Grades on a standardized test are known to have a mean of 1000 for students in the United States. The test is administered to 453 randomly selected students in Florida; in this sample, the mean is 1013, and the standard deviation \((s)\) is 108 .

a. Construct a \(95 \%\) confidence interval for the average test score for Florida students.

b. Is there statistically significant evidence that Florida students perform differently than other students in the United States?

c. Another 503 students are selected at random from Florida. They are given a three-hour preparation course before the test is administered. Their average test score is 1019 , with a standard deviation of 95 .

i. Construct a 95\% confidence interval for the change in mean test scores associated with the prep course.

ii. Is there statistically significant evidence that the prep course helped?

d. The original 453 students are given the prep course and then are asked to take the test a second time. The average change in their test scores is 9 points, and the standard deviation of the change is 60 points.

i. Construct a \(95 \%\) confidence interval for the change in average test scores.

ii. Is there statistically significant evidence that students will perform better on their second attempt, after taking the prep course?

iii. Students may have performed better in their second attempt because they took the prep course or because they gained test-taking experience in their first attempt. Describe an experiment that would quantify these two effects.