Chapter PII, Problem 17RE

(a)

To determine

To describe what you see in the scatterplot of intervals between eruptions versus duration of the previous eruption.

Explanation of Solution

It is given in the question that there is evidence that eruptions of old faithful can best be predicted by knowing the duration of the previous eruption. The scatterplot of eruption versus duration of the previous eruption is also given. Thus, as we see in the scatterplot the points are moving in upward direction so there is a positive relationship also it is fairly moderate to high in strength and there are two main groups of points.

(b)

To determine

To write the equation of the line of best fit.

Answer to Problem 17RE

   .

Explanation of Solution

It is given in the question that there is evidence that eruptions of old faithful can best be predicted by knowing the duration of the previous eruption. Now, from the regression analysis we have,

  $\begin{array}{l}\text{Intercept}=33.9668\\ \text{Duration}=10.3582\\ R^2=77\%\end{array}$

Thus, the regression line will be as:

  

(c)

To determine

To carefully explain what the slope of the line means in this context.

Explanation of Solution

It is given in the question that there is evidence that eruptions of old faithful can best be predicted by knowing the duration of the previous eruption. And the regression line is as:

  

Thus, the slope of the line means in this context that every minute the eruption lasts the interval until the next eruption will be about $10.3582$ minutes longer.

(d)

To determine

To explain how accurate do you expect predictions based on this model to be.

Answer to Problem 17RE

It is fairly accurate.

Explanation of Solution

It is given in the question that there is evidence that eruptions of old faithful can best be predicted by knowing the duration of the previous eruption. And the regression line is as:

  

Thus, we expect that the prediction based on this model is to be fairly accurate because $R^2=77\%$ of the variability of the interval length is accounted for by the duration.

(e)

To determine

To explain how long do you predict you will have to wait to see the next eruption if you just witnessed an eruption that lasted four minutes.

Answer to Problem 17RE

We have to wait $75.4$ minutes to see the next eruption if you just witnessed an eruption that lasted four minutes.

Explanation of Solution

It is given in the question that there is evidence that eruptions of old faithful can best be predicted by knowing the duration of the previous eruption. And the regression line is as:

  

Thus, we will predict the duration to wait to see the next eruption if you just witnessed an eruption that lasted four minutes is calculated as:

  

So, we have to wait $75.4$ minutes to see the next eruption if you just witnessed an eruption that lasted four minutes.

(f)

To determine

To define a residual by using the example mentioned.

Answer to Problem 17RE

There is a difference of $3.6$ minutes.

Explanation of Solution

It is given in the question that there is evidence that eruptions of old faithful can best be predicted by knowing the duration of the previous eruption. And the regression line is as:

  

Thus, it is given that the next eruption came in $79$ minutes and we have predicted that it came in $75.4$ minutes, so the residual will be as:

  $\begin{array}{l}\text{Residual}=\text{Observed}-\text{Predicted}\\ =79-75.4\\ =3.6\end{array}$

Thus, there is a difference of $3.6$ minutes between them.