A college wants to investigate whether students perform better on mornings that they have had a nutritious breakfast compared to mornings that they have not.

You randomly recruit 100 students and tell each of them to come to the lab in two different mornings: each morning they will be offered some breakfast and will take a test. Each student will receive the light breakfast one time and the nutritious breakfast the other time (some will receive the light breakfast first, the others will receive the nutritious breakfast first, and the second time they switch).

The average difference in a student's score after eating the nutritious breakfast and their score after eating a light breakfast is 4.5, and the standard deviation of this difference is 3.6. Thus we are doing students nutritious breakfast scores minus the light breakfast scores.

a. Construct a 95% confidence interval for the difference in the average score of students after having a nutritious breakfast and the average score of students after having a light breakfast.

b. Interpret the interval from part a. in context of the study.