8-1 Basics of Hypothesis Testing 379 Step 6: Find the Value of the Test Statistic, Then Find Either the P-Value or the Critical Value(s) The test statistic gives us a measure of the amount of the discrepancy between a sample statistic and the claimed value of the population parameter assumed in the null hypothesis. For proportions, the test statistic is a measure of the discrepancy between the sample proportion pn and the claimed proportion p. Sign used in H1: Þ Two-tailed test Sign used in H1: , Left-tailed test Sign used in H1: . Right-tailed test FIGURE 8-2 Critical Region in Two-Tailed, Left-Tailed, and Right-Tailed Tests DEFINITION The test statistic is a value used in making a decision about the null hypothesis. It is found by converting the sample statistic (such as pn, x, or s) to a score (such as z, t, or x2) with the assumption that the null hypothesis is true. In this chapter we use the test statistics listed in the last column of Table 8-2. Example: From Example 1 we have a claim made about the population proportion p, we have n = 926 and pn = 0.52. With the null hypothesis of H0: p = 0.5, we are working with the assumption that p = 0.5, and it follows that q = 1 - p = 0.5. We can evaluate the test statistic as shown below (or technology can find the test statistic for us). The test statistic of z = 1.25 from each of the previous technology displays is more accurate than the result of z = 1.22 shown below. (If we replace 0.52 with 482>926 = 0.5205183585, we get z = 1.25.) z = pn - p Apq n = 0.52 - 0.5 B10.5210.52 926 = 1.22 Finding the P-value and>or critical value(s) requires that we first consider whether the hypothesis test is two-tailed, left-tailed, or right-tailed, which are described as follows. CP Depending on the claim being tested, the critical region could be in the two extreme tails, it could be in the left tail, or it could be in the right tail. ■ Two-tailed test: The critical region is in the two extreme regions (tails) under the curve (as in the top graph in Figure 8-2). ■ Left-tailed test: The critical region is in the extreme left region (tail) under the curve (as in the middle graph in Figure 8-2). ■ Right-tailed test: The critical region is in the extreme right region (tail) under the curve (as in the bottom graph in Figure 8-2). HINT Look at the symbol used in the alternative hypothesis H1. • The symbol 7 points to the right and the test is right-tailed. • The symbol 6 points to the left and the test is left-tailed. • The symbol ≠ is used for a two-tailed test. Example: With H0: p = 0.5 and H1: p 7 0.5, we reject the null hypothesis and support the alternative hypothesis only if the sample proportion is greater than 0.5 by a significant amount, so the hypothesis test in this case is right-tailed. DEFINITION The critical region (or rejection region) is the area corresponding to all values of the test statistic that cause us to reject the null hypothesis. Two-Tailed, Left-Tailed, Right-Tailed
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