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**Testing the Difference Between Means (Dependent Samples)**

Section 8.3 Testing the Difference Between Means (Dependent Samples) © 2012 Pearson Education, Inc. All rights reserved. 1 of 70

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Section 8.3 Objectives Perform a t-test to test the mean of the differences for a population of paired data © 2012 Pearson Education, Inc. All rights reserved. 2 of 70

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**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 = x1 – x2 Difference between entries for a data pair The test statistic is the mean of these differences. Mean of the differences between paired data entries in the dependent samples © 2012 Pearson Education, Inc. All rights reserved. 3 of 70

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**t-Test for the Difference Between Means**

Three conditions are required to conduct the test. The samples must be randomly selected. The samples must be dependent (paired). Both populations must be normally distributed. If these requirements are met, then the sampling distribution for is approximated by a t-distribution with n – 1 degrees of freedom, where n is the number of data pairs. -t0 t0 μd © 2012 Pearson Education, Inc. All rights reserved. 4 of 70

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**Symbols used for the t-Test for μd**

Symbol Description n The number of pairs of data d The difference between entries for a data pair, d = x1 – x2 The hypothesized mean of the differences of paired data in the population © 2012 Pearson Education, Inc. All rights reserved. 5 of 70

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**Symbols used for the t-Test for μd**

Symbol Description The mean of the differences between the paired data entries in the dependent samples sd The standard deviation of the differences between the paired data entries in the dependent samples © 2012 Pearson Education, Inc. All rights reserved. 6 of 70

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**t-Test for the Difference Between Means**

The test statistic is The standardized test statistic is The degrees of freedom are d.f. = n – 1. © 2012 Pearson Education, Inc. All rights reserved. 7 of 70

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**t-Test for the Difference Between Means (Dependent Samples)**

In Words In Symbols State the claim mathematically and verbally. Identify the null and alternative hypotheses. Specify the level of significance. Determine the degrees of freedom. Determine the critical value(s). State H0 and Ha. Identify α. d.f. = n – 1 Use Table 5 in Appendix B if n > 29 use the last row (∞) . © 2012 Pearson Education, Inc. All rights reserved. 8 of 70

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**t-Test for the Difference Between Means (Dependent Samples)**

In Words In Symbols Determine the rejection region(s). Calculate and Find the standardized test statistic. © 2012 Pearson Education, Inc. All rights reserved. 9 of 70

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**t-Test for the Difference Between Means (Dependent Samples)**

In Words In Symbols Make a decision to reject or fail to reject the null hypothesis. Interpret the decision in the context of the original claim. If t is in the rejection region, reject H0. Otherwise, fail to reject H0. © 2012 Pearson Education, Inc. All rights reserved. 10 of 70

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**Example: t-Test for the Difference Between Means**

A shoe manufacturer claims that athletes can increase their vertical jump heights using the manufacturer’s new Strength Shoes®. The vertical jump heights of eight randomly selected athletes are measured. After the athletes have used the Strength 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 = 0.10? (Adapted from Coaches Sports Publishing) Athletes 1 2 3 4 5 6 7 8 Height (old) 24 22 25 28 35 32 30 27 Height (new) 26 29 33 34 © 2012 Pearson Education, Inc. All rights reserved. 11 of 70

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**Solution: Two-Sample t-Test for the Difference Between Means**

d = (jump height before shoes) – (jump height after shoes) H0: Ha: α = d.f. = Rejection Region: μd ≥ 0 μd < 0 (claim) 0.10 8 – 1 = 7 © 2012 Pearson Education, Inc. All rights reserved. 12 of 70

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**Solution: Two-Sample t-Test for the Difference Between Means**

d = (jump height before shoes) – (jump height after shoes) Before After d d2 24 26 –2 4 22 25 –3 9 28 29 –1 1 35 33 2 32 34 30 –5 27 Σ = –14 Σ = 56 © 2012 Pearson Education, Inc. All rights reserved. 13 of 70

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**Solution: Two-Sample t-Test for the Difference Between Means**

d = (jump height before shoes) – (jump height after shoes) H0: Ha: α = d.f. = Rejection Region: Test Statistic: µd ≥ 0 μd < 0 (claim) 0.10 8 – 1 = 7 Decision: Reject H0. 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 new Strength Shoes®. t ≈ –2.333 © 2012 Pearson Education, Inc. All rights reserved. 14 of 70

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**To use the calculator for testing the difference between the**

means of dependent samples, enter the data in two columns and form a third Column ( 𝐿 3 ) in which you calculate the difference for each pair ( 𝐿 1 − 𝐿 2 ). You can then perform a one-sample t-test on the difference column ( 𝐿 3 ).

