Suppose that the proportion, p, of defective items in a large population of item

Question

Suppose that the proportion, p, of defective items in a large population of items is unknown, and that one wishes to test the following hypotheses. H_0 : p = 0.2, H_1 : p is not equal to 0.2. Suppose also that a random sample of n = 20 items is drawn from the population. Let X denote the number of defective items in the sample, and consider a test procedure that rejects the null hypothesis if either X greater than or equal to 7 or X less than or equal to 1. Are the hypotheses simple or composite? Provide a brief explanation. Write down the critical region of the test. Compute the size of the test and the probability of Type I error. Calculate the value of the power function II_(p) at the points p sigma {0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1}

Explanation / Answer

a) Simple hypothesis

b) It is p +/- t*sqrt[(p)*(1-p)/n]

c) size of a test is the probability of falsely rejecting the null hypothesis. That is, it is the probability of making a Type I error. It is denoted by the Greek letter

The size of the test is the value of the power function when p = 0.2 = 0.3941

d) Calculate for any given value of p, (p) = Pr(Y 7) + Pr(Y 1) where Y has a
binomial distribution with parameters n and p. You can use the binomial table. For
p = 0, (0) = 1. For p = 0:1, (0:1) = 1 Pr(Y < 7) + Pr(Y 1) = 1 (Pr(Y =
6) + Pr(Y = 5) + ::: + Pr(Y = 1) + Pr(Y = 0)) + P(Y = 1) + Pr(Y = 0) = 0:3941.
For p = 0:2 (0:2) = 0:1558, and so on.

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