ACT, a college entrance exam used for admission, looked at historical records and established 21 as the minimum score on the ACT reading portion of the exam for a student to be considered prepared for social science in college. (Note: "Being prepared" means there is a 75% probability of successfully completing a social science course in college.) An official with the Illinois State Department of Education wonders whether a majority of the students in her state who took the ACT are prepared to take social science. She obtains a simple random sample of 500 records of students who have taken the ACT and finds that 269 are prepared. Does this represent significant evidence that a majority (more than 50%) of the students in the state of
Illinois are prepared for social science in college upon graduation?
(a) What does it mean to make a Type II error for this test?
(b) If the researcher decides to test this hypothesis at the a = 0.05 level of significance,
determine the probability of making a Type II error if the true population proportion is
0.52. What is the power of the test?
(c) Redo part (b) if the true proportion is 0.55.