Section 9.2 Hypothesis Tests for a Population Mean, Standard Deviation Known 411 530 Critical region: Area = 0.05 ¯ x = 562 z = 2.76 z = 1.645 H0 value for μ Critical value Test statistic Figure 9.2 The critical value is 1.645. The critical region contains all z-scores greater than or equal to 1.645. The value of our test statistic is z = 2.76. This value is in the critical region, so we reject H0. The probability that we use to determine whether an event is unusual is called the significance level of the test, and is denoted with the letter ��. In Figure 9.2, we used �� = 0.05. This is the most commonly used value for ��, but other values are sometimes used as well. Next to �� = 0.05, the most commonly used value is �� = 0.01. The choice of �� is determined by how strong we require the evidence against H0 to be in order to reject it. The smaller the value of ��, the stronger we require the evidence to be. For example, if we choose �� = 0.05, we will reject H0 if the test statistic is in the most extreme 5% of its distribution. However, if we choose �� = 0.01, we will not reject H0 unless the test statistic is in the most extreme 1% of its distribution. DEFINITION If we reject H0 after choosing a significance level ��, we say that the result is statistically significant at the �� level. We also say that H0 is rejected at the �� level. In our SAT example, we rejected H0 at the �� = 0.05 level, and the result was statistically significant at the �� = 0.05 level. Our alternate hypothesis of �� > 530 was a right-tailed alternative. For this reason, the critical region was in the right tail of the distribution. The location of the critical region depends on whether the alternate hypothesis is left-tailed, right-tailed, or two-tailed. Critical Values for Hypothesis Tests Let �� denote the chosen significance level. The critical value depends on whether the alternate hypothesis is left-tailed, right-tailed, or two-tailed. Critical region: −zα Area = α For left-tailed H1: The critical value is −z��, which has area �� to its left. Reject H0 if z ≤ −z��. Critical region: zα Area = α For right-tailed H1: The critical value is z��, which has area �� to its right. Reject H0 if z ≥ z��. Critical region: Area = α/2 Critical region: Area = α/2 −zα/2 zα/2 For two-tailed H1: The critical values are z��∕2, which has area ��∕2 to its right, and −z��∕2, which has area ��∕2 to its left. Reject H0 if z ≥ z��∕2 or z ≤ −z��∕2.
navidi_monk_elementary_statistics_2e_ch7-9
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