Chapter 12.2, Problem 42E

(a)

To determine

To explain would an exponential model or a power model provide a better description of the relationship between bounce number and height.

Answer to Problem 42E

The exponential model.

Explanation of Solution

In the question the data about the bounce number and height in given. So, the model that gives the better description of the relationship between the bounce number and height needs to have a roughly linear pattern in the scatterplot. Thus, the exponential model will provide a better description of the relationship between bounce number and height since the top graph corresponds with an exponential model and the bottom graph corresponds with a power model.

(b)

To determine

To give the equation of the least squares regression line.

Answer to Problem 42E

  $\log \widehat{y}=0.45374-0.117160x$ .

Explanation of Solution

Now, it is given in the question that variable x be the bounce and the variable y be height. Thus, the general equation will be as:

  $\log \widehat{y}=a+bx$

And the slope and the constant of the regression line is give in the question as:

  $\begin{array}{l}a=0.45374\\ b=-0.117160\end{array}$

Thus, the regression line is as:

  $\begin{array}{l}\log \widehat{y}=a+bx\\ \Rightarrow \log \widehat{y}=0.45374-0.117160x\end{array}$

(c)

To determine

To use your model in part (b) to predict the highest point the ball reaches on its seventh bounce.

Answer to Problem 42E

It is $0.43015$ feet.

Explanation of Solution

Now, the regression line is as:

  $\begin{array}{l}\log \widehat{y}=a+bx\\ \Rightarrow \log \widehat{y}=0.45374-0.117160x\end{array}$

Thus, we need to predict the highest point the ball reaches on its seventh bounce thus, evaluating the equation in part (b), we have,

  $\begin{array}{l}\log \widehat{y}=a+bx\\ \Rightarrow \log \widehat{y}=0.45374-0.117160x\\ \Rightarrow \log \widehat{y}=0.45374-0.117160\times 7\\ \Rightarrow \log \widehat{y}=-0.36638\\ \Rightarrow \widehat{y}={10}^{-0.36638}\\ \Rightarrow \widehat{y}=0.43015\end{array}$

(d)

To determine

To explain do you expect your prediction part (c) to be too high, too low or just right.

Answer to Problem 42E

The prediction is too low.

Explanation of Solution

Now, the regression line is as:

  $\begin{array}{l}\log \widehat{y}=a+bx\\ \Rightarrow \log \widehat{y}=0.45374-0.117160x\end{array}$

And the residual plot of the linear regression in part (b) is shown in the question. Thus, we note that for bounce $\text{3,4 and 5}$ , there is an increasing pattern in the residual plot and thus the residual will most likely increase even more for bounce seventh. And the residual is then expected to be positive. Therefore, the residual is the actual value decreased by the predicted value. Since the residual is expected to be positive, the predicted value is then expected to be lower than the actual value. Thus, our prediction is too low.