Chapter 25, Problem 28E

To determine

To estimate the mean weight gain in first semester freshmen and comment on the “freshman $15$ ”.

Answer to Problem 28E

We are $90\%$ confident that the mean decrease in weight is between $-2.4280\text{ and }-1.3956$ pounds.

Explanation of Solution

The table of the freshmen’s initial weight and the terminal weight is given in the question. Now, we have to calculate the mean and the standard deviation of the difference between the two samples as:

  $\begin{array}{l}\overline{d}=\frac{3-1-2+\mathrm{....}-3+0}{68}\\ =-1.9118\\ s_d=\sqrt{\frac{{(3-(-1.9118))}^2+{(-1-(-1.9118))}^2+{(-2-(-1.9118))}^2+\mathrm{....}+{(0-(-1.9118))}^2}{68-1}}\\ =2.1282\end{array}$

The degrees of freedom is then will be as:

  $\begin{array}{l}df=n-1\\ =68-1\\ =67>60\end{array}$

Now, let us find out the critical value using table T in appendix F with the degree of freedom as:

  $t=2$

Now the confidence interval will be as:

  $\begin{array}{l}\overline{d}-t_{\alpha /2}\times \frac{s_d}{\sqrt{n}}\\ =-1.9118-2\times \frac{2.1282}{\sqrt{68}}\\ =-2.4280\\ \overline{d}+t_{\alpha /2}\times \frac{s_d}{\sqrt{n}}\\ =-1.9118+2\times \frac{2.1282}{\sqrt{68}}\\ =-1.3956\end{array}$

Thus, we are $90\%$ confident that the mean decrease in weight is between $-2.4280\text{ and }-1.3956$ pounds.