A hypothesis test is performed to test the claim that a population proportion is greater than 0.7. Find the probability of a type II error, β, given that the true value of the population proportion is 0.72. The sample size is
50 and the significance level is 0.05.
A) 0.4129
B) 0.7123
C) 0.9706
D) 0.5754

Question

Quizplus · Accepted Answer

To find the probability of a type II error, we first need to determine the alternative hypothesis. Since the null hypothesis is that the population proportion is less than or equal to 0.7, the alternative hypothesis would be that the population proportion is greater than 0.7.

Based on this information, we can set up the following hypotheses:

H0: p ≤ 0.7
Ha: p > 0.7

Next, we need to find the critical value for the test. Since the significance level is 0.05 and the test is one-tailed (right-tailed), the critical value is 1.645 (found using a standard normal distribution table or calculator).

Using the true value of the population proportion (p = 0.72), we can find the test statistic:

z = (0.72 - 0.7) / sqrt(0.7 * 0.3 / 50) = 1.039

Since the test statistic (1.039) is less than the critical value (1.645), we fail to reject the null hypothesis.

The probability of a type II error is the probability of failing to reject the null hypothesis given that the alternative hypothesis is true. In other words, it is the probability of not detecting a difference when one actually exists.

We can find this probability by calculating the area under the null distribution that falls to the right of the test statistic (1.039):

β = P(Z ≤ 1.039 | p = 0.72)

Using a standard normal distribution table or calculator, we can find that this probability is approximately 0.9706. Therefore, the answer is C.

A Hypothesis Test Is Performed to Test the Claim That