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**Testing the Difference Between Proportions**

Section 8.4 Testing the Difference Between Proportions © 2012 Pearson Education, Inc. All rights reserved. 16 of 70

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Section 8.4 Objectives Perform a z-test for the difference between two population proportions p1 and p2 © 2012 Pearson Education, Inc. All rights reserved. 17 of 70

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**Two-Sample z-Test for Proportions**

Used to test the difference between two population proportions, p1 and p2. Three conditions are required to conduct the test. The samples must be randomly selected. The samples must be independent. The samples must be large enough to use a normal sampling distribution. That is, n1p1 ≥ 5, n1q1 ≥ 5, n2p2 ≥ 5, and n2q2 ≥ 5. © 2012 Pearson Education, Inc. All rights reserved. 18 of 70

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**Two-Sample z-Test for the Difference Between Proportions**

If these conditions are met, then the sampling distribution for is a normal distribution. Mean: A weighted estimate of p1 and p2 can be found by using Standard error: © 2012 Pearson Education, Inc. All rights reserved. 19 of 70

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**Two-Sample z-Test for the Difference Between Proportions**

The test statistic is The standardized test statistic is where © 2012 Pearson Education, Inc. All rights reserved. 20 of 70

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**Two-Sample z-Test for the Difference Between Proportions**

In Words In Symbols State the claim. Identify the null and alternative hypotheses. Specify the level of significance. Determine the critical value(s). Determine the rejection region(s). Find the weighted estimate of p1 and p2. Verify that State H0 and Ha. Identify α. Use Table 4 in Appendix B. © 2012 Pearson Education, Inc. All rights reserved. 21 of 70

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**Two-Sample z-Test for the Difference Between Proportions**

In Words In Symbols Find the standardized test statistic and sketch the sampling distribution. Make a decision to reject or fail to reject the null hypothesis. Interpret the decision in the context of the original claim. If z is in the rejection region, reject H0. Otherwise, fail to reject H0. © 2012 Pearson Education, Inc. All rights reserved. 22 of 70

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**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 = 0.10 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 © 2012 Pearson Education, Inc. All rights reserved. 23 of 70

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**Solution: Two-Sample z-Test for the Difference Between Means**

Ha: α = n1= , n2 = Rejection Region: p1 = p2 (claim) p1 ≠ p2 0.10 150 200 © 2012 Pearson Education, Inc. All rights reserved. 24 of 70

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**Solution: Two-Sample z-Test for the Difference Between Means**

Note: 𝑛 1 𝑝 = ≥ 𝑛 1 𝑞 = ≥5 𝑛 2 𝑝 = ≥ 𝑛 2 𝑝 = ≥5 © 2012 Pearson Education, Inc. All rights reserved. 25 of 70

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**Solution: Two-Sample z-Test for the Difference Between Means**

© 2012 Pearson Education, Inc. All rights reserved. 26 of 70

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**Solution: Two-Sample z-Test for the Difference Between Means**

Ha: α = n1= , n2 = Rejection Region: p1 = p2 (claim) p1 ≠ p2 Test Statistic: 0.10 150 200 Decision: Reject H0 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. © 2012 Pearson Education, Inc. All rights reserved. 27 of 70

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**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 = 0.01 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 © 2012 Pearson Education, Inc. All rights reserved. 28 of 70

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**Solution: Two-Sample z-Test for the Difference Between Means**

Ha: α = n1= , n2 = Rejection Region: p1 ≥ p2 p1 < p2 (claim) 0.01 4700 4300 z –2.33 0.01 © 2012 Pearson Education, Inc. All rights reserved. 29 of 70

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**Solution: Two-Sample z-Test for the Difference Between Means**

© 2012 Pearson Education, Inc. All rights reserved. 30 of 70

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**Solution: Two-Sample z-Test for the Difference Between Means**

© 2012 Pearson Education, Inc. All rights reserved. 31 of 70

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**Solution: Two-Sample z-Test for the Difference Between Means**

Ha: α = n1= , n2 = Rejection Region: p1 ≥ p2 p1 < p2 (claim) Test Statistic: 0.01 4700 4300 Decision: Reject H0 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. z –2.33 0.01 –2.33 –3.46 © 2012 Pearson Education, Inc. All rights reserved. 32 of 70

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Section 8.4 Summary Performed a z-test for the difference between two population proportions p1 and p2 © 2012 Pearson Education, Inc. All rights reserved. 33 of 70

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