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Lecture 25-2007 - Examples and Multivariate Testing Lecture...

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1 Examples and Multivariate Testing Lecture XXV I. Examples. A. Example 9.6.1 (mean of a binomial distribution) Assume that we want to know whether a coin toss is biased based on a sample of ten tosses. B. Our null hypothesis is that the coin is fair ( 0 1 : 2 H p ) versus an alternative hypothesis that the coin toss is biased towards heads ( 1 1 : 2 H p ). 1. Assume that you tossed the coin ten times and observed eight heads. What is the probability of drawing eight heads from ten tosses of a fair coin? 0 1 2 10 9 8 10 10 10 8 1 1 1 10 9 8 P n p p p p p p If 1 2 p , 8 0.054688 P n . Thus, we reject 0 H at a confidence level of 0.10 and fail to reject 0 H at a 0.05 confidence level. 2. Moving to the Likelihood ratio test: 2 8 2 8 .5 .5 1 .5 .1455 ˆ .8 .8 1 .8 MLE p p Given that 2 1 2ln ~ We reject the hypothesis of a fair coin toss at a 0.05 confidence level. ( 2ln 3.854 ) and the critical region for a chi-squared distribution at one degree of freedom is 3.84. C. Example 9.6.2. Suppose the heights of male Stanford students is distributed 2 , N with a known variance of 0.16. 1. Assume that we want to test whether the mean of this distribution is 5.8 against the hypothesis that the mean of the distribution is 6. What is the test statistic for a 5 percent level of confidence and a 10 percent level of confidence?
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