Check Your Understanding A high school student is planning a very large graduation party. She needs to know how many people are planning on attending in order to estimate the cost of the party. She plans to select a random sample of 25 guests that were invited to the party and record whether or not each person plans to attend the party. She will then test H_(0):p=0.75 versus H_(a):pneq 0.75 where p=the true proportion of all invited guests that will attend the party. She will use alpha =0.05. a. Suppose the true proportion of invited guests that will attend the party is 0.70 Then the power of the given significance test is 0.10. Interpret the power. b. Find the probability of a Type l error and the probability of a Type II error for the test in part (a). c. Determine whether each of the following changes would increase or decrease the power of the test. Explain your answers. - Use alpha =0.01 instead of alpha =0.05 - Use n=100 instead of n=25

a. The power of a significance test is the probability that the test correctly rejects the null hypothesis when the alternative hypothesis is true. In this case, the power of the test is 0.10, which means that there is a 10% chance that the test will correctly reject the null hypothesis (H0: p = 0.75) when the true proportion of invited guests that will attend the party is 0.70.<br /><br />b. The probability of a Type I error is the probability that the test incorrectly rejects the null hypothesis when it is true. In this case, the probability of a Type I error is equal to the significance level, which is 0.05.<br /><br />The probability of a Type II error is the probability that the test fails to reject the null hypothesis when the alternative hypothesis is true. In this case, the probability of a Type II error is equal to 1 - the power of the test, which is 1 - 0.10 = 0.90.<br /><br />c. <br />- Using α = 0.01 instead of α = 0.05 would decrease the power of the test. This is because a smaller significance level means that the test is more conservative, making it less likely to reject the null hypothesis. Therefore, the probability of correctly rejecting the null hypothesis when it is false (i.e., the power of the test) would be lower.<br />- Using n = 100 instead of n = 25 would increase the power of the test. This is because a larger sample size provides more information about the population, making it more likely that the test will correctly reject the null hypothesis when it is false. Therefore, of correctly rejecting the null hypothesis when it is false (i.e., the power of the test) would be higher.