The two major college entrance exams that a majority of colleges accept for admission are the SAT and ACT. ACT looked at historical records and established 22 as the minimum ACT math score for a student to be considered prepared for college mathematics. [Note: "Being prepared" means there is a \(75 \%\) probability of successfully completing College Algebra in college.] An official with the Illinois State Department of Education wonders whether less than half of the students in her state are prepared for College Algebra. She obtains a simple random sample of 500 records of students who have taken the ACT and finds that 219 are prepared for college mathematics (that is, scored at least 22 on the ACT math test). Does this represent significant evidence that less than half of Illinois students who have taken the ACT are prepared for college mathematics upon graduation from a high school? Use the \(\alpha=0.05\) level of significance. Source: ACT High School Profile Report.

Approach This problem deals with a hypothesis test of a proportion. We want to determine if the sample evidence shows that less than half of the students are prepared for college mathematics. Symbolically, we represent this as \(p<\frac{1}{2}\) or \(p<0.5\).

Verify the three requirements to perform the hypothesis test: the sample must be a simple random sample, \(n p_{0}\left(1-p_{0}\right) \geq 10\), and the sample size cannot be more than \(5 \%\) of the population size (for independence). Then follow Steps 1 through 5.