This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full Document
 Spring '11
 Ho
 Statistics, Normal Distribution, Null hypothesis, Statistical hypothesis testing, Xij

Click to edit the document details