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Problem 2-2, Comparing a single mean to a specified value (second example)
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===Section B: Z-values=== To compare the mean of the true distribution to that of the theoretical distribution, we test the null hypothesis with a z-test. The z-value is calculated as in problem 2-1: <center><math>z=\frac{\bar{y}-\mu_0}{\sigma/\sqrt{n}}=1.92</math><br/></center> Since we have a two-tailed alternative hypothesis, we must define a rejection region at both extremes of our theoretical distribution. Our value for <math>\alpha</math> determines the total size of the rejection region, so we simply declare that 2.5% (since <math>\alpha</math> = .05, or 5%) of the area on the left of our theoretical distribution is a rejection region, and 2.5% of the area on the right is also a rejection region (see Figure 2). We calculate <math>z_{\alpha/2}</math> to determine the x-value that corresponds to the rightmost edge of the rejection region on the left, and <math>z_{1-\alpha/2}</math> to find the leftmost edge of the rejection region on the right (see Figure 2 again). <center><math>z_{\alpha/2}=-1.96</math>, <math>z_{1-\alpha/2} = 1.96</math></center> If z is between <math>{z_\alpha}</math> and <math>z_{1-\alpha/2}</math>, it is not in the rejection region and we claim the null hypothesis to be true (with a confidence of 95%). Otherwise we claim the alternative hypothesis to be true (with the same confidence). Note that <math>|z_{\alpha/2}| = z_{1-\alpha/2}</math>. This is always true. You can simply compare your z-value to <math>|z_{\alpha/2}|</math> to perform a z-test. If <math>z < |z_{\alpha/2}|</math>, claim your null hypothesis to be true, otherwise claim that your alternative hypothesis is true.
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