Question

Q11111)

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: y=3.68−0.43x​, r2 = 022. Complete parts a and b below.

a. Give a practical interpretation of the coefficient of​ determination r2 Choose the correct answer below.

A. There is little or no relationship between numbers of yards to the opposing goal line and numbers of points scored because 0.22 is near to zero.

B. Sample variations in the numbers of yards to the opposing goal line explain 78​% of the sample variation in the numbers of points scored using the least squares line.

C. Sample variations in the numbers of yards to the opposing goal line explain 22​% of the sample variation in the numbers of points scored using the least squares line.

D. There is a positive linear relationship between numbers of yards to the opposing goal line and numbers of points scored because 0.22 is positive.

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.

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

B. The coefficient of​ correlation, r = ------​, is negative because the estimator of β1 is negative.

C. The coefficient of​ correlation, r = ------​, is positive because the estimator of β1 is positive.

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

Answer 1

a. Give a practical interpretation of the coefficient of​ determination r2 Choose the correct answer below.

The correct choice is A.

A. There is little or no relationship between numbers of yards to the opposing goal line and numbers of points scored because 0.22 is near to zero.

Here we need to test the linearity of correlation .

Null hypothesis : There is no correlation between x and y

Alternative hypothesis : There is correlation between x and y

Here r² = 0.22

Therefore | r | = 0.469

From pearson correlation table we get critical r value for two tailed test and n - 2 = 3 degrees of freedom ( with 0.05 level of significance) is as follows:

critical r = 0.878

Since critical r = 0.878 > r = 0.469

So we fail to reject null hypothesis and conclude that there was no correlation between x and y.

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.

The correct choice is B

B. The coefficient of​ correlation, r = -469​, is negative because the estimator of β1 is negative.

Note that the sign of cooefficient of correlation( r ) and regression coefficient( ) in simple linear regression are same.