Chapter 12.2, Problem 38E

(a)

To determine

To make a reasonable accurate scatterplot of the data using the depth as the explanatory variable and describe what you see.

Explanation of Solution

Now, the scatterplot of the data using the depth as the explanatory variable is as follows:

  

Thus, by looking at the scatterplot we can see that,

Direction: Negative, because the scatterplot slopes downwards.

Form: Curved, because the points do not lie on a straight line.

Strength: Strong, because all points lie very close together in the same pattern.

(b)

To determine

To explain why it would be reasonable to use an exponential model to describe the relationship between the light intensity and depth.

Explanation of Solution

Now, the data of the light intensity and depth is given in the question. Thus, we can say that it would be reasonable to use an exponential model to describe the relationship between the light intensity and depth because the corresponding scatterplot has a roughly linear pattern with no notable strong outliers.

(c)

To determine

To give the equation of the least squares regression line.

Answer to Problem 38E

  $\ln \widehat{y}=6.78910-0.333021x$ .

Explanation of Solution

Now, it is given that x is the depth and y is the light intensity. Thus the transformation is the natural logarithm of the light intensity as $\ln y$ .

Thus, as we know that the general equation of the regression line is as follows:

  $\ln \widehat{y}=a+bx$

And the slope and the constant given in the computer output is as:

  $\begin{array}{l}a=6.78910\\ b=-0.333021\end{array}$

Thus, the regression line is as follows:

  $\begin{array}{l}\ln \widehat{y}=a+bx\\ \Rightarrow \ln \widehat{y}=6.78910-0.333021x\end{array}$

(d)

To determine

To use your model to predict the light intensity at $12$ meters and explain did this surprise you.

Answer to Problem 38E

  $\widehat{y}=16.3275$ and yes, it is surprising.

Explanation of Solution

Now, the regression line is as follows:

  $\ln \widehat{y}=6.78910-0.333021x$

Thus, we will evaluate the equation as:

  $\begin{array}{l}\ln \widehat{y}=a+bx\\ \Rightarrow \ln \widehat{y}=6.78910-0.333021x\\ \Rightarrow \ln \widehat{y}=6.78910-0.333021(12)\\ \Rightarrow \ln \widehat{y}=2.792848\\ \Rightarrow \widehat{y}=e^{2.792848}\\ =16.3275\end{array}$

Thus the residual is as:

  $\begin{array}{l}\text{Residual}=\ln y-\ln \widehat{y}\\ =\ln 16.2-2.792848\\ =-0.007837\end{array}$

Since the residual is larger than $s=0.000055$ it is a surprising result for us.