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Chapter. 8. Hypothesis Testing with Two Samples. Chapter Outline. 8.1 Testing the Difference Between Means (Independent Samples,  1 and  2 Known ) 8.2 Testing the Difference Between Means (Independent Samples,  1 and  2 Unknown )

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  1. Chapter 8 Hypothesis Testing with Two Samples

  2. Chapter Outline • 8.1 Testing the Difference Between Means (Independent Samples, 1 and 2Known) • 8.2 Testing the Difference Between Means (Independent Samples, 1 and 2Unknown) • 8.3 Testing the Difference Between Means (Dependent Samples) • 8.4 Testing the Difference Between Proportions .

  3. Section 8.1 Testing the Difference Between Means (Independent Samples, 1 and 2Known) .

  4. Section 8.1 Objectives • How to determine whether two samples are independent or dependent • An introduction to two-sample hypothesis testing for the difference between two population parameters • Perform a two-sample z-test for the difference between two means μ1 and μ2 using independent samples with 1 and 2known .

  5. 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. .

  6. Independent and Dependent Samples Independent Samples Dependent Samples Sample 1 Sample 2 Sample 1 Sample 2 .

  7. 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) .

  8. 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.) .

  9. Two Sample Hypothesis Test with Independent Samples • Null hypothesis H0 • A statistical hypothesis that usually states there is no difference between the parameters of two populations. • Always contains the symbol , =, or . • Alternative hypothesis Ha • A statistical hypothesis that is true when H0 is false. • Always contains the symbol >, , or <. .

  10. Two Sample Hypothesis Test with Independent Samples H0: μ1 = μ2 Ha: μ1 ≠ μ2 H0: μ1 ≤ μ2 Ha: μ1 > μ2 H0: μ1 ≥ μ2 Ha: μ1 < μ2 Regardless of which hypotheses you use, you always assume there is no difference between the population means, orμ1 = μ2. .

  11. 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 μ1 and μ2. • The population standard deviations are known. • The samples are randomly selected. • The samples are independent. • The populations are normally distributed or each sample size is at least 30. .

  12. Sampling distribution for : Two Sample z-Test for the Difference Between Means If these requirements are met, the sampling distribution for (the difference of the sample means) is a normal distribution with Mean: When no difference is assumed, the mean is 0. Standard error: .

  13. Two Sample z-Test for the Difference Between Means • Test statistic is • The standardized test statistic is Can be used when these conditions are met: • Both σ1 and σ2 are known. • The samples are random. • The samples are independent. • The populations are normally distributed or both n1≥ 30 and n2≥ 30. .

  14. Using a Two-Sample z-Test for the Difference Between Means (Large Independent Samples) In Words In Symbols • Verify that σ1 and σ2 are known, the samples are random and independent, and either the pops. are normally dist. or both n1  30and n2 30 . • State the claim mathematically. Identify the null and alternative hypotheses. • Specify the level of significance. State H0 and Ha. Identify . .

  15. Using a Two-Sample z-Test for the Difference Between Means (Large Independent Samples) In Words In Symbols Use Table 4 in Appendix B. • Determine the critical value(s). • Determine the rejection region(s). • Find the standardized test statistic and sketch the sampling distribution. .

  16. Using a Two-Sample z-Test for the Difference Between Means (Large Independent 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 z is in the rejection region, reject H0.Otherwise, fail to reject H0. .

  17. 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 σ1 = $1045 for California and σ2 = $1350 for Illinois. Do the results support the group’s claim? Use α = 0.05.(Source: PlasticEconomy.com) .

  18. μ1 = μ2 μ1 ≠ μ2 0.05 250 250 Solution: Two-Sample z-Test for the Difference Between Means • H0: • Ha: •  • n1= , n2 = • Rejection Region: • Test Statistic: Fail to Reject H0 • Decision: 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. .

  19. 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 σ1 = $19 for Texas and σ2 = $24 for Virginia, and that both populations are normally distributed. Atα = 0.01, is there enough evidence to support the claim? (Adapted from American Automobile Association) .

  20. μ1 ≥ μ2 μ1 < μ2 Solution: Using Technology to Perform a Two-Sample z-Test • H0: • Ha: .

  21. Solution: Using Technology to Perform a Two-Sample z-Test • Rejection Region: • Decision: Fail to Reject H0 At the 1% level of significance, there is not enough evidence to support the travel agency’s claim. -2.33 0.01 -0.91 z 0 .

  22. Section 8.1 Summary • Determined whether two samples are independent or dependent • Introduced to two-sample hypothesis testing for the difference between two population parameters • Performed a two-sample z-test for the difference between two means μ1 and μ2 using independent samples with σ1 and σ2known .

