A fan of television game shows has observed that, in general, the more educated

Question

A fan of television game shows has observed that, in general, the more educated the contestant, the less money he or she wins. To test her belief she gathers data about the last five winners of her favorite game show. She records their winnings in dollars and the number of years of education. The data are as follows.

She estimated a regression model using winnings as the dependent variable (Y) and years of education as the independent variable (X). She used the ordinary least squares estimator. What is the predicted change in winnings when the education of the winner increases by a year? Please Show Work! :)

decrease by $50

decrease by $37.5

decrease by $75

decrease by $87.5

decrease by $62.5

Explanation / Answer

here winnings is the dependent variable (Y) and years of education is the independent variable (X)

there are n=5 observations

using ordinary least square of Y on X the regression equation of Y on X is

Y = a+b*X where b=r*sy/sx a=ybar-b*xbar

where r=correlation coefficient between x and y

sy= standard deviation of y sx= standard deviation of x

ybar=mean of y and xbar=mean of x

now r=-0.949 ybar=500 xbar=14 sy=158.1 sx=2 hence b=-0.949*158.1/2=-75

hence a=500+75*14=1550

hence the regression equation of Y on X is Y=1550-75*X

so if education is increased by 1 year that is new Xn=X+1

then new Y is Yn=1550-75*Xn=1550-75*(X+1)=1550-75-75=Y-75

hence Yn=Y-75

hence the predicted change will decrease by $75 [answer] [option C]

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