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Section 8.1 Testing the Difference Between Means (Large Independent Samples)

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1 Section 8.1 Testing the Difference Between Means (Large Independent Samples)

2 Section 8.1 Objectives Determine whether two samples are independent or dependent Perform a two-sample z-test for the difference between two means μ 1 and μ 2 using large independent samples

3 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.

4 Independent and Dependent Samples Independent Samples Sample 1Sample 2 Dependent Samples Sample 1Sample 2

5 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)

6 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.)

7 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 <.

8 Two Sample Hypothesis Test with Independent Samples H 0 : μ 1 = μ 2 H a : μ 1 ≠ μ 2 H 0 : μ 1 ≤ μ 2 H a : μ 1 > μ 2 H 0 : 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. Note that if μ 1 = μ 2, that is the same as μ 1 - μ 2 = 0

9 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 The samples must be randomly selected. 2. The samples must be independent. 3. Each sample size must be at least 30, or, if not, each population must have a normal distribution with a known standard deviation.

10 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 Sampling distribution for : Mean: Standard error:

11 Two Sample z - Test for the Difference Between Means Test statistic is. The standardized test statistic is µ 1 - µ 2 = 0 When the samples are large, you can use s 1 and s 2 in place of σ 1 and σ 2. If the samples are not large, you can still use a two-sample z-test, provided the populations are normally distributed and the population standard deviations are known.

12 Using a Two - Sample z - Test for the Difference Between Means (Large Independent Samples) 1.State the claim mathematically and verbally. Identify the null and alternative hypotheses. 2.Specify the level of significance. 3.Determine the critical value(s). 4.Determine the rejection region(s). State H 0 and H a. Identify α. Use Table 4 in Appendix B. In WordsIn Symbols

13 Using a Two - Sample z - Test for the Difference Between Means (Large Independent Samples) 5.Find the standardized test statistic and sketch the sampling distribution. 6.Make a decision to reject or fail to reject the null hypothesis. 7.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. µ 1 - µ 2 = 0 In WordsIn Symbols

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 New York and Texas. The results of a random survey of 250 households from each state are shown below. The two samples are independent. Do the results support the group’s claim? Use α = New YorkTexas σ 1 = $ σ 2 = $ n 1 = 250n 2 = 250

15 Solution: Two - Sample z - Test for the Difference Between Means H 0 : H a : α  n 1 =, n 2 = Rejection Region: Test Statistic: μ 1 = μ 2 μ 1 ≠ μ 2 (claim) 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. Fail to Reject H 0. z = –1.11

16 Example: Two - Sample z - Test for the Difference Between Means An energy company wants to choose between two regions to install energy producing wind turbines. A researcher claims the wind speed in Region C and Region D are equal. The average wind speeds for 75 days are observed in Region C and for 80 days in Region D. Do the results support the claim? Use α = Region CRegion D σ 1 = 2.9 mphσ 2 = 3.3 mph n 1 = 75n 2 = 80

17 Solution: Two - Sample z - Test for the Difference Between Means H 0 : H a : α  n 1 =, n 2 = Rejection Region: Test Statistic: μ 1 = μ 2 (claim) μ 1 ≠ μ 2

18 Solution: Two - Sample z - Test for the Difference Between Means Test Statistic: μ 1 = μ 2 (claim) μ 1 ≠ μ 2

19 Solution: Two - Sample z - Test for the Difference Between Means H 0 : H a : α  n 1 =, n 2 = Rejection Region: α/2 =.015 So, z 0 = -2.17, (from table) Decision: Reject H 0 Interpret? Test Statistic: z= μ 1 = μ 2 (claim) μ 1 ≠ μ 2

20 Section 8.1 Summary Determined whether two samples are independent or dependent Performed a two-sample z-test for the difference between two means μ 1 and μ 2 using large independent samples


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