Chapter PII, Problem 19RE

(a)

To determine

To find the correlation between diameter and age and explain does this suggest that a linear model may be appropriate.

Answer to Problem 19RE

No, it cannot say a model is appropriate.

Explanation of Solution

It is given in the question that the researchers want to find out can we estimate the tree’s age simply from its diameter or not. Thus, the diameter and ages of $27$ trees are given in the table. So, we will find out the correlation for this by using the excel function for the two arrays by the function:

  $=\text{CORREL}(\text{array1, array2})$

Thus, the calculation for the correlation is as follows:

Correlation =

=CORREL(C32:C58,D32:D58)

Thus, the result for the correlation will be as:

Correlation =

0.8882

So, we have $r=0.888$ and by the correlation we cannot tell about that the linear is appropriate or not as correlation cannot say a model is appropriate.

(b)

To determine

To create a scatterplot and describe the association.

Explanation of Solution

It is given in the question that the researchers want to find out can we estimate the tree’s age simply from its diameter or not. Thus, the diameter and ages of $27$ trees are given in the table. Thus, now we will construct a scatterplot for the diameter and ages for the trees by using the Excel as:

  

Thus, by looking at the scatterplot we can say that the points in the scatterplot is from left to right in an upward direction so there is a positive association and there is fairly linear and moderately strong association between the variables.

(c)

To determine

To create a linear model.

Answer to Problem 19RE

The linear model is .

Explanation of Solution

It is given in the question that the researchers want to find out can we estimate the tree’s age simply from its diameter or not. Thus, the diameter and ages of $27$ trees are given in the table. Thus, for constructing a linear model we will use Excel as we will first select the data in Excel and then go to the insert tab from where we will select the scatterplot option in the charts and when the scatterplot is constructed we will go to the design tab under the chart tools, from where we will select the quick layout option and select the layout with the equation option then the linear model is constructed in the scatterplot as:

  

Thus, the regression line is as:

  

(d)

To determine

To check the residuals and explain why a linear model is probably not appropriate.

Explanation of Solution

It is given in the question that the researchers want to find out can we estimate the tree’s age simply from its diameter or not. Thus, the diameter and ages of $27$ trees are given in the table. The residuals for the following data are constructed by using the Excel going to the data tab and then selecting the option of data analysis. And then we will select the regression from the dialogue box that will appear. Thus putting the values in the second dialogue box for regression and selecting the residual plot the graph is as follows:

  

Thus, by looking at the scatterplot we can say that the residuals show an upward curve with a couple of outliers so the linear model is not appropriate.

(e)

To determine

To explain would it generally overestimate or underestimate the ages of very large trees if you used this model.

Answer to Problem 19RE

It will generally underestimate the ages of very large trees if you used this model.

Explanation of Solution

It is given in the question that the researchers want to find out can we estimate the tree’s age simply from its diameter or not. Thus, the diameter and ages of $27$ trees are given in the table. Thus, we can say that it will generally underestimate the ages of very large trees if you used this model because $\text{5 of the 7}$ last trees have positive residuals.