Question

Each week coaches in a certain football league face a decision during the game. On​ fourth-down, should the team punt the ball or go for a​ first-down? To aid in the​ decision-making process, statisticians at a particular university developed a regression model for predicting the number of points scored​ (y) by a team that has a​ first-down with a given number of yards​ (x) from the opposing goal line. One of the models fit to data collected on five league teams from a recent season was the simple linear regression​ model, E(y)=β0+β1x.The regression yielded the following​results: −0.59x​,  r2=0.24.

Complete parts a and b below.

a. Give a practical interpretation of the coefficient of​ determination,r2.

Choose the correct answer below.

A.Sample variations in the numbers of yards to the opposing goal line explain 24​% of the sample variation in the numbers of points scored using the least squares line.Your answer is correct.

B.There is a positive linear relationship between numbers of yards to the opposing goal line and numbers of points scored because 0.24 is positive.

C.There is little or no relationship between numbers of yards to the opposing goal line and numbers of points scored because 0.24 is near to zero.

D.Sample variations in the numbers of yards to the opposing goal line explain 76%of the sample variation in the numbers of points scored using the least squares line.

b. Compute the value of the coefficient of​ correlation, r, from the value of r2. Is the value of r positive or​ negative? Why? Select the correct choice below and fill in the answer box within your choice.

​(Round to three decimal places as​ needed.)

A.The coefficient of​ correlation, r=______,is positive because the estimator of β1is positive.

B.The coefficient of​ correlation,

r=______,is positive because the estimator of β1 is negative.

C.The coefficient of​ correlation,

r=________is negative because the estimator of β1is negative.

D.The coefficient of​ correlation, r=_______,is negative because the estimator of β1 is positive

Answer 1

Answer
It is given that the coefficient of determination is 0.24
You have already answered the first part correctly.
I am explaining the solution for part B

(b) we have r square = 0.24
Relationship between r square and r is given as

putting the value of r square in the above formula, we get

now, we need to select the negative or positive value for correlation coefficient r based on the sign of slope of regression of equation. Signs for slope of regression equation and coefficient correlation are always same.

Slope of regression equation is -0.59 (i.e. negative value)

So, the correlation coefficient must also be negative, i.e. r = -0.490

Option C is correct as r value is negative because the estimator of beta1 is negative.