Chapter 8, Problem R8.2RE

a)

To determine

To find point estimate, margin of error, standard error and the sample standard deviation.

Answer to Problem R8.2RE

  $\overline{x}=450,E=20,SE=11.3095,s=61.9447$

Explanation of Solution

Given:

The 95% confidence interval for mean:

(430, 470)

Sample size = n =30

Formula:

Sample mean:

  $\overline{x}=\frac{Lowerlimit+Upperlimit}{2}$

Margin of error:

  $E=\frac{Upperlimit-Lowerlimit}{2}$

Standard error:

  $SE=\frac{s}{\sqrt{n}}$

Using the confidence interval,

Sample mean is,

  $\overline{x}=\frac{430+470}{2}=450$

Margin of error is,

  $E=\frac{470-430}{2}=20$

Now first need to find critical-t value:

  $t_{\alpha /2}=2.042$ ..Using excel formula, =TINV(0.05,29)

Standard deviation can be found using the formula of margin of error,

  $\begin{array}{l}E=t_{\alpha /2}\times \frac{s}{\sqrt{n}}\\ 20=2.042\times \frac{s}{\sqrt{30}}\\ s=\frac{20}{0.3229}=61.9447\end{array}$

So, standard deviation = 61.9447

Now, standard error is,

  $SE=\frac{61.9447}{\sqrt{30}}=11.3095$

b)

To determine

To check whether the interpretation of confidence interval is incorrect or not.

Answer to Problem R8.2RE

The interpretation is incorrect.

Explanation of Solution

Given:

Reporter’s comment: 95% of this company’s AA batteries last between 430 and 470 minutes.

We cannot say sure about the confidence interval as battery will either last between 430 and 470 minutes or not. The correct interpretation is, we are 95% confident that the mean battery life time is between 430 and 470 minutes.

c)

To determine

To check whether the claim is correct or not.

Answer to Problem R8.2RE

Not agree with claim.

Explanation of Solution

Given:

Claim: There is 95% probability that the mean lifetime will fall between 430 and 470 minutes.

We cannot say sure about the confidence interval as battery will either last between 430 and 470 minutes or not. If the sample mean 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%. Hence, do not agree with the claim.

d)

To determine

To give the interpretation of confidence interval.

Answer to Problem R8.2RE

We are 95% confident that the true population mean is between 430 and 470 minutes.

Explanation of Solution

Given:

The 95% confidence interval for mean:

(430, 470)

We are 95% confident that the true population mean is between 430 and 470 minutes.