Chapter 18, Problem 33E

To determine

To explain what might the mean GPA of one of these seminar groups be and also describe the appropriate sampling distribution model- shape, center and spread- with attention to assumptions and conditions and also make a sketch using the $68-95-99.7$ rule.

Answer to Problem 33E

The mean GPA will be $3.4$ .

Explanation of Solution

It is given in the question that a college’s data about the incoming freshmen indicates that the mean of their high school GPAs was $3.4$ with a standard deviation of $0.35$ and the distribution was roughly mounded and only slightly skewed. The students are randomly assigned to freshmen writing seminars in group of $25$ . Thus, let us check the conditions and assumptions for the following as:

Random condition: It is satisfied as the freshmen are randomly assigned to writing seminars.

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

Success/failure condition: It is satisfied as the distribution for this sample size is uni-modal and approximately normal so, $25$ is a large enough sample.

Thus, all the conditions and assumptions are met. Thus, the sampling distribution model is Normal model with the mean and the standard deviation as:

  $\begin{array}{l}\mu ={\mu }_{\overline{x}}=3.4\\ {\sigma }_{\overline{x}}=\frac{\sigma }{\sqrt{n}}\\ =\frac{0.35}{\sqrt{25}}\\ =0.07\end{array}$

And the sketch using the $68-95-99.7$ rule will be as follows: