Testing the Difference Between Means (Dependent Samples) 8.3 SECTION 8.3 Testing the Difference Between Means (Dependent Samples) 437 What You Should Learn How to perform a t-test to test the mean of the differences for a population of paired data The t-Test for the Difference Between Means The t-Test for the Difference Between Means In Sections 8.1 and 8.2, you performed two-sample hypothesis tests with independent samples using the test statistic x1 - x2 (the difference between the means of the two samples). To perform a two-sample hypothesis test with dependent samples, you will use a different technique. You will first find the difference d for each data pair. d = 1data entry in first sample2 - 1corresponding data entry in second sample2 The test statistic is the mean d of these differences d = Σd n . Mean of the differences between paired data entries in the dependent samples These conditions are necessary to conduct the test. 1. The samples are randomly selected. 2. The samples are dependent (paired). 3. The populations are normally distributed or the number n of pairs of data is at least 30. When these conditions are met, the sampling distribution for d, the mean of the differences of the paired data entries in the dependent samples, is approximated by a t@distribution with n - 1 degrees of freedom, where n is the number of data pairs. dμ d t0 −t0 The symbols listed in the table are used for the t@test for md. Although formulas are given for the mean and standard deviation of differences, you should use technology to calculate these statistics. Symbol Description n The number of pairs of data d The difference between entries in a data pair md The hypothesized mean of the differences of paired data in the population d The mean of the differences between the paired data entries in the dependent samples d = Σd n sd The standard deviation of the differences between the paired data entries in the dependent samples sd = BΣ1d - d22 n - 1 Study Tip Recall from Section 8.1 that two samples are dependent when each member of one sample corresponds to a member of the other sample. Study Tip You can also calculate the standard deviation of the differences between paired data entries using the alternative formula sd = HΣd2 - c 1Σd22 n d n - 1 .
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