The proportion of adults living in a small town who are college graduates is est

Question

The proportion of adults living in a small town who are college graduates is estimated to be
p = 0:6: To test this hypothesis, a random sample of 15 adults is selected. If the number of
college graduates in our sample is anywhere from 6 to 12, we shall not reject the null hypothesis
that p = 0:6; otherwise, we shall conclude that p 6= 0:6.

(a) Evaluate assuming that p = 0:6. Use the binomial distribution.
(b) Evaluate for the alternatives p = 0:5.
(c) Is this a good test procedure?

Any help would be greatly appreciated.

Explanation / Answer

Let's have X have a binomial distribution with number of trials n = 15 and probability of success p.

We will not reject the null hypothesis if 6 <= X <= 12, so we should reject the null if X < 6 or X > 12.

a. You commit a type I error if you reject the null hypothesis, however the null was true (meaning p = 0.6).
P(type I error) = P(Reject null | null is true)
= P(X < 6 or X > 12 | p = 0.6)
= 0.060947304.

b. You commit a type II error if you do not reject the nul hypothesis, but the null is not true (meaning that p is something different from 0.6).

If p = 0.5 in reality, then the null is not true, so
P(type II error) = P(do not reject null | null is not true)
= P(6 <= X <= 12 | p = 0.5)
= 0.845428467 again using a binomial with n = 15 and p = 0.5

c.The probability of a type I error is reasonable. The probabilities of a type II error is quite large, even though 0.5 is fairly far from 0.6, so I would say that this is not a good test procedure. I would suggest taking a larger sample size.

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