Chapter PII, Problem 20RE

(a)

To determine

To find out does the scatterplot looks better.

Answer to Problem 20RE

Yes, it looks better.

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, the researchers are recreating the model by squaring the diameter since age is measured in square units. So, the scatterplot will be constructed as:

  

Thus, now the scatterplot is straightened and yes it looks better than the previous scatterplot as the points are more near and in a straight line.

(b)

To determine

To create a model that predicts age from the square of the diameter.

Answer to Problem 20RE

   .

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, the researchers are recreating the model by squaring the diameter since age is measured in square units. Thus, 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 follows:

  

(c)

To determine

To check the residual plot for this new model and explain is this model more appropriate and why.

Answer to Problem 20RE

Yes, it is more 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. Thus, the researchers are recreating the model by squaring the diameter since age is measured in square units.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, we can say by looking at the new residual plot that yes, this model is more appropriate model because the residual do not form a curve anymore as in the previous plot.

(d)

To determine

To estimate the age of a tree $\text{18}$ inches in diameter.

Answer to Problem 20RE

The estimated age of the tree is $43.8576$ years old.

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, the researchers are recreating the model by squaring the diameter since age is measured in square units. And the regression line is as:

  

Thus, the estimated the age of a tree $\text{18}$ inches in diameter is calculated as:

  

The estimated age of the tree is $43.8576$ years old.