Introduction to Probability and Statistics for Engineers and Scientists

Problems 419

13.Suppose in the simple linear regression model

Y =α+βx+e

that 0<β<1.

(a) Show that ifx<α/(1−β), then

x<E[Y]<

α

1 −β

(b) Show that ifx>α/(1−β), then

x>E[Y]>

α

1 −β

and conclude thatE[Y]is always betweenxandα/(1−β).

14.A study has shown that a good model for the relationship betweenXandY, the

first and second year batting averages of a randomly chosen major league baseball

player, is given by the equation

Y=.159+.4X+e

wheree is a normal random variable with mean 0. That is, the model is

a simple linear regression with a regression toward the mean.

(a) If a player’s batting average is .200 in his first year, what would you predict

for the second year?

(b) If a player’s batting average is .265 in his first year, what would you predict

for the second year?

(c) If a player’s batting average is .310 in his first year, what would you predict

for the second year?

15.Experienced flight instructors have claimed that praise for an exceptionally fine

landing is typically followed by a poorer landing on the next attempt, whereas

criticism of a faulty landing is typically followed by an improved landing. Should

we thus conclude that verbal praise tends to lower performance levels, whereas

verbal criticism tends to raise them? Or is some other explanation possible?

16.Verify Equation 9.4.3.

17.The following data represent the relationship between the number of alignment

errors and the number of missing rivets for 10 different aircrafts.

(a) Plot a scatter diagram.

(b) Estimate the regression coefficients.

(c) Test the hypothesis thatα=1.