Chapter 4.5, Problem 6E

For a freshman taking introductory statistics and majoring in psychology, let X equal the student’s A(1 mathematics score and Y the student’s ACT verbal score. Assume that X and Y have a bivariate normal distribution with ${\mu }_X=22.7,{\sigma }_X^2=17.64,{\mu }_Y=22.7,{\sigma }_Y^2=12.25$, and $\rho =0.78$.

(a) Find $P\left(19.9<Y<26.9\right)$.

(b) Find $E\left(Y|x\right)$.

(c) Find $Var\left(Y|x\right)$.

(d) Find $P\left(19.9<Y<26.9|X=23\right)$.

(e) Find $P\left(19.9<Y<26.9|X=25\right)$.

(f) For $x=21,23,$ and 25, draw a graph of $z=h\left(y|x\right)$ similar to Figure 4.5-1.