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Unformatted text preview: Statistical Hypothesis Testing Dr. Phillip YAM 2010/2011 Spring Semester Reference: Chapter 7 of “Tests of Statistical Hypotheses” by Hogg and Tanis. Section 7.1 Tests about Proportions I A statistical hypothesis test is a formal method of making decisions, upon the probabilistic structure of a random mathematical model, by analyzing the available sample I Example (Simple) Null hypothesis: H : p = 0 . 06 completely specifies the distribution Against (Composite) Alternative hypothesis: H 1 : p < . 06 does not completely specify the distribution; it is composed of many simple hypotheses I Possible error: Type I error: Rejecting H and accepting H 1 when H is true; Type II error: Failing to reject H when H 1 is true (i.e., when H is false). Section 7.1 Tests about Proportions I Consider the test of H : p = p against H 1 : p > p , where p = probability of success. I Base our test upon the number of successes Y in n independent Bernoulli trials. Using CLT, Y / n has an approximate normal distribution N [ p , p (1 p ) / n ], provided that H : p = p is true and n is large. I We intend to reject H and accepts H 1 if and only if Z = Y / n p p p (1 p ) / n ≥ z α . That is to say: if Y / n exceeds p by z α standard deviations of Y / n , we reject H and accept the hypothesis H 1 : p > p . The approximate probability of this occurring when H : p = p is true is α . The significance level of this test is approximately α . Section 7.1 Tests about Proportions I Example 7.11: Many commercially manufactured dice are not fair because the “spots” are really indentations, so that, for example, the 6side is lighter then the 1side. Let p = the probability of rolling a 6. I To test H : p = 1 / 6 against the alternative hypothesis H 1 : p > 1 / 6. Suppose that we have a total of n = 8000 observations. Let Y equal the number of times that 6 resulted in the 8000 trials. The results of the experiment yielded y = 1389, so the calculated value of the test statistic is z = 1389 / 8000 1 / 6 p (1 / 6)(5 / 6) / 8000 = 1 . 670 > 1 . 645 = z . 05 . and hence, the null hypothesis is rejected, and the experimental results indicate that these dice favor a 6 more than a fair die would be. Section 7.1 Tests about Proportions I Formal Statistical Hypothesis Testing can be regarded as a statistical version of “Mathematical Proof by Contradiction”. An example of the latter is “Euclid’s proof of infinitely many primes”. I Analogy: I (1) H 1 VS H ( C 1 :“Infinitely Many Primes” VS C “Finitely Many Primes”) I (2) A random sequence of sample X 1 ,..., X n under H (A finite deterministic sequence of primes p 1 ,..., p n under C ) I (3) Functional inequality Z = Y / n p p p (1 p ) / n ≥ z α (a new positive integer p = p 1 ··· p n + 1) I (4) Definite conclusion (conclusion subject to chance) I A “reasonably good” test for a parameter normally relies on the maximum likelihood estimator (more precisely, the sufficient statistic) for the parameter. Section 7.1 Tests about ProportionsSection 7....
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 Spring '11
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 Statistics

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