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

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Examples and Multivariate Examples and Multivariate Testing Testing Lecture XXV
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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. Our null hypothesis is that the coin is fair ( H 0 : p =1/2) versus an alternative hypothesis that the coin toss is biased towards heads ( H 1 : p >1/2).
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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? If p =1/2, P [ n 8]=.054688. Thus, we reject H 0 at a confidence level of .10 and fail to reject H 0 at a .05 [ ] ( 29 ( 29 ( 29 2 8 1 9 0 10 1 8 10 1 9 10 1 10 10 8 p p p p p p n P - + - + - =
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Moving to the Likelihood ratio test: Given that ( 29 ( 29 = = = - - = Λ 8 . ˆ 5 . 1455 . 8 . 1 8 . 5 . 1 5 . 2 8 2 8 MLE p p ( 29 2 1 ~ ln 2 χ Λ -
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We reject the hypothesis of a fair coin toss at a .05 confidence level. (-2 ln( Λ )=3.854 and the critical region for a chi-squared distribution at one degree of freedom is 3.84. Example 9.6.2. Suppose the heights of male Stanford students is distributed N ( μ , σ 2 ) with a known variance of .16.
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Assume that we want to test whether the mean of this distribution is 5.8 against the
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Lecture 25-2007 - Examples and Multivariate Testing Lecture...

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