Chapter 18, Problem 17E

To determine

To use the $68-95-99.7$ rule to describe the sampling distribution model for the percentage of those students we expect to return to that school of their sophomore years and explain do you think the appropriate conditions are met.

Answer to Problem 17E

Yes, all the appropriate conditions are met.

Explanation of Solution

It is given in the question that the company reported that the national college freshmen-to-sophomore retention rate held steady at $74\%$ over the previous four years. So, we will consider the random sample of $400$ freshmen who took the ACT. Thus, using the rule $68-95-99.7$ , we get,

  $\begin{array}{l}\mu =p=0.74\\ \sigma =\sqrt{\frac{pq}{n}}\\ =\sqrt{\frac{0.74(0.26)}{400}}\\ =0.023\end{array}$

Thus, about $68\%$ of the sample proportions are expected to be between $\text{0}\text{.718 and 0}\text{.762}$ and about $95\%$ are expected to be between $\text{0}\text{.696 and 0}\text{.784}$ and about

  $99.7\%$ are expected to be between $\text{0}\text{.674 and 0}\text{.806}$ , by using the rule $68-95-99.7$ to describe the sampling distribution model. And also, the appropriate conditions necessary for your analysis are as:

Random condition: It is satisfied as we have a random sample of $400$ students who took ACT.

  $10\%$ condition: The sample size less than $10\%$ of the population size.

Success/failure condition: as $np=296,n(1-p)=104$ and both are greater than ten then this is also true.

Thus, the conditions are all met. So, the sampling distribution model for the percentage of those students we expect to return to that school of their sophomore years is $N(0.74,0.023)$ .