  23. Section 8.2 Testing the Difference Between Means (Independent Samples, 1 and 2Unknown) .

  24. Section 8.2 Objectives • How to perform a t-test for the difference between two means μ1 and μ2 with independent samples with 1 and 2unknown .

  25. Two Sample t-Test for the Difference Between Means A two-samplet-test is used to test the difference between two population means μ1 and μ2 when • σ2 and σ2 are unknown, • the samples are random, • the samples are independent, and • the populations are normally distributed or both n1≥ 30 and n2≥ 30. .

  26. Two Sample t-Test for the Difference Between Means • The • The standardized test statistic is • The standard error and the degrees of freedom of the sampling distribution depend on whether the population variances and are equal. .

  27. 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 • The standard error for the sampling distribution of is • d.f.= n1 + n2 – 2 .

  28. 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 • d.f = smaller of n1 – 1 or n2 – 1 .

  29. Normal (z) or t-Distribution? .

  30. Two-Sample t-Test for the Difference Between Means (Small Independent Samples) In Words In Symbols • Verify that 1 and 2 are unknown, the samples are random and independent, and either the populations are normally distributed or both n1  30and n2 30. • State the claim mathematically. Identify the null and alternative hypotheses. • Specify the level of significance. State H0 and Ha. Identify . .

  31. Two-Sample t-Test for the Difference Between Means (Small Independent Samples) In Words In Symbols d.f. = n1+ n2 – 2 or d.f. = smaller of n1 – 1 or n2 – 1. • Determine the degrees of freedom. • Determine the critical value(s). • Determine the rejection region(s). Use Table 5 in Appendix B. .

  32. Two-Sample t-Test for the Difference Between Means (Small Independent Samples) 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 t is in the rejection region, reject H0.Otherwise, fail to reject H0. .

  33. 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 = 0.10. Assume the populations are normally distributed and the population variances are not equal. .

  34. μ1 = μ2 μ1 ≠ μ2 0.10 8 – 1 = 7 Solution: Two-Sample t-Test for the Difference Between Means • Test Statistic: • H0: • Ha: •  • d.f. = • Rejection Region: Fail to Reject H0 • Decision: 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. .

  35. Example: Two-Sample t-Test for the Difference Between Means A manufacturer claims that the calling range (in feet) of its 2.4-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 = 0.05, can you support the manufacturer’s claim? Assume the populations are normally distributed and the population variances are equal. .

  36. μ1 ≤ μ2 μ1 > μ2 0.05 14 + 16 – 2 = 28 Solution: Two-Sample t-Test for the Difference Between Means • H0: • Ha: •  • d.f. = • Rejection Region: • Test Statistic: • Decision: 0.05 t 0 1.701 .

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

  38. μ1 ≤ μ2 μ1 > μ2 0.05 14 + 16 – 2 = 28 Solution: Two-Sample t-Test for the Difference Between Means • H0: • Ha: •  • d.f. = • Rejection Region: • Test Statistic: Reject H0 • Decision: 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. 0.05 t 0 1.701 1.811 .

  39. Section 8.2 Summary • Performed a t-test for the difference between two means μ1 and μ2 with independent samples with 1 and 2unknown .

  40. Section 8.3 Testing the Difference Between Means (Dependent Samples) .

  41. Section 8.3 Objectives • How to perform a t-test to test the mean of the difference for a population of paired data .

  42. 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 – x2Difference 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 .

  43. -t0 μd t0 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 or the number n of pairs of data is at least 30. If these conditions 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. .

  44. Symbols used for the t-Test for μd Symbol Description n The number of pairs of data The difference between entries for a data pair,d = x1 – x2 d The hypothesized mean of the differences of paired data in the population .

  45. 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 .

  46. 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. • The samples are random. • The samples are dependent (paired). • The populations are normally distributed or the number n ≥30. .

  47. t-Test for the Difference Between Means • The test statistic is • The standardizedteststatistic is • The degrees of freedom are d.f. = n – 1 .

  48. t-Test for the Difference Between Means (Dependent Samples) In Words In Symbols • Verify that the samples are random and dependent, and either the populations are normally distributed orn 30. • State the claim mathematically. Identify the null and alternative hypotheses. • Specify the level of significance. State H0 and Ha. Identify . .

  49. t-Test for the Difference Between Means (Dependent Samples) In Words In Symbols • Identify the degrees of freedom. • Determine the critical value(s). • Determine the rejection region(s). • Calculate and d.f. = n – 1 Use Table 5 in Appendix B. .

  50. t-Test for the Difference Between Means (Dependent Samples) 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 t is in the rejection region, then reject H0 . Otherwise, fail to reject H0 . .

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