# Power For a hypothesis test with a specified significance level α, the probability of a type I error is a, whereas the probability β of a type II error depends on the particular value of p that is used as an alternative to the null hypothesis.a. Using an alternative hypothesis of p < 0.4, a sample size of n = 50, and assuming that the true value of p is 0.25, find the power of the test. See Exercise 1 in Section 8-2. ( Hint: Use the values p = 0.25 and  pq/n = (0.25)(0.75)/50.)b. Find the value of ß, the probability of making a type II error.c. Given the conditions cited in part (a), what do the results indicate about the effectiveness of the hypothesis test?Exercise 1 Using Confidence Intervals to Test Hypotheses When analyzing the last digits of telephone numbers in Port Jefferson, it is found that among 1000 randomly selected digits, 119 are zeros. If the digits are randomly selected, the proportion of zeros should be 0.1.

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Power For a hypothesis test with a specified significance level α, the probability of a type I error is a, whereas the probability β of a type II error depends on the particular value of p that is used as an alternative to the null hypothesis.

a. Using an alternative hypothesis of p < 0.4, a sample size of n = 50, and assuming that the true value of p is 0.25, find the power of the test. See Exercise 1 in Section 8-2. ( Hint: Use the values p = 0.25 and  pq/n = (0.25)(0.75)/50.)

b. Find the value of ß, the probability of making a type II error.

c. Given the conditions cited in part (a), what do the results indicate about the effectiveness of the hypothesis test?

Exercise 1

Using Confidence Intervals to Test Hypotheses When analyzing the last digits of telephone numbers in Port Jefferson, it is found that among 1000 randomly selected digits, 119 are zeros. If the digits are randomly selected, the proportion of zeros should be 0.1.

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Step 1

a)
Given
Null Hypothesis Ho p = 0.40
Alternate Hypothesis Ha p < 0.40
Level of significance is assumed as 0.05 (when nothing is mentioned in question usually we take α = 0.05)
sample size n = 50
With reference to below Z table the probability for Z score -1.64 and -1.65 are 0.505 and 0.495 respectively. So the Z score for 0.05 probability value will be approximately -1.645 (average of both Z scores).
Type - I error is rejecting the true null hypothesis.

Step 2

probability of type I error = 0.05. we calculate the following
P(p^ < 0.286) = 0.05

Type - II error is non rejection of false hypothesis
Power of the test is probability of rejecting the null hypothesis when it is actually false.
So power = 1 - β

Step 3

The power of test is calculated as shown below.
power = P(Z < 0.5879) = P( Z < 0.59) (rounding Z score to two decim...

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