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: y=4.83−0.41x​, r2=0.19.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 19% of the sample variation in the numbers of points scored using the least squares line.

B.There is little or no relationship between numbers of yards to the opposing goal line and numbers of points scored because 0.19 is near to zero.

C.There is a positive linear relationship between numbers of yards to the opposing goal line and numbers of points scored because 0.19 is positive.

D.Sample variations in the numbers of yards to the opposing goal line explain 81​%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 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 negative.

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

Answer 1

Given regression equation is y=4.83−0.41x​ and r^{​​​​​​2}​​​​​ = 0.19.

a) given coefficient of determination r2 = 0.19 which represents the model only explains the 19% variability of the data. In the above problem the answer is option A.

b) coefficient of correlation is the square root of coefficient of determination.

r = = = +-0.436

We have to consider negative sign because the slope is negative which says increase in x represents the reduction in y. Option B.