Chapter 23, Problem 15E

(a)

To determine

To explain would a $90\%$ confidence interval be wider or narrower than the $98\%$ confidence interval you calculated before.

Answer to Problem 15E

It will be narrower.

Explanation of Solution

It is given in the question that the researcher described in previous exercise also measured the body temperature of that randomly selected group of adults. And all conditions are met.

It is given:

  $\begin{array}{l}n=52\\ \overline{x}=98.285\\ s=0.6824\end{array}$

Thus, we have, a $90\%$ confidence interval be narrower than the $98\%$ confidence interval you calculated before because if the confidence level is decreased, then we become less sure that the confidence interval contains the true population mean and thus the confidence interval would become narrower.

(b)

To determine

To find out what are the advantages and disadvantages of the $98\%$ confidence interval.

Explanation of Solution

It is given in the question that the researcher described in previous exercise also measured the body temperature of that randomly selected group of adults. And all conditions are met.

It is given:

  $\begin{array}{l}n=52\\ \overline{x}=98.285\\ s=0.6824\end{array}$

Now, the advantage of the $98\%$ confidence interval is that more confident that the confidence interval contains the true population mean.The advantage of the $98\%$ confidence interval is thatconfidence interval is wider than a confidence interval with a lower confidence level.

(c)

To determine

To explain how would you expect the $98\%$ confidence interval to change, if we conduct further research this time using a sample of $500$ adults.

Answer to Problem 15E

It will become narrower.

Explanation of Solution

It is given in the question that the researcher described in previous exercise also measured the body temperature of that randomly selected group of adults. And all conditions are met.

It is given:

  $\begin{array}{l}n=52\\ \overline{x}=98.285\\ s=0.6824\end{array}$

Thus, if we conduct further research this time using a sample of $500$ adults then if the sample size increases, then we have more information about the population and thus the estimates will be more precise.If the estimates are more precise, the then confidence interval will become narrower.

(d)

To determine

To explain how large a sample might allow you to estimate the mean body temperature to within $0.1$ degrees with $98\%$ confidence.

Answer to Problem 15E

The large sample size is $253$ .

Explanation of Solution

It is given in the question that the researcher described in previous exercise also measured the body temperature of that randomly selected group of adults. And all conditions are met.

It is given:

  $\begin{array}{l}n=52\\ \overline{x}=98.285\\ s=0.6824\end{array}$

Now, we have to estimate the mean body temperature to within $0.1$ degrees with $98\%$ confidence, this can be done as:

It is given,

  $\begin{array}{l}E=0.1\\ \sigma =0.6824\end{array}$

Mow, the formula to calculate the sample size will be:

  $n={(\frac{z_{\alpha /2}\sigma }{E})}^2$

Thus, for confidence level, $1-\alpha =0.98$ , we have,

  $\begin{array}{l}{z}_{\alpha /2}=z_{0.01}\\ \Rightarrow z_{\alpha /2}=2.33\end{array}$

Now, the sample size is:

  $\begin{array}{l}n={(}^{\frac{z_{\alpha /2}\sigma }{E}}\\ ={(}^{\frac{2.33\times 0.6824}{0.1}}\\ \approx 253\end{array}$

Thus, the large sample size is $253$ .