# Chapter8: Hypothesis Testing with Two Samples

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Hypothesis Testing with Two SamplesChapter 8

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

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Two Sample Hypothesis TestingIn a two-sample hypothesis test, two parameters from two populations are compared. For a two-sample hypothesis test, the null hypothesis H0 is a statistical hypothesis that usually states there is no difference between the parameters of two populations. The null hypothesis always contains the symbol , =, or . the alternative hypothesis Ha is a statistical hypothesis that is true when H0 is false. The alternative hypothesis always contains the symbol >, , or <.

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Two Sample Hypothesis TestingTo write a null and alternative hypothesis for a two-sample hypothesis test, translate the claim made about the population parameters from a verbal statement to a mathematical statement.Regardless of which hypotheses used, μ1 = μ2 is always assumed to be true.

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Two Sample z-TestThree 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. Two samples are independent if the sample selected from one population is not related to the sample selected from the second population. 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-TestIf these requirements are met, the sampling distribution for (the difference of the sample means) is a normal distribution with mean and standard error of and

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Two Sample z-TestTwo-Sample z-Test for the Difference Between Means A two-sample z-test can be used to test the difference between two population means μ1 and μ2 when a large sample (at least 30) is randomly selected from each population and the samples are independent. The test statistic is and 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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Two Sample z-Test for the MeansState the claim mathematically. Identify the null and alternative hypotheses. Specify the level of significance. Sketch the sampling distribution. Determine the critical value(s). Determine the rejection regions(s).Continued.Using a Two-Sample z-Test for the Difference Between Means (Large Independent Samples)In Words In SymbolsState H0 and Ha. Identify .Use Table 4 in Appendix B.

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Two Sample z-Test for the MeansIn Words In SymbolsIf z is in the rejection region, reject H0. Otherwise, fail to reject H0.Find the standardized test statistic. Make a decision to reject or fail to reject the null hypothesis. Interpret the decision in the context of the original claim.Using a Two-Sample z-Test for the Difference Between Means (Large Independent Samples)

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Two Sample z-Test for the MeansExample: A high school math teacher claims that students in her class will score higher on the math portion of the ACT then students in a colleague’s math class. The mean ACT math score for 49 students in her class is 22.1 and the standard deviation is 4.8. The mean ACT math score for 44 of the colleague’s students is 19.8 and the standard deviation is 5.4. At  = 0.10, can the teacher’s claim be supported?Ha: 1 > 2 (Claim)H0: 1  2Continued.z0 = 1.28

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Two Sample z-Test for the MeansExample continued:The standardized error isHa: 1 > 2 (Claim)H0: 1  2The standardized test statistic isz0 = 1.28z0123-3-2-1Reject H0.There is enough evidence at the 10% level to support the teacher’s claim that her students score better on the ACT.

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

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Two Sample t-TestThe samples must be randomly selected. The samples must be independent. Two samples are independent if the sample selected from one population is not related to the sample selected from the second population. Each population must have a normal distribution.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.

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Two Sample t-TestTwo-Sample t-Test for the Difference Between Means A two-sample t-test is used to test the difference between two population means μ1 and μ2 when a sample is randomly selected from each population. Performing this test requires each population to be normally distributed, and the samples should be independent. The standardized test statistic is If the population variances are equal, then information from the two samples is combined to calculate a pooled estimate of the standard deviation Continued.

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Two Sample t-TestTwo-Sample t-Test (Continued) The standard error for the sampling distribution of is and d.f.= n1 + n2 – 2. If the population variances are not equal, then the standard error is and d.f = smaller of n1 – 1 or n2 – 1.Variances equalVariances not equal

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Normal or t-Distribution?Are both sample sizes at least 30?Are both populations normally distributed?You cannot use the z-test or the t-test. Are both population standard deviations known?Use the z-test.Are the population variances equal? Use the z-test.

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Two Sample t-Test for the MeansState the claim mathematically. Identify the null and alternative hypotheses. Specify the level of significance. Identify the degrees of freedom and sketch the sampling distribution. Determine the critical value(s).Continued.Using a Two-Sample t-Test for the Difference Between Means (Small Independent Samples)In Words In SymbolsState H0 and Ha. Identify .Use Table 5 in Appendix B.d.f. = n1+ n2 – 2 or d.f. = smaller of n1 – 1 or n2 – 1.

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Two Sample t-Test for the MeansIn Words In SymbolsIf t is in the rejection region, reject H0. Otherwise, fail to reject H0.Determine the rejection regions(s). Find the standardized test statistic. Make a decision to reject or fail to reject the null hypothesis. Interpret the decision in the context of the original claim.Using a Two-Sample t-Test for the Difference Between Means (Small Independent Samples)

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Two Sample t-Test for the MeansExample: A random sample of 17 police officers in Brownsville has a mean annual income of \$35,800 and a standard deviation of \$7,800. In Greensville, a random sample of 18 police officers has a mean annual income of \$35,100 and a standard deviation of \$7,375. Test the claim at  = 0.01 that the mean annual incomes in the two cities are not the same. Assume the population variances are equal.Ha: 1  2 (Claim)H0: 1 = 2Continued.–t0 = –2.576d.f. = n1 + n2 – 2 = 17 + 18 – 2 = 33t0 = 2.576

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Two Sample t-Test for the MeansExample continued:The standardized error isHa: 1  2 (Claim)H0: 1 = 2Continued.

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Two Sample t-Test for the MeansExample continued:The standardized test statistic isFail to reject H0.There is not enough evidence at the 1% level to support the claim that the mean annual incomes differ.Ha: 1  2 (Claim)H0: 1 = 2

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§ 8.3Testing the Difference Between Means (Dependent Samples)

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Two samples are independent if the sample selected from one population is not related to the sample selected from the second population. Two samples are dependent if each member of one sample corresponds to a member of the other sample. Dependent samples are also called paired samples or matched samples.Independent and Dependent SamplesIndependent SamplesDependent Samples

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Example: Classify each pair of samples as independent or dependent.Independent and Dependent SamplesSample 1: The weight of 24 students in a first-grade class Sample 2: The height of the same 24 studentsThese samples are dependent because the weight and height can be paired with respect to each student.Sample 1: The average price of 15 new trucks Sample 2: The average price of 20 used sedansThese samples are independent because it is not possible to pair the new trucks with the used sedans. The data represents prices for different vehicles.

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t-Test for the Difference Between MeansTo perform a two-sample hypothesis test with dependent samples, the difference between each data pair is first found:Three conditions are required to conduct the test.d = x1 – x2Difference between entries for a data pair.Mean of the differences between paired data entries in the dependent samples.

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t-Test for the Difference Between MeansThe samples must be randomly selected. The samples must be dependent (paired). Both populations must be normally distributed.

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Last Updated: 8th March 2018