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**Testing the Difference Between Means (Large Independent Samples)**

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

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Section 8.1 Objectives Determine whether two samples are independent or dependent Perform a two-sample z-test for the difference between two means μ1 and μ2 using large independent samples

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**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.

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**Independent and Dependent Samples**

Independent Samples Dependent Samples Sample 1 Sample 2 Sample 1 Sample 2

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**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)

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**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.)

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**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 <.

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**Two Sample Hypothesis Test with Independent Samples**

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, or μ1 = μ2. Note that if μ1 = μ2, that is the same as μ1 - μ2 = 0

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

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**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: Standard error: Sampling distribution for :

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

Test statistic is The standardized test statistic is µ1 - µ2 = 0 When the samples are large, you can use s1 and s2 in place of σ1 and σ2. 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.

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**Using a Two-Sample z-Test for the Difference Between Means (Large Independent Samples)**

In Words In Symbols State the claim mathematically and verbally. Identify the null and alternative hypotheses. Specify the level of significance. Determine the critical value(s). Determine the rejection region(s). State H0 and Ha. Identify α. Use Table 4 in Appendix B.

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**Using a Two-Sample z-Test for the Difference Between Means (Large 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 z is in the rejection region, reject H0. Otherwise, fail to reject H0. µ1 - µ2 = 0

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**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 = 0.05. New York Texas σ1 = $ σ2 = $ n1 = 250 n2 = 250

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

Ha: α = n1= , n2 = Rejection Region: μ1 = μ2 μ1 ≠ μ2 (claim) Test Statistic: 0.05 250 250 Decision: Fail to Reject H0. 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. z = –1.11

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**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 = 0.03. Region C Region D 𝑥 1 = 14.0 mph 𝑥 2 = 15.1 mph σ1 = 2.9 mph σ2 = 3.3 mph n1 = 75 n2 = 80

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

Ha: α = n1= , n2 = Rejection Region: μ1 = μ2 (claim) μ1 ≠ μ2 Test Statistic: 0.03 75 80

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

Ha: α = n1= , n2 = 𝒛= 𝟏𝟒−𝟏𝟓.𝟏 𝟐.𝟗 𝟐 𝟕𝟓 + 𝟑.𝟑 𝟐 𝟖𝟎 μ1 = μ2 (claim) μ1 ≠ μ2 Test Statistic: 0.03 75 80 𝒛= −𝟏.𝟏 𝟖.𝟒𝟏𝟎 𝟕𝟓 + 𝟏𝟎.𝟖𝟗𝟎 𝟖𝟎 = − = − = − =−2.21

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

Ha: α = n1= , n2 = Rejection Region: α/2 = .015 So, z0 = -2.17, +2.17 (from table) Decision: Reject H0 Interpret? μ1 = μ2 (claim) μ1 ≠ μ2 Test Statistic: z= -2.21 0.03 75 80

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Section 8.1 Summary Determined whether two samples are independent or dependent Performed a two-sample z-test for the difference between two means μ1 and μ2 using large independent samples

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