Chapter PVII, Problem 25RE

(a)

To determine

To explain which analysis would you use to answer the CEO’s question.

Answer to Problem 25RE

Regression analysis should be used.

Explanation of Solution

It is given in the question that Peninsula sells both cottage cheese and ice cream. The scatterplot for the same is shown. And also the two tables for regression analysis and t -test analysis for differences are also shown. As the CEO wants to know about the association between the cottage cheese and ice cream, then he should use the regression analysis. Thus, regression analysis would be used to answer the CEO’s question.

(b)

To determine

To explain what would you tell the CEO.

Explanation of Solution

It is given in the question that Peninsula sells both cottage cheese and ice cream. The scatterplot for the same is shown. And also the two tables for regression analysis and t -test analysis for differences are also shown. Thus, we would tell the CEO that there is a moderate relationship between cottage cheese and ice cream sales as for every million pounds of cottage cheese, $1.19$ million pounds of ice cream are sold on average.

(c)

To determine

To explain which analysis would you use to test whether the company sells more cottage cheese or ice cream in a typical year and why.

Answer to Problem 25RE

Mean difference test is used to test whether the company sells more cottage cheese or ice cream in a typical year.

Explanation of Solution

It is given in the question that Peninsula sells both cottage cheese and ice cream. The scatterplot for the same is shown. And also the two tables for regression analysis and t -test analysis for differences are also shown. Thus, in this we want to test whether the company sells more cottage cheese or ice cream in a typical year or not. Therefore, for this testing the mean difference is equal to zero will be used as the regression analysis would not be able to answer this question.

(d)

To determine

To explain what would you tell the CEO about this other result.

Answer to Problem 25RE

The company sells more cottage cheese than ice cream, on average.

Explanation of Solution

It is given in the question that Peninsula sells both cottage cheese and ice cream. The scatterplot for the same is shown. And also the two tables for regression analysis and t -test analysis for differences are also shown. Thus, from the mean difference test we have given that,

  $\begin{array}{l}P=0.0003\\ t=5.113\end{array}$

As we know that if the P-value is less than or equal to the significance level then the null hypothesis is rejected, so we have,

  $P<0.05\Rightarrow \text{Reject }{H}_0$

Thus, we conclude that there is evidence that the company sells more cottage cheese than ice cream, on average.

(e)

To determine

To explain what assumptions are you making in the analysis you chose in part (a) and in part (c).

Explanation of Solution

It is given in the question that Peninsula sells both cottage cheese and ice cream. The scatterplot for the same is shown. And also the two tables for regression analysis and t -test analysis for differences are also shown. Thus, for part (a), the assumption we will make in analysis is that it is a linear relationship, the residuals have a normal distribution and the residuals are independent with equal variations about the line.

And for the part (c), we will make the assumptions that the observations are independent, differences are approximately Normal and the sample is less than $10\%$ of all possible month’s data.

(f)

To determine

To find out how much ice cream do you predict Peninsula Creameries will sell.

Answer to Problem 25RE

  $71.32$ million pounds will be the sale of ice cream.

Explanation of Solution

It is given in the question that Peninsula sells both cottage cheese and ice cream. The scatterplot for the same is shown. And also the two tables for regression analysis and t -test analysis for differences are also shown.As from the regression analysis, the regression equation will come out to be as:

  $\begin{array}{l}\widehat{y}=b_0+b_{1}x\\ \Rightarrow \widehat{I}\mathrm{ce cream}=-26.5306+1.19334\times \text{Cottage cheese}\end{array}$

So, it is given next moth cottage cheese sales are $\text{82}$ million pounds thus, the sale of ice cream we predict will be:

  $\begin{array}{l}\widehat{I}\mathrm{ce cream}=-26.5306+1.19334\times \text{Cottage cheese}\\ =-26.5306+1.19334\times 82\\ =71.32\end{array}$

(g)

To determine

To give a $95\%$ confidence interval for the true slope of the regression equation of ice cream sales by cottage cheese sales.

Answer to Problem 25RE

$(0.09,2.29)$ is the $95\%$ confidence interval for the true slope of the regression equation of ice cream sales by cottage cheese sales.

Explanation of Solution

It is given in the question that Peninsula sells both cottage cheese and ice cream. The scatterplot for the same is shown. And also the two tables for regression analysis and t -test analysis for differences are also shown. Thus, the $95\%$ confidence interval for the true slope of the regression equation of ice cream sales by cottage cheese sales can be calculated from the regression analysis as:

  $\begin{array}{l}{b}_1\pm t*\times S{E}_1\\ =1.19334\pm 2.23\times 0.4936\\ =(0.09,2.29)\end{array}$

(h)

To determine

To explain what your interval means.

Explanation of Solution

It is given in the question that Peninsula sells both cottage cheese and ice cream. The scatterplot for the same is shown. And also the two tables for regression analysis and t -test analysis for differences are also shown. Thus, the $95\%$ confidence interval for the true slope of the regression equation of ice cream sales by cottage cheese sales can be calculated from the regression analysis as:

  $\begin{array}{l}{b}_1\pm t*\times S{E}_1\\ =1.19334\pm 2.23\times 0.4936\\ =(0.09,2.29)\end{array}$

Thus, from this regression, every million pounds of cottage cheese sold is associated with an increase in ice cream sales of between $0.09\text{ to 2}\text{.29}$ million pounds.