Chapter 12.1, Problem 7E

(a)

To determine

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

Answer to Problem 7E

  ${\text{SE}}_b=0.002456$ .

Explanation of Solution

In the question, researchers examined the nature in which it regulates the size of animal predators which remove higher proportion of the population than when the density of the prey is low. The explanatory variable is the number of perch in a confined area and the response variable is the proportion of perch killed by bass in two hours when the bass are allowed access to the perch. The scatterplot is given and it shows a linear relationship. And the residual plot for least square regression line is given. 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.”:

  ${\text{SE}}_b=0.002456$

This means that the slope of the sample regression line is expected to vary by about $0.002456$ the true slope of the population regression line.

(b)

To determine

To find the critical value for a $90\%$ confidence interval for the slope of the true regression line and then calculate the confidence interval.

Answer to Problem 7E

  $(0.004244,0.012894)$ .

Explanation of Solution

In the question, researchers examined the nature in which it regulates the size of animal predators which remove higher proportion of the population than when the density of the prey is low. The explanatory variable is the number of perch in a confined area and the response variable is the proportion of perch killed by bass in two hours when the bass are allowed access to the perch. The scatterplot is given and it shows a linear relationship. And the residual plot for least square regression line is given. The computer output is given for this data. Thus, it is given,

  $\begin{array}{l}n=16\\ b=0.008569\\ {\text{SE}}_b=0.002456\end{array}$

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 $c=90\%$ as:

  $t^{*}=1.761$

Thus, the confidence interval is as:

  $\begin{array}{l}b-t^{*}\times {\text{SE}}_b\\ =0.008569-1.761\times 0.002456\\ =0.004244\\ b+t^{*}\times {\text{SE}}_b\\ =0.008569+1.761\times 0.002456\\ =0.012894\end{array}$

(c)

To determine

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

Explanation of Solution

In the question, researchers examined the nature in which it regulates the size of animal predators which remove higher proportion of the population than when the density of the prey is low. The explanatory variable is the number of perch in a confined area and the response variable is the proportion of perch killed by bass in two hours when the bass are allowed access to the perch. The scatterplot is given and it shows a linear relationship. And the residual plot for least square regression line is given. The computer output is given for this data. Thus from part (b) we have,

  $(0.004244,0.012894)$

Thus, we are $90\%$ confident that the true slope of the population regression line lies between $(0.004244,0.012894)$ , thus we are $90\%$ confident that the proportion killed increases between $(0.004244,0.012894)$ per peach.

(d)

To determine

To explain the meaning of “ $90\%$ confident” in context.

Explanation of Solution

In the question, researchers examined the nature in which it regulates the size of animal predators which remove higher proportion of the population than when the density of the prey is low. The explanatory variable is the number of perch in a confined area and the response variable is the proportion of perch killed by bass in two hours when the bass are allowed access to the perch. The scatterplot is given and it shows a linear relationship. And the residual plot for least square regression line is given. The computer output is given for this data. The “ $90\%$ confident” in this context means that in $90\%$ of all possible samples the confidence interval will contain the true value of the population parameter.