Question

Tree Diameter and Age To estimate ages of trees, forest rangers use a linear model that relates tree diameter to age. The model is useful because tree diameter is much easier to measure than tree age (which requires special tools for extracting a representative cross section of the tree and counting the rings). To find the model, use the data in the table, which were collected for a certain variety of oaks.

(a) Make a scatter plot of the data.

(b) Find and graph a linear function that models the data.

(c) Use the model to estimate the age of an oak whose diameter is 18 in.

Short answer

(a) The scatter plot is

(b) The regression line equation is $y=6.45x-0.15$and the graph is,

(c) Age of tree is when its diameter is 18 in.

Step-by-step solution

Part a. Step 1. Concept of plotting the data.

Among the given variables we assign one of them is abscissa and the other at ordinate. Thus, we get some coordinates,

To get the scatter plot we plot the point.

Part a. Step 2. Assumption of variables.

Let us assign the diameter as abscissa and age as ordinate.

Part a. Step 3. Plotting the graph.

The given table is,

So, the scatter plot is,

Part b. Step 1. Concept of regression line.

A regression line is a simple line that best fits the data.

Part b. Step 2. Given conditions.

The given table is,

So, the scatter plot is,

Part b. Step 3. Find the equation of line.

By using graphing ability, the equation of regression line here is $y=6.45x-0.15$.

Part c. Step 1. Concept of finding of variables.

The value of one is known and the other is not. Then we substitute the known value in the equation. So, we get the value of the other variable.

Part c. Step 2. Given conditions.

Since, the regression lineis$y=6.45x-0.15$

We have to find the age of tree when its diameter is 18 in.

Part c. Step 3. Finding the age.

When $x=18$ in .

Then$$\begin{gathered} y=6.45\times 16-0.15 \\ =116 \end{gathered}$$