SECTION 8.1 Testing the Difference Between Means (Independent Samples, s1 and s2 Known) 419 An Overview of Two-Sample Hypothesis Testing In this section and the next, you will learn how to test a claim comparing the means of two different populations using independent samples. For instance, an advertiser is developing a marketing plan and wants to determine whether there is a difference in the amounts of time adults ages 18 to 34 and adults ages 35 to 49 spend on social media each day. The only way to conclude with certainty that there is a difference is to take a census of all adults in both age groups, calculate their mean daily times spent on social media, and find the difference. Of course, it is not practical to take such a census. However, it is possible to determine with some degree of certainty whether such a difference exists. To determine whether a difference exists, the advertiser begins by assuming that there is no difference in the mean times of the two populations. That is, m1 - m2 = 0. Assume there is no difference. Then, by taking a random sample from each population, a two-sample hypothesis test is performed using the test statistic x1 - x2 = 0. Test statistic The advertiser obtains the results shown in the next two figures. Adults 18 to 34 that are not in the sample Sample 1 x1 = 55 min s1 = 13 min n1 = 200 Adults 18 to 34 Adults 35 to 49 that are not in the sample Sample 2 x2 = 59 min s2 = 15 min n2 = 150 Adults 35 to 49 The figure below shows the sampling distribution of x1 - x2 for many similar samples taken from two populations for which m1 - m2 = 0. The figure also shows the test statistic and the standardized test statistic (assuming that the population variances are not equal). 0 x1 − x2 t Difference in sample means (in minutes) 0 Sampling Distribution −1 −2 −3 −4 −5 1 2 3 4 5 −1 −2 −3 1 2 3 Test statistic: x1 − x2 = 55 − 59 = −4 Standardized test statistic: t ≈ −2.612 From the figure, you can see that it is quite unlikely to obtain sample means that differ by 4 minutes, assuming the actual difference is 0. The difference of the sample means would be more than 2.5 standard errors from the hypothesized difference of 0. Performing a two-sample hypothesis test using a level of significance of a = 0.10, the advertiser can conclude that there is a difference in the amounts of time adults ages 18 to 34 and adults ages 35 to 49 spend on social media each day. Study Tip In the figures at the right, the members in the two samples, adults ages 18 to 34 and adults ages 35 to 49, are not matched or paired, so the samples are independent. Study Tip In the figure at the right, the standardized test statistic is from a two-sample t-test. You will study this test in the next section.
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