Chapter 9, Problem 26E

(a)

To determine

To explain what does the $R^2$ mean for this regression.

Explanation of Solution

The researcher wants to study that are the swimmers getting faster or slower if the swim many times. The regression of the crossing time against the year of crossing and the residual plot is given in the question. Also, it is given that:

  $\begin{array}{l}{R}^2=1.3\%\\ s=443.8\\ \text{Intercept}=-8950.40\\ \text{Year}=5.14171\end{array}$

Thus, $R^2$ mean for this regression that the linear model accurately predicts the time $1.3\%$ of the time and it also explains the variation between the variables, that is between the crossing time and the year.

(b)

To determine

To explain are the swimmers getting faster or slower.

Answer to Problem 26E

They are getting slower.

Explanation of Solution

The researcher wants to study that are the swimmers getting faster or slower if the swim many times. The regression of the crossing time against the year of crossing and the residual plot is given in the question. Also, it is given that:

  $\begin{array}{l}{R}^2=1.3\%\\ s=443.8\\ \text{Intercept}=-8950.40\\ \text{Year}=5.14171\end{array}$

Now, from this we see that the swimmers are getting slower because more of the residuals are becoming negative. This means that there are more predicted times that are faster than actual times, indicating that swimmers are slowing down.

(c)

To determine

To explain would removing it change the model a lot.

Answer to Problem 26E

Yes, the model will change.

Explanation of Solution

The researcher wants to study that are the swimmers getting faster or slower if the swim many times. The regression of the crossing time against the year of crossing and the residual plot is given in the question. Also, it is given that:

  $\begin{array}{l}{R}^2=1.3\%\\ s=443.8\\ \text{Intercept}=-8950.40\\ \text{Year}=5.14171\end{array}$

Now, it is mentioned that the outlier seen in the residual plot is a crossing time by Vicki Keith in which she swam a round trip, north to south and then back again. Thus, by removing this outlier the model will change because the linear model fit the rest of the data much more accurately.