US News and World Report Ranks Colleges and Universities Annually

Essay

U.S. News and World Report ranks colleges and universities annually. You randomly sample 100 of the national universities and liberal arts colleges from the year 2000 issue. The average cost, which includes tuition, fees, and room and board, is $23,571.49 with a standard deviation of $7,015.52.
(a)Based on this sample, construct a 95% confidence interval of the average cost of attending a university/college in the United States.
(b)Cost varies by quite a bit. One of the reasons may be that some universities/colleges have a better reputation than others. U.S. News and World Reports tries to measure this factor by asking university presidents and chief academic officers about the reputation of institutions. The ranking is from 1 ("marginal")to 5 ("distinguished"). You decide to split the sample according to whether the academic institution has a reputation of greater than 3.5 or not. For comparison, in 2000, Caltech had a reputation ranking of 4.7, Smith College had 4.5, and Auburn University had 3.1. This gives you the statistics shown in the accompanying table. $\begin{array} { | c | c | c | c | } \hline \begin{array} { c } \text { Reputation } \\\text { Category }\end{array} & \text { Average Cost } \bar { Y } & \begin{array} { c } \text { Standard deviation } \\\text { of Cost } \left( S _ { Y } \right)\end{array} & \mathrm { N } \\\hline \text { Ranking } > 3.5 & \$ 29,311.31 & \$ 5,649.21 & 29 \\\hline \text { Ranking } \leq 3.5 & \$ 21,227.06 & \$ 6,133.38 & 71 \\\hline\end{array}$ Test the hypothesis that the average cost for all universities/colleges is the same independent of the reputation. What alternative hypothesis did you use?
(c)What other factors should you consider before making a decision based on the data in (b)?