Chapter 15, Problem 24E

(a)

To determine

To find: The equation for the weight of the rat after x weeks with its birth weight of 110 g.

Answer to Problem 24E

Solution: The required equation is

$y\left(\text{weight}\right)=110+39x\left(\text{weeks}\right)$

Explanation of Solution

Calculation:

The regression equation is the statistical method that models the data to predict a response variable from the explanatory variables. The equation of the regression line is of the form

$y=a+bx$

The number b is the slope of the regression line that indicates the amount by which the response variable changes when the explanatory variable increases by one unit.

Hence, in the provided problem, 39 g is the slope since this is the amount by which the weight of the rat increases per week. The constant a is the birth weight which is 110 g. Therefore, the equation is obtained as follows:

$y\left(\text{weight}\right)=110+39x\left(\text{weeks}\right)$

Therefore, this is the required equation for the rat’s weight after x weeks.

To find: The slope of the obtained equation.

Solution: The slope of the obtained equation is 39.

Explanation:

Calculation:

The equation of the regression line for the rat’s weight after x weeks is obtained as $y\left(\text{weight}\right)=110+39x\left(\text{weeks}\right)$ in the previous part.

This shows that the slope of the line is 39 since this the quantity by which the weight of the rat increases per week.

Therefore, the slope of the line is 39.

Interpretation: The slope of the regression equation is 39, which indicates that with a unit change in the week the weight increases by 39 units.

(b)

To determine

To graph: The regression line for the obtained equation from the birth to 10 weeks of age.

Explanation of Solution

Calculation:

The equation for the weight of the rat after x weeks with its birth weight of 110 g is obtained as $y\left(\text{weight}\right)=110+39x\left(\text{weeks}\right)$ in the previous part.

Choose two values of x as 0 and 10 for the birth and 10 weeks, respectively. To draw the regression line, use the provided equation to predict the weight y for the number of weeks as 0 as follows:

$\begin{array}{c}y\left(\text{weight}\right)=110+39x\left(\text{weeks}\right)\\ =110+\left(39\times 0\right)\\ =110+0\\ =110\end{array}$

The predicted weight y for the number of weeks as 10 is calculated as follows:

$\begin{array}{c}y\left(\text{weight}\right)=110+39x\left(\text{weeks}\right)\\ =110+\left(39\times 10\right)\\ =110+390\\ =500\end{array}$

Therefore, the points are $\left(0,110\right)\text{ and }\left(10,500\right)$. Use Minitab to plot these points and to draw a line through these points.

Graph: The steps followed to obtain the regression line are as follows:

Step 1: Open the Minitab file and enter the obtained two points in the worksheet.

Step 2: Go to Graph and then select Scatterplot.

Step 3: Select With regression and click Ok.

Step 4: Enter “Weight” in the Y variables column and “Weeks” in the X variables column and click Ok.

The regression line appears as obtained in the Minitab file.

In the graph, the horizontal axis shows the weeks and the vertical axis shows the weights.

Interpretation: The graph clearly displays a positive association between the number of weeks and the weight. Thus, as the number of weeks increases the weight also increases.

(c)

To determine

Can the obtained line be willingly used to predict the rat’s weight at the age of 2 years.

Answer to Problem 24E

Solution: No, the obtained line cannot be used to predict the rat’s weight at the age of 2 years.

Explanation of Solution

It is provided that the birth weight of male white rat is 110 g and its weight increases exactly 39 grams per week. A least-squares regression equation to find the weight of rat from the number of weeks is obtained. However, the required rat’s weight at the age of 2 years cannot be predicted with the same equation as it will be outside the range of values.

To find: The weight of rat at the age of 2 years.

Solution: The predicted weight of rat at the age of 2 years is 4166 g.

Explanation:

Calculation:

The equation for the weight of the rat after x weeks with its birth weight of 110 g is obtained as $y\left(\text{weight}\right)=110+39x\left(\text{weeks}\right)$ in the previous part.

For 2 years, the number of weeks is 104. Substitute the number of weeks as 104 in the obtained equation to determine the predicted weight for 2 years of age.

$\begin{array}{c}y\left(\text{weight}\right)=110+39x\left(\text{weeks}\right)\\ =110+\left(39\times 104\right)\\ =110+4056\\ =4166\end{array}$

Therefore, the predicted rat’s weight at the age of 2 years would be 4166 g.

To explain: If the obtained results are reasonable or not.

Solution: The predicted rat’s weight of 4166 g at 2 years is not reasonable as there is no possibility of the weight of 4166 g for a rat. The rats do not grow at a constant rate throughout their lives.

Explanation: The predicted rat’s weight for 2 years is obtained as 4166 g. The rats are very small creatures and this weight is not reasonable in real world. The rats also do not grow at the same constant rate throughout their lives.

Therefore, it can be concluded that the obtained regression line is only reliable for “young” rats.