You are considering testing a new drug that is supposed to facilitate learning in mentally retarded children. Because there is relatively little known about the drug, you plan to use a nondirectional alternative hypothesis. Since your resources are limited, you can test only 15 subjects. The subjects will be run in a repeated measures design and the data will be analyzed with the sign test, using a = 0.05_{2 tail}. What is the probability of a Type II error? A) 0.7032 B) 0.7031 C) 0.9824 D) 0.9648

Question

Quizplus · Accepted Answer

To find the probability of a Type II error, we need to calculate the power of the test. Since we are using a sign test, we can use the binomial distribution to calculate the power. Using a significance level of 0.05, a sample size of 15, and a nondirectional alternative hypothesis, the critical values for the sign test are -4 and +4. The probability of a Type II error (β) is equal to the probability of observing fewer than 5 or more than 11 sign changes, given a true population proportion of 0.5.

Using the binomial distribution, we can calculate the probability of observing fewer than 5 sign changes as:

P(X < 5) = Σ (15 choose k) * 0.5^15, for k = 0 to 4

This is equal to 0.7031. Similarly, the probability of observing more than 11 sign changes is also 0.7031. Therefore, the total probability of a Type II error is:

β = 0.7031 + 0.7031 = 0.1406

So the probability of a Type II error is 0.1406, which is closest to option B.

You Are Considering Testing a New Drug That Is Supposed