Chapter 8, Problem 3CRE

(a)

To determine

To determine the sample mean and standard deviation.

Answer to Problem 3CRE

The mean is $450$ and standard deviation is $61.9447$ .

Explanation of Solution

It is given in the question,

  $\begin{array}{l}n=40\\ 95\%\text{ confidence interval: 430 to 470 minutes}\end{array}$

The sample mean lies in the middle of the confidence interval, thus we have,

  $\begin{array}{l}\overline{x}=\frac{430+470}{2}\\ =450\end{array}$

The margin of error will be as:

  $\begin{array}{l}ME=450-430\\ =20\end{array}$

Now, we will find out the critical value using table B with degree of freedom as:

  $\begin{array}{l}df=40-1\\ =39>30\\ t_{\alpha /2}=t_{0.025}\\ =2.042\end{array}$

We also know that the margin of error of a confidence interval for the population mean is:

  $\begin{array}{l}ME=t_{\alpha /2}\times \frac{s}{\sqrt{n}}\\ =2.042\times \frac{s}{\sqrt{40}}\\ =0.3229s\end{array}$

Thus, from above we have,

  $\begin{array}{l}20=0.3229s\\ \Rightarrow s=\frac{20}{0.3229}\\ =61.9447\end{array}$

(b)

To determine

To comment on the interpretation.

Explanation of Solution

The interpretation given in the question is correct because the battery will either last between $\text{430 to 470}$ minutes or not, thus the probability is either zero or one. A better interpretation will be that we are $95\%$ confident that the mean battery life time is between $\text{430 to 470}$ minutes.

(c)

To determine

To explain do you agree with the statement given in the question.

Answer to Problem 3CRE

No, we do not agree.

Explanation of Solution

No, we do not agree with the statement given in the question because if the sample mean on which the confidence interval is based, was very unlikely to occur by chance for a given population mean then the probability that a different sample mean is in the same confidence interval will be less than $95\%$ .

(d)

To determine

To find out a statistically correct interpretation of the confidence interval that could be published in a newspaper report.

Explanation of Solution

It is given in the question,

  $\begin{array}{l}n=40\\ 95\%\text{ confidence interval: 430 to 470 minutes}\end{array}$

And the mean is $450$ and standard deviation is $61.9447$ . Thus, a statistically correct interpretation of the confidence interval that could be published in a newspaper report could be that we are $95\%$ confident that the population mean is between $\text{430 to 470}$ minutes.