Problem
A travel agency is interested in finding out if different age groups frequent different Spring Break destinations, in order to better target the appropriate audiences. A random sarfhple of college Spring Break vacationers produces the results given in the table below. Is there enough evidence at the 0.005 level of significance to show that there is a relationship between age (by college classification) and destination? \begin{tabular}{|c|c|c|c|c|c|} \hline \multicolumn{7}{|c|}{ Observed Sample of College Students } \\ \hline & Beach & Mountains & City & Home & Total \\ \hline Freshman & 21 & 12 & 2 & 19 & 54 \\ \hline Sophomore & 26 & 16 & 22 & 23 & 87 \\ \hline Junior & 22 & 9 & 2 & 18 & 51 \\ \hline Senior & 18 & 7 & 9 & 13 & 47 \\ \hline Total & 87 & 44 & 35 & 73 & 239 \\ \hline \end{tabular} Copy Data Step 1 of 4: Calculate the expected value for the number of freshmen going to the beach during Spring Break. Round your answer to six decimal places.
Solution
Step 1 ：Given the total number of freshmen is 54, the total number of students going to the beach is 87, and the total number of students is 239.
Step 2 ：The expected value can be calculated by multiplying the row total by the column total and then dividing by the grand total.
Step 3 ：Using the formula \(\frac{{row\_total \times column\_total}}{{grand\_total}}\), we substitute the given values to get \(\frac{{54 \times 87}}{{239}}\).
Step 4 ：Calculating the above expression, we get an expected value of approximately 19.656903765690377.
Step 5 ：Rounding this to six decimal places, we get 19.656904.
Step 6 ：Thus, the expected value for the number of freshmen going to the beach during Spring Break is \(\boxed{19.656904}\).
