The best university in the world has an amazing dining commons (aka cafeteria) that serves 3 meals a day.  Students have the choice of paying based on a meal plan.  For example, the "reducto" meal plan is for the student who doesn't care to eat much dining commons and allows them to eat up to 125 meals at the cafeteria per quarter.  The director of the dining commons enjoys saying hello to all the students who come to dine and noticed that he rarely sees premed students.  He's wondering if premed students are too busy with studying to eat the dining commons and by comparison looks at Economics majors who have more time on their hands to see how many times they visit the dining commons to eat in a quarter. The table below shows the mean and standard deviation of the amount of time spent on homework each night (in minutes) for a random sample of 25 pre-med majors and a separate random sample of 25 economics majors.

Based on dotplots of these data, it is not unreasonable to assume that the distribution of times for each major were approximately normally distributed.

(a) Estimate the difference in mean amount of meals consumed at the dining commons in a quarter between a pre-med and economics majoring student by building a 90% confidence interval. Be sure to interpret your interval.

(b) An assistant director at the dining commons reasoned that a much narrower confidence interval could be obtained if the students were paired based on their responses; for example, pairing the pre-med and the econ students with the highest number of dining commons visits, the pre-med and econ students with the next highest number of visits to the dining commons, and so on. Is the assistant director correct in thinking that matching students in this way and then computing a matched-pairs confidence interval for the mean difference in visits to the dining commons in a quarter is a better procedure than the one used in part (a)? Justify your answer.