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Class 18, June 23, 2014 Plan for Class 18: 1. The Problem of Hypothesis Testing (The case of Large Samples) Hypothesis Testing for Large Samples Problem. Example 7.5, p.373. X = {The amount of cornflakes discharged by a filling machine into a standard box, in ounces} The machine is design to discharge on average 12 ounces. A statistical test below should decide whether there are deviations from 12 ounces. Test H0: µx= 12 versus H1: µx≠ 12 at the level of significance α=0.01. First we construct the Rejection Region corresponding to α=0.01. These are numbers to the left of z0.005 2.576 and to the right of z0.005 2.576 . Now, given sample we find (the book gives this data): n 100, X 11.851, s 0.512 and calculate X 0 2.91. s/ n If Z gets int o the rejection region ( RR ), we reject H 0 in favor the value of test statistic Z of the alternative hypothesis H a at the level of significan ce . Since indeed Z gets int o RR , we reject H 0 in favor of the alternative hypothesis H a . As we have shown in class, this procedure is justified by the fact that the probability of type I error (we reject the correct hypothesis) is α, a small number. The above procedure can be restated (equivalently) by the notion of the p-value. p-value is the sum of tail areas corresponding to the points X 0 s/ n and X 0 . s/ n In our case this will be to the left of X 0 2.91. s/ n X 0 2.91 and s/ n to the right of The sum of these two areas (the p-value) is 0.0036. Once the p-value is found, we can easily say whether or not we reject the null hypothesis for any level α. Namely, if p value, we reject, if p value, we don' t reject. . Since our p-value is 0.0036, we can say for example that for 0.0025 we don' t reject, while for 0.005 we do reject. Homework: Read Example 7.5 on page 373.