2 1 Use Table 5 in Appendix B Copyright 2015, 2012, and 2009 Pearson Education, Inc 25

26 Two-Sample t-test for the Difference Between Means (Small Independent Samples) In Words 7 Find the standardized test statistic and sketch the sampling distribution In Symbols ( ) - ( m 1 - m ) 2 t = x 1 - x 2 s x1 -x 2 8 Make a decision to reject or fail to reject the null hypothesis 9 Interpret the decision in the context of the original claim If t is in the rejection region, reject H 0 Otherwise, fail to reject H 0 Copyright 2015, 2012, and 2009 Pearson Education, Inc 26

27 Example: Two-Sample t-test for the Difference Between Means The results of a state mathematics test for random samples of students taught by two different teachers at the same school are shown below Can you conclude that there is a difference in the mean mathematics test scores for the students of the two teachers? Use α = 010 Assume the populations are normally distributed and the population variances are not equal Teacher 1 Teacher 2 x1 473 x2 459 s 1 = 397 s 2 = 245 n 1 = 8 n 2 = 18 Copyright 2015, 2012, and 2009 Pearson Education, Inc 27

Solution: Two-Sample t-test for the Difference Between Means H 0 : μ 1 = μ 2 H a : μ 1 μ 2 010 df = 8 1 = 7 Rejection Region: Test Statistic: (473 459) 0 t 0922 2 2 397 245 8 18 Decision: Fail to

28 Solution: Two-Sample t-test for the Difference Between Means H 0 : μ 1 = μ 2 H a : μ 1 μ df = 8 1 = 7 Rejection Region: Test Statistic: ( ) 0 t Decision: Fail to Reject H 0 At the 10% level of significance, there is not enough evidence to support the claim that the mean mathematics test scores for the students of the two teachers are different Copyright 2015, 2012, and 2009 Pearson Education, Inc 28

29 Example: Two-Sample t-test for the Difference Between Means A manufacturer claims that the calling range (in feet) of its 24-GHz cordless telephone is greater than that of its leading competitor You perform a study using 14 randomly selected phones from the manufacturer and 16 randomly selected similar phones from its competitor The results are shown below At α = 005, can you support the manufacturer s claim? Assume the populations are normally distributed and the population variances are equal Manufacturer (1) Competition (2) x1 1275ft x2 1250ft s 1 = 45 ft s 2 = 30 ft n 1 = 14 n 2 = 16 Copyright 2015, 2012, and 2009 Pearson Education, Inc 29

30 Solution: Two-Sample t-test for the Difference Between Means H 0 : μ 1 μ 2 H a : μ 1 > μ df = = 28 Rejection Region: Test Statistic: Decision: t Copyright 2015, 2012, and 2009 Pearson Education, Inc 30

31 Solution: Two-Sample t-test for the Difference Between Means s x x n1 1 s1 n2 1 s2 1 1 n n 2 n n ( ) - ( m 1 - m ) 2 t = x - x 1 2 = s x1 -x 2 ( ) »1811 Copyright 2015, 2012, and 2009 Pearson Education, Inc 31

32 Solution: Two-Sample t-test for the Difference Between Means H 0 : μ 1 μ 2 H a : μ 1 > μ df = = 28 Rejection Region: t Test Statistic: t 1811 Decision: Reject H 0 At the 5% level of significance, there is enough evidence to support the manufacturer s claim that its phone has a greater calling range than its competitors Copyright 2015, 2012, and 2009 Pearson Education, Inc 32

33 t-test for the Difference Between Means To perform a two-sample hypothesis test with dependent samples, the difference between each data pair is first found: d = x 1 x 2 The test statistic is the mean d d n Difference between entries for a data pair d of these differences Mean of the differences between paired data entries in the dependent samples Copyright 2015, 2012, and 2009 Pearson Education, Inc 33

t-test for the Difference Between Means Three conditions are required to conduct the test 1 The samples must be randomly selected 2 The samples must be dependent (paired) 3 Both populations must be

34 t-test for the Difference Between Means Three conditions are required to conduct the test 1 The samples must be randomly selected 2 The samples must be dependent (paired) 3 Both populations must be normally distributed or the number n of pairs of data is at least 30 If these conditions are met, then the sampling distribution for d is approximated by a t-distribution with n 1 degrees of freedom, where n is the number of data pairs -t 0 μ d t 0 d Copyright 2015, 2012, and 2009 Pearson Education, Inc 34

35 Symbols used for the t-test for μ d Symbol n d d Description The number of pairs of data The difference between entries for a data pair, d = x 1 x 2 The hypothesized mean of the differences of paired data in the population Copyright 2015, 2012, and 2009 Pearson Education, Inc 35

36 Symbols used for the t-test for μ d Symbol d s d Description The mean of the differences between the paired data entries in the dependent samples d d n The standard deviation of the differences between the paired data entries in the dependent samples 2 2 ( d) 2 ( d d) d s n d n1 n1 Copyright 2015, 2012, and 2009 Pearson Education, Inc 36

