Chapter PVI, Problem 36RE

(a)

To determine

To create and interpret a $90\%$ confidence interval.

Answer to Problem 36RE

We are $90\%$ confident that the mean number of baby hamster per liter is between $7.1077\text{ and 8}\text{.3323}$ .

Explanation of Solution

It is given in the question that the researchers want to find out how large are hamster litters. Thus, to find the confidence interval for mean we will use the calculator $\text{TI}-89$ . In the STAT Intsmenu, choose $2:$ TInterval. Specify whether you are using data stored in a list or whether you will enter the mean, standard deviation, and sample size. You must also specify the desired level of confidence. So, we have,

  $\begin{array}{l}\overline{x}=7.72\\ s=2.5\\ n=47\\ c=90\%=0.90\end{array}$

Thus, by using the calculator $\text{TI}-89$ , we have,

  $\begin{array}{l}t=1.679\\ \text{Margin of error}=0.6123\end{array}$

And the confidence interval is as:

  $(7.1077,8.3323)$ .

Thus, we are $90\%$ confident that the mean number of baby hamster per liter is between $7.1077\text{ and 8}\text{.3323}$ .

(b)

To determine

To explain would a $98\%$ confidence interval have a larger or smaller margin of error.

Answer to Problem 36RE

  $98\%$ confidence interval have a larger margin of error.

Explanation of Solution

It is given in the question that the researchers want to find out how large are hamster litters. Thus, we can say that increasing the confidence level, makes us more confident that the confidence interval contains the population mean and thus the confidence interval needs to contain more possible values for the population mean, which makes the confidence interval wider. Since the margin of error is half the width of the confidence interval the margin of error will be larger. So, $98\%$ confidence interval have a larger margin of error.

(c)

To determine

To find out how many litters must be used to estimate the average litter size to within one baby hamster with $95\%$ confidence.

Answer to Problem 36RE

  $25$ litters must be used to estimate the average litter size to within one baby hamster with $95\%$ confidence.

Explanation of Solution

It is given in the question that the researchers want to find out how large are hamster litters. Thus, as the margin of error is one, then the number of litters must be used to estimate the average litter size to within one baby hamster with $95\%$ confidence is calculated as:

  $\begin{array}{l}ME=t*\times \frac{s}{\sqrt{n}}\\ \Rightarrow 1=1.96\times \frac{2.5}{\sqrt{n}}\\ \Rightarrow \frac{1}{1.96}=\frac{2.5}{\sqrt{n}}\\ \Rightarrow \sqrt{n}=\frac{2.5\times 1.96}{1}\\ \Rightarrow n=25\end{array}$

Thus, $25$ litters must be used to estimate the average litter size to within one baby hamster with $95\%$ confidence.