LINEAR REGRESSION – Use the sample to complete this section. Remember, the X var

Question

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). _______________

Runs Scored (X) Wins (Y) 708 69 875 90 654 79 704 80 787 95 730 71 667 86 619 63 867 97 645 74 556 67 707 91 855 96 743 81 731 94 641 89 654 71 735 79 735 73 615 56

Explanation / Answer

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*sy/sx   where r is the correlation coefficient between X and Y. sy is the standard deviation of Y. sx 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 Yn=a+bXn

then Yn+1=a+bXn+1     where (Xn,Yn) denote the nth pair of X and Y

so Yn+1-Yn=b(Xn+1-Xn)

or, b=(Yn+1-Yn)/(Xn+1-Xn)

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]

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