To solve the problem, we will go through the steps of performing a linear regression analysis on the given data points and then use the resulting regression equation to predict the average score for a given height.
The data set provided consists of the following pairs [tex]\((x, y)\)[/tex]:
[tex]\[
(65, 20), (68, 28), (73, 29), (77, 32), (78, 30)
\][/tex]
Step 1: Perform Linear Regression
We need to determine the slope [tex]\( m \)[/tex] and y-intercept [tex]\( b \)[/tex] of the best fit line, which can be expressed in the form:
[tex]\[
y \approx mx + b
\][/tex]
Using the regression analysis for the given data set, we find:
[tex]\[
m \approx 0.703 \quad \text{(slope rounded to the nearest thousandth)}
\][/tex]
[tex]\[
b \approx -22.991 \quad \text{(intercept rounded to the nearest thousandth)}
\][/tex]
So, the linear regression equation that models the data is:
[tex]\[
y \approx 0.703x - 22.991
\][/tex]
Step 2: Predict the Average Score for a Height of 70 Inches
We use the regression equation to predict the average points per game for a player who is 70 inches tall. Substitute [tex]\( x = 70 \)[/tex] into the equation:
[tex]\[
y \approx 0.703(70) - 22.991
\][/tex]
Calculating this:
[tex]\[
y \approx 49.21 - 22.991
\][/tex]
[tex]\[
y \approx 26.219 \quad \text{(rounded to the nearest thousandth)}
\][/tex]
Thus, according to the regression model, the predicted average score for a basketball player who is 70 inches tall is:
[tex]\[
26.219 \quad \text{points per game}
\][/tex]
Summary:
The linear regression equation modeling the data is:
[tex]\[
y \approx 0.703x - 22.991
\][/tex]
And the predicted average score for a basketball player who is 70 inches tall is:
[tex]\[
26.219 \quad \text{points per game}
\][/tex]
