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STA 4502 Chapter 2 - Revised Chapter 2 The Dichotomous...

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Revised August 12, 2004 Chapter 2 The Dichotomous Problem ( 1 ) Introduction The data in this chapter is of the dichotomous type of independent repeated Bernoulli trials having constant probability of success p. Thus, we have an experiment that has two possible outcomes , “success 2 ” and “failure” (called a Bernoulli trial). This experiment is repeated n times, each repetition is independent of the others and the probability of “success” is assumed to be equal to p and it does not change from trial to trial. The random variable 3 of interest is B = the number of “success”s in n independent repetitions of this Bernoulli trial. These are summarized in the following assumptions Assumptions: A1. The experiment has only two possible outcomes , called “success” and “failure.” [Bernoulli experiment.] A2. Probability of “success,” denoted by p, remains constant from trial to trial. A3. The n repetitions of the experiment [trials] are independen t of each other. A4. The random variable of interest is B = number of “success”s in n independent trials. 2.1 A binomial Test The null hypothesis is Ho: p = p 0 , where p 0 is some specified number, 0 < p 0 < 1. The level of significance of the test = α = P(Type I Error) is fixed at α . We have three types of alternative hypotheses (i.e., tests): a) One-Sided, Upper-Tail Test, Ha: p > p 0 b) One-Sided Lower-Tail Test, Ha: p < p 0 c) Two-Sided Test Ha: p ≠ p 0 1 Students are strongly urged to review their previous knowledge of statistical inference (point and interval estimation as well as hypothesis testing). Also, a quick reading of chapter 1 is strongly suggested to get the reasons for learning nonparametric (or distribution free) statistical inference techniques, its differences from what you already (should) have learned as well as an overview of the text. 2 In this context, “success” may have nothing to do with success as used in our daily lives, it is simply observing the outcome of the experiment that we are interested in. 3 In your previous courses you may have used X, Y, Z as symbols for the random variables. Here B is used as the name of the random variable to remind you that the random variable has the B inomial distribution. Stat 4502 Chap 2, Page 1 of 7
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These are summarized in Table – 1. Table – 1. A Brief Summary of the Binomial Test Type of Test Hypotheses to be tested Decision Rule (Rejection Region) Remarks Upper-Tail H 0 : p = p 0 vs. H A : p > p 0 Reject H 0 if B b α See Footnote 4 Lower-Tail H 0 : p = p 0 vs. H A : p < p 0 Reject H 0 if B c α See Footnote 5 Two-Sided H 0 : p = p 0 vs. H A : p p 0 Reject H 0 if B 1 b α or B 2 c α See Footnote 6 Large Sample Approximation Her we will use the normal approximation to the binomial distribution, which states that when n × p 0 10 AND n × (1 – p 0 ) 10 7 , B has an approximate normal distribution with mean n × p 0 and standard deviation 0 0 (1 ) n p p × × - ( 8 ). Hence we may use the tables of the standard normal distribution for testing the hypotheses after standardizing B, as, * 0 0 (1 ) o B n B np p - = - . Here B * has the standard normal distribution 9 , B * ~ N(0,1). Under these conditions, the above summary table changes to
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