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Слайды и текст этой презентации

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Chapter 8 Hypothesis Testing with Two Samples

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Chapter Outline 8.1 Testing the Difference Between Means (Large Independent Samples) 8.2 Testing the Difference Between Means (Small Independent Samples) 8.3 Testing the Difference Between Means (Dependent Samples) 8.4 Testing the Difference Between Proportions

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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

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Example: Independent and Dependent Samples Classify the pair of samples as independent or dependent. Sample 1: Resting heart rates of 35 individuals before drinking coffee. Sample 2: Resting heart rates of the same individuals after drinking two cups of coffee.

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Example: Independent and Dependent Samples Classify the pair of samples as independent or dependent. Sample 1: Test scores for 35 statistics students. Sample 2: Test scores for 42 biology students who do not study statistics.

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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 =. Alternative hypothesis Ha A statistical hypothesis that is supported when H0 is rejected. Always contains the symbol >, , or

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Two Sample Hypothesis Test with Independent Samples

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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

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Two Sample z-Test for the Difference Between Means Test statistic is The standardized test statistic is When the samples are large, you can use s1 and s2 in place of 1 and 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.

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

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

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

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Example: 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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Solution: Using Technology to Perform a Two-Sample z-Test

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Solution: Using Technology to Perform a Two-Sample z-Test

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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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Section 8.2 Testing the Difference Between Means (Small Independent Samples)

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Section 8.2 Objectives Perform a t-test for the difference between two means μ1 and μ2 using small independent samples

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Two Sample t-Test for the Difference Between Means If samples of size less than 30 are taken from normally-distributed populations, 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 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.

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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 .

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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

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Normal or t-Distribution?

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Two-Sample t-Test for the Difference Between Means (Small Independent Samples)

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Two-Sample t-Test for the Difference Between Means (Small Independent Samples)

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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. (Adapted from Consumer Reports)

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

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

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Section 8.2 Summary Performed a t-test for the difference between two means μ1 and μ2 using small independent samples

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Section 8.3 Testing the Difference Between Means (Dependent Samples)

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Section 8.3 Objectives Perform a t-test to test the mean of the difference for a population of paired data

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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

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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 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.

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Symbols used for the t-Test for μd

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Symbols used for the t-Test for μd

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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

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

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

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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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Solution: Two-Sample 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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Section 8.3 Summary Performed a t-test to test the mean of the difference for a population of paired data

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Section 8.4 Testing the Difference Between Proportions

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Section 8.4 Objectives Perform a z-test for the difference between two population proportions p1 and p2

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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.

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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:

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Two-Sample z-Test for the Difference Between Proportions The test statistic is The standardized test statistic is where

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Two-Sample z-Test for the Difference Between Proportions

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Two-Sample z-Test for the Difference Between Proportions

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Example: Two-Sample z-Test for the Difference Between Proportions In a study of 200 randomly selected adult female and 250 randomly selected adult male 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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Solution: Two-Sample z-Test for the Difference Between Means

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

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

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

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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)