37 t-test for the Difference Between Means A t-test can be used to test the difference of two population means when these conditions are met 1The samples are random 2The samples are dependent (paired) 3The populations are normally distributed or the number n 30 Copyright 2015, 2012, and 2009 Pearson Education, Inc 37

38 t-test for the Difference Between Means The test statistic is d d n The standardized test statistic is d t d sd n The degrees of freedom are df = n 1 Copyright 2015, 2012, and 2009 Pearson Education, Inc 38

39 t-test for the Difference Between Means (Dependent Samples) In Words 1 Verify that the samples are random and dependent, and either the populations are normally distributed or n 30 In Symbols 2 State the claim mathematically Identify the null and alternative hypotheses 3 Specify the level of significance State H 0 and H a Identify Copyright 2015, 2012, and 2009 Pearson Education, Inc 39

40 t-test for the Difference Between Means (Dependent Samples) In Words 4 Identify the degrees of freedom 5 Determine the critical value(s) 6 Determine the rejection region(s) d s 7 Calculate and d s d In Symbols df = n 1 Use Table 5 in Appendix B d d n 2 ( d) 2 ( d d) d n n1 n1 2 Copyright 2015, 2012, and 2009 Pearson Education, Inc 40

41 t-test for the Difference Between Means (Dependent Samples) In Words 8 Find the standardized test statistic and sketch the sampling distribution In Symbols t d s d d n 9 Make a decision to reject or fail to reject the null hypothesis 10 Interpret the decision in the context of the original claim If t is in the rejection region, then reject H 0 Otherwise, fail to reject H 0 Copyright 2015, 2012, and 2009 Pearson Education, Inc 41

42 Example: t-test for the Difference Between Means A shoe manufacturer claims that the athletes can increase their vertical jump heights using the manufacturer s training shoes The vertical jump heights of eight randomly selected athletes are measured After the athletes have used the training shoes for 8 months, their vertical jump heights are measured again The vertical jump heights (in inches) for each athlete are shown in the table Assuming the vertical jump heights are normally distributed, is there enough evidence to support the manufacturer s claim at α = 010? (Adapted from Coaches Sports Publishing) Athletes Height (old) Height (new) Copyright 2015, 2012, and 2009 Pearson Education, Inc 42

43 Solution: Two-Sample t-test for the Difference Between Means d = (jump height before shoes) (jump height after shoes) H 0 : μ d < 0 H a : μ d >= df = 8 1 = 7 Rejection Region: Test Statistic: Decision: Copyright 2015, 2012, and 2009 Pearson Education, Inc 43

44 Solution: Two-Sample t-test for the Difference Between Means d = (jump height before shoes) (jump height after shoes) Before After d d Σ = -14 Σ = 56 d s d d n d ( d) n n 1 ( 14) Copyright 2015, 2012, and 2009 Pearson Education, Inc 44

45 Solution: Two-Sample t-test for the Difference Between Means d = (jump height before shoes) (jump height after shoes) H 0 : μ d < 0 H a : μ d >= df = 8 1 = 7 Rejection Region: t Test Statistic: d s d d n Decision: Reject H 0 At the 10% level of significance, there is enough evidence to support the shoe manufacturer s claim that athletes can increase their vertical jump heights using the manufacturer s training shoes Copyright 2015, 2012, and 2009 Pearson Education, Inc 45

46 Two-Sample z-test for Proportions Used to test the difference between two population proportions, p 1 and p 2 Three conditions are required to conduct the test 1 The samples must be randomly selected 2 The samples must be independent 3 The samples must be large enough to use a normal sampling distribution That is, n 1 p 1 5, n 1 q 1 5, n 2 p 2 5, and n 2 q 2 5 Copyright 2015, 2012, and 2009 Pearson Education, Inc 46

47 Two-Sample z-test for the Difference Between Proportions If these conditions are met, then the sampling distribution for pˆ1 pˆ2 is a normal distribution Mean: pˆ1 pˆ p1 p2 2 A weighted estimate of p 1 and p 2 can be found by using x1 x p 2, where x1 n1 pˆ1 and x2 n2pˆ n n Standard error: 1 1 pˆ ˆ 1 p pq 2 n 1 n 2 Copyright 2015, 2012, and 2009 Pearson Education, Inc 47

48 Two-Sample z-test for the Difference Between Proportions The test statistic is The standardized test statistic is where x z x pˆ pˆ 1 2 ( pˆ1 pˆ2 ) ( p1 p2) 1 1 pq n 1 n 2 p 1 2 and q 1 p n1 n2 Note: n p, n q, n p, and n q must be at least Copyright 2015, 2012, and 2009 Pearson Education, Inc 48

49 Two-Sample z-test for the Difference Between Proportions In Words 1 Verify that the samples are random and independent In Symbols 2 Find the weighted estimate of p 1 and p 2 Verify that are at least 5 n p, n q, n p, n q State the claim Identify the null and alternative hypotheses 4 Specify the level of significance x x p 1 2, q 1 p n1 n2 State H 0 and H a Identify Copyright 2015, 2012, and 2009 Pearson Education, Inc 49

