Chapter 12.1, Problem 8E

(a)

To determine

To interpret the value of ${\text{SE}}_b$ in context.

Answer to Problem 8E

  ${\text{SE}}_b=0.0024$ .

Explanation of Solution

In the question researchers examined how well does the number of beers a person drinks predict his or her blood alcohol content. Sixteen volunteers with an initial BAC of zero drank randomly assigned number of cans of beer. Least squared regression line was performed on the data. A residual plot and a histogram of the residual are shown in the question. The computer output is given for this data. ${\text{SE}}_b$ is the standard deviation of the slope which is given in the row with “Perch” and in the column with “Stdev.”:

  ${\text{SE}}_b=0.0024$

This means that the slope of the sample regression line is expected to vary by about $0.0024$ the true slope of the population regression line.

(b)

To determine

To find the critical value for a $99\%$ confidence interval for the slope of the true regression line and then calculate the confidence interval.

Answer to Problem 8E

  $(0.0108192,0.0251088)$ .

Explanation of Solution

In the question researchers examined how well does the number of beers a person drinks predict his or her blood alcohol content. Sixteen volunteers with an initial BAC of zero drank randomly assigned number of cans of beer. Least squared regression line was performed on the data. A residual plot and a histogram of the residual are shown in the question. The computer output is given for this data. Thus, it is given,

  $\begin{array}{l}n=16\\ b=0.017964\\ {\text{SE}}_b=0.0024\end{array}$

The degrees of freedom is the sample size decreased by $2$ :

  $\begin{array}{l}df=n-2\\ =16-2\\ =14\end{array}$

The critical t -value can be found in table B in the row of $df=14$ and in the column of $c=99\%$ as:

  $t^{*}=2.977$

Thus, the confidence interval is as:

  $\begin{array}{l}b-t^{*}\times {\text{SE}}_b\\ =0.017964-2.977\times 0.0024\\ =0.0108192\\ b+t^{*}\times {\text{SE}}_b\\ =0.017964+2.977\times 0.0024\\ =0.0251088\end{array}$

(c)

To determine

To interpret the interval from part (b) in context.

Explanation of Solution

In the question researchers examined how well does the number of beers a person drinks predict his or her blood alcohol content. Sixteen volunteers with an initial BAC of zero drank randomly assigned number of cans of beer. Least squared regression line was performed on the data. A residual plot and a histogram of the residual are shown in the question. The computer output is given for this data. Thus from part (b) we have,

  $(0.0108192,0.0251088)$

Thus, we are $99\%$ confident that the true slope of the population regression line lies between $(0.0108192,0.0251088)$ .

(d)

To determine

To explain the meaning of “ $99\%$ confident” in context.

Explanation of Solution

In the question researchers examined how well does the number of beers a person drinks predict his or her blood alcohol content. Sixteen volunteers with an initial BAC of zero drank randomly assigned number of cans of beer. Least squared regression line was performed on the data. A residual plot and a histogram of the residual are shown in the question. The computer output is given for this data. The “ $99\%$ confident” in this context means that in $99\%$ of all possible samples the confidence interval will contain the true value of the population parameter.