LINEAR REGRESSION Use the sample to complete this section R

LINEAR REGRESSION – Use the sample to complete this section. Remember, the X variable is NUMBER OF RUNS SCORED, and the Y variable is NUMBER OF WINS.

Interpreting the regression output.

i. The regression equation is: ____________________________________

ii. Explain the exact meaning of the slope of the regression equation:

iii. Explain the exact meaning of the y-intercept of the regression equation:

iv. Explain the exact meaning of the standard error of the estimate:

v. Explain the exact meaning of the coefficient of determination:

vi. Predict the number of wins for a team that scores 670 runs (round off to the nearest integer). _______________

Solution

let the linear regression equation of Y on X be Y=a+bX where Y=number of wins ,X=number of runs scored, a=y intercept ,b=slope

a and b are estimated by method of least squares.

using method of least squares

b=r*s_y/s_x   where r is the correlation coefficient between X and Y. s_y is the standard deviation of Y. s_x is the standard deviation of X.

and using method least squares a=ybar-b*xbar where ybar and xbar denote the mean of Y and X respectively.

i) hence the regression equation comes out to be

Y=10.9+0.0972*X [answer]

ii)we have Y_n=a+bX_n

then Y_n+1=a+bX_n+1     where (X_n,Y_n) denote the nth pair of X and Y

so Y_n+1-Y_n=b(X_n+1-X_n)

or, b=(Y_n+1-Y_n)/(X_n+1-X_n)

hence the exact meaning of the slope is the unit increase of the value of Y per unit increase value of X.

here b=0.0972. which means Y increases by 0.0972 amount per unit increase value of X

iii) now we have Y=a+bX

when X=0 we have Y=a

so exact meaning of a means the value of Y when the value of X is zero.

here a=10.9 which means that when X=0 the value of Y is 10.9 [answer]

iv) here the estimated value is Y. because we estimating Y on the basis of a regression equation.

now standard error of Y means the dispersion in the values of Y from its mean.

now this dispersion is due to the dispersion in original value of y and the dispersion of the error values, the error that arises in estimating y using the regression equation.

so the standard error of the estimate means the dispersion of the original values of y and the dispersion of the error values,the error that arises in estimating y using the regression equation.

v) coefficient of determination is R²=r² where r is the correlation coefficient.

we know R²=V(Y)/V(y)   where y is the original variable and Y is the predicted values of y on the basis of regression equation.

so R² means the proportion of total variation explained by the linear regression equation of Y on X.

the higher the values of R² we can say that the prediction is getting better.

here R²=0.489

means that only 48.9% of the variation of y can be explained by the regression equation.

vi) number of wins for a team that scores 670 runs is [by putting X=670 in the regression equation]

Y=10.9+0.0972*670=76.024=76 [nearest integer] [answer]