50 Two-Sample z-test for the Difference Between Proportions In Words 5 Determine the critical value(s) 6 Determine the rejection region(s) In Symbols Identify Use Table 4 in Appendix B 7 Find the standardized test statistic and sketch the sampling distribution z ( pˆ1 pˆ2 ) ( p1 p2) 1 1 pq n 1 n 2 Copyright 2015, 2012, and 2009 Pearson Education, Inc 50

51 Two-Sample z-test for the Difference Between Proportions In Words 8 Make a decision to reject or fail to reject the null hypothesis 9 Interpret the decision in the context of the original claim In Symbols If z is in the rejection region, reject H 0 Otherwise, fail to reject H 0 Copyright 2015, 2012, and 2009 Pearson Education, Inc 51

52 Example: Two-Sample z-test for the Difference Between Proportions In a study of 150 randomly selected occupants in passenger cars and 200 randomly selected occupants in pickup trucks, 86% of occupants in passenger cars and 74% of occupants in pickup trucks wear seat belts At α = 010 can you reject the claim that the proportion of occupants who wear seat belts is the same for passenger cars and pickup trucks? (Adapted from National Highway Traffic Safety Administration) Solution: 1 = Passenger cars 2 = Pickup trucks Copyright 2015, 2012, and 2009 Pearson Education, Inc 52

53 Solution: Two-Sample z-test for the Difference Between Means H 0 : p 1 = p 2 H a : p 1 p n 1 = 150, n 2 = 200 Rejection Region: Test Statistic: Decision: Copyright 2015, 2012, and 2009 Pearson Education, Inc 53

54 Solution: Two-Sample z-test for the Difference Between Means x ˆ x2 n2 p n1 p1 129 ˆ p x n x n q 1 p Note: n p 150(07914) 5 n q 150(02086) n p 200(07914) 5 n q ( 02086) 5 Copyright 2015, 2012, and 2009 Pearson Education, Inc 54

Solution: Two-Sample z-test for the Difference Between Means H 0 : p 1 = p 2 H a : p 1 p 2 010 n 1 = 150, n 2 = 200 Rejection Region: Test Statistic: z 273 Decision: Reject H 0 At the 10% level of

56 Solution: Two-Sample z-test for the Difference Between Means H 0 : p 1 = p 2 H a : p 1 p n 1 = 150, n 2 = 200 Rejection Region: Test Statistic: z 273 Decision: Reject H 0 At the 10% level of significance, there is enough evidence to reject the claim that the proportion of occupants who wear seat belts is the same for passenger cars and pickup trucks Copyright 2015, 2012, and 2009 Pearson Education, Inc 56

57 Example: Two-Sample z-test for the Difference Between Proportions A medical research team conducted a study to test the effect of a cholesterol reducing medication At the end of the study, the researchers found that of the 4700 randomly selected subjects who took the medication, 301 died of heart disease Of the 4300 randomly selected subjects who took a placebo, 357 died of heart disease At α = 001 can you conclude that the death rate due to heart disease is lower for those who took the medication than for those who took the placebo? (Adapted from New England Journal of Medicine) Solution: 1 = Medication 2 = Placebo Copyright 2015, 2012, and 2009 Pearson Education, Inc 57

58 Solution: Two-Sample z-test for the Difference Between Means H 0 : p 1 p 2 H a : p 1 < p n 1 = 4700, n 2 = 4300 Rejection Region: Test Statistic: Decision: z Copyright 2015, 2012, and 2009 Pearson Education, Inc 58

59 Solution: Two-Sample z-test for the Difference Between Means pˆ 1 x n pˆ 2 x n p x n x 1 2 n q 1 p Note: n p 1 1 n p 4700(00731) 5 n q 4700(09269) (00731) 5 n q 4300(09269) Copyright 2015, 2012, and 2009 Pearson Education, Inc 59

61 Solution: Two-Sample z-test for the Difference Between Means H 0 : p 1 p 2 H a : p 1 < p n 1 = 4700, n 2 = 4300 Rejection Region: z Test Statistic: z 346 Decision: Reject H 0 At the 1% level of significance, there is enough evidence to conclude that the death rate due to heart disease is lower for those who took the medication than for those who took the placebo Copyright 2015, 2012, and 2009 Pearson Education, Inc 61

hypotheses. P-value Test for a 2 Sample z-test (Large Independent Samples) n > 30 P-value Test for a 2 Sample t-test (Small Samples) n < 30 Identify α

hypotheses. P-value Test for a 2 Sample z-test (Large Independent Samples) n > 30 P-value Test for a 2 Sample t-test (Small Samples) n < 30 Identify α Chapter 8 Notes Section 8-1 Independent and Dependent Samples Independent samples have no relation to each other. An example would be comparing the costs of vacationing in Florida to the cost of vacationing