Chapter PII, Problem 40RE

(a)

To determine

To explain why are the re-expressed data better for regression.

Explanation of Solution

It is given in the question researchers want to find how are a company’s profits related to its sales and the data from large U.S. corporations are examined. All the amounts are in millions of dollars. The histograms of profits and sales and histograms of logarithms of profits and sales are also given. Thus, the re-expressed data better for regression because it allows us to better see the shape of the graph.

(b)

To determine

To explain do you think this model is appropriate.

Answer to Problem 40RE

Yes, the model is appropriate.

Explanation of Solution

It is given in the question researchers want to find how are a company’s profits related to its sales and the data from large U.S. corporations are examined. All the amounts are in millions of dollars.The histograms of profits and sales and histograms of logarithms of profits and sales are also given. Thus, the scatterplot shows an association that is positive, linear, and strong. Therefore, the model is appropriate because the regression shows no pattern.

(c)

To determine

To write the equation for regression.

Answer to Problem 40RE

  $\text{Log(profit)}=-0.106259+0.647798\times \log (\text{sales)}$ .

Explanation of Solution

It is given in the question researchers want to find how are a company’s profits related to its sales and the data from large U.S. corporations are examined. All the amounts are in millions of dollars. And the regression analysis is as:

  $\begin{array}{l}{R}^2=48.1\%\\ \text{Intercept}=-0.106259\\ \text{Internet use}=0.647798\end{array}$

Thus, the equation will be as:

  $\begin{array}{l}\widehat{y}=a+bx\\ \Rightarrow \text{Log(profit)}=-0.106259+0.647798\times \log (\text{sales)}\end{array}$

(d)

To determine

To use your equation to estimate profits earned by a company with scales of $2.5$ billion dollars.

Answer to Problem 40RE

The profit is $124.4285$ million dollars.

Explanation of Solution

It is given in the question researchers want to find how are a company’s profits related to its sales and the data from large U.S. corporations are examined. All the amounts are in millions of dollars. And the regression analysis is as:

  $\begin{array}{l}{R}^2=48.1\%\\ \text{Intercept}=-0.106259\\ \text{Internet use}=0.647798\end{array}$

Thus, the equation will be as:

  $\text{Log(profit)}=-0.106259+0.647798\times \log (\text{sales)}$

Thus, using our equation to estimate profits earned by a company with scales of $2.5$ billion dollars as:

  $\begin{array}{l}\text{Log(profit)}=-0.106259+0.647798\times \log (2500)\\ \Rightarrow \text{Log(profit)}=-0.106259+0.647798\times 3.39794\\ \Rightarrow \text{Log(profit)}=2.0949197\\ \Rightarrow \text{profit}={10}^{2.0949197}\\ \Rightarrow \text{profit}=124.4285\end{array}$

Thus, the profit is $124.4285$ million dollars.