Chapter 10, Problem 10.59RE

(a)

To determine

To Calculate: the sample mean and standard deviation.

Answer to Problem 10.59RE

Mean=7.5 hours or 45 minutes

Standard deviation = 64.5363 minutes

Explanation of Solution

Given:

  $\begin{array}{c}\overline{x}=7.5\text{hours}\\ \text{E=20}\text{minutes}\\ \text{n=40}\end{array}$

95% confidence interval: 7.5 hour =20 minutes

Formula used:

  $E=z_{\alpha /2}\times \frac{s}{\sqrt{n}}$

Calculation:

The sample mean is the first value given in the confidence interval:

  $\overline{x}=7.5\text{hours =450 minutes}$

Since the sample size of $\text{n=40}$ is large, using the normal distribution. For confidence level $1-\alpha =0.95,$

  $z_{\alpha /2}=z_{0.025}$ using the normal probability table

  $z_{\alpha /2}=1.96$

Margin of error:

  $E=z_{\alpha /2}\times \frac{s}{\sqrt{n}}$

Solving the formula to the sample standard deviation

  $s=\frac{\sqrt{n}\cdot E}{z_{\alpha /2}}=\frac{\sqrt{40}\cdot 20}{1.96}=64.5363$

(b)

To determine

To Explain: that a researcher translates the statistical announcement into plain English, comment on this interpretation.

Answer to Problem 10.59RE

Incorrect interpretation

Explanation of Solution

The interpretation of the researcher is not correct, because the interpretation is a single observation, where the confidence interval is about the population mean.

(c)

To determine

To Explain: that about the claim agree or not.

Answer to Problem 10.59RE

No

Explanation of Solution

The interpretation researcher is not correct, because the interpretation is about the sample mean, where the confidence interval is about the population mean.

(d)

To determine

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

Answer to Problem 10.59RE

There is 95%confident that the mean life time of all AA batteries is between 430 minutes and 470 minutes.

Explanation of Solution

The statistically correct interpretation of the confidence interval is that could be published in a newspaper report. There is 95%confident that the mean life time of all AA batteries is between 430 minutes and 470 minutes.