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=4.12−0.59x​, r squared equals 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. This 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 beta 1β1 is positive.

B.The coefficient of​ correlation,r=__________, is positive because the estimator of beta 1β1is negative.

C.The coefficient of​ correlation,r=___________,is negative because the estimator of beta 1β1is negative.

D.The coefficient of​ correlation,r=_________,is negative because the estimator of beta 1β1 is positive.

Answer 1

(A) we know that the R squared explains us about the variation in the dependent variable using the independent variable. Here dependent variable is number of points scored and independent variable is the number of yards from the opposing goal line.

So, r squared value of 0.24 means that 24% of variation in the y values can be explained by the regression line.

option A is correct answer

(B) we know that the relationship between r sqaured(coefficient of determination) and r(coefficient of correlation) is given as

setting the r-sqaured value, we get

(rounded to 3 decimals)

it is positive, but the slope is negative. As we know that the relationship between slope beta1 and coefficient of correlation r is given as

= r*(Sy/Sx)

so, slope and coefficient of correlation has same sign. This means that the correlation coefficient must be negative

(we know that the square of negative value is positive, so for r square value of 0.24, we can have a correlation coefficient of -0.490

option C is correct.