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Ch8: Hypothesis Testing (2 Samples) 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) 8.3 Testing the Difference Between Means (Dependent Samples) 8.4 Testing the Difference Between Proportions

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8.1 Two Sample Hypothesis Test Compares two parameters from two populations. Two types of sampling methods: Independent (unrelated) Samples Dependent Samples (paired or matched samples) Each member of one sample corresponds to a member of the other sample.

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Stating a Hypotheses in 2-Sample Hypothesis Test Null hypothesis A statistical hypothesis H0 Statement of equality (, =, or ). No difference between the parameters of two populations.

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Two Sample z-Test for the Difference Between Means (μ1 and μ2.) Three conditions are necessary The samples must be randomly selected. The samples must be independent. Each population must have a normal distribution with a known population standard deviation OR each sample size must be at least 30.

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Using a Two-Sample z-Test for the Difference Between Means (Independent Samples 1 and 2 known or n1 and n2  30 )

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Example1: Two-Sample z-Test for the Difference Between Means A consumer education organization claims that there is a difference in the mean credit card debt of males and females in the United States. The results of a random survey of 200 individuals from each group are shown below. The two samples are independent. Do the results support the organization’s claim? Use α = 0.05.

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Example2: Using Technology to Perform a Two-Sample z-Test The American Automobile Association claims that the average daily cost for meals and lodging for vacationing in Texas is less than the same average costs for vacationing in Virginia. The table shows the results of a random survey of vacationers in each state. The two samples are independent. At α = 0.01, is there enough evidence to support the claim?

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8.2 Two Sample t-Test for the Difference Between Means (1 or 2 unknown) If (1 or 2 is unknown and samples are taken from normally-distributed) OR If (1 or 2 is unknown and both sample sizes are greater than or equal to 30) THEN a t-test may be used to test the difference between the population means μ1 and μ2. Three conditions are necessary to use a t-test for small independent samples. The samples must be randomly selected. The samples must be independent. Each population must have a normal distribution.

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Two Sample t-Test for the Difference Between Means Equal Variances Information from the two samples is combined to calculate a pooled estimate of the standard deviation .

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Normal or t-Distribution?

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Two-Sample t-Test for the Difference Between Means - Independent Samples (1 or 2 unknown)

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Example: Two-Sample t-Test for the Difference Between Means The braking distances of 8 Volkswagen GTIs and 10 Ford Focuses were tested when traveling at 60 miles per hour on dry pavement. The results are shown below. Can you conclude that there is a difference in the mean braking distances of the two types of cars? Use α = 0.01. Assume the populations are normally distributed and the population variances are not equal. (Consumer Reports)

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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.

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8.3 t-Test for the Difference Between Means (Paired Data/Dependent Samples) 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

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Symbols used for the t-Test for μd

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

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Example: t-Test for the Difference Between Means

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

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8.4 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.

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Two-Sample z-Test for the Difference Between Proportions

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Example1: Two-Sample z-Test for the Difference Between Proportions In a study of 200 randomly selected adult female(1) and 250 randomly selected adult male(2) Internet users, 30% of the females and 38% of the males said that they plan to shop online at least once during the next month. At α = 0.10 test the claim that there is a difference between the proportion of female and the proportion of male Internet users who plan to shop online.

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Example2: 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(1). 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(2), 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? (New England Journal of Medicine)