Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

554 Inferences About Normal Linear Models

withρ=0sothat^12 log[(1 +ρ)/(1−ρ)] = 0. However, usingW, we can also test

a hypothesis likeH 0 :ρ=ρ 0 againstH 1 :ρ =ρ 0 ,whereρ 0 is not necessarily zero.

In that case, the hypothesized mean ofWis

1

2

log

(

1+ρ 0

1 −ρ 0

)

.

Furthermore, as outlined in Exercise 9.7.6,Zcanbeusedtoobtainanasymptotic
confidence interval forρ.

EXERCISES

9.7.1.Show that

R=

∑n

1

(Xi−X)(Yi−Y)

√√

√

√

∑n

1

(Xi−X)^2

∑n

1

(Yi−Y)^2

=

∑n

1

XiYi−nXY

√√

√

√

(n

∑

1

Xi^2 −nX

2

)(n

∑

1

Yi^2 −nY

2

).

9.7.2.A random sample of sizen= 6 from a bivariate normal distribution yields
a value of the correlation coefficient of 0.89. Would we accept or reject, at the 5%
significance level, the hypothesis thatρ=0?

9.7.3.Verify Equation (9.7.3) of this section.

9.7.4.Verify the pdf (9.7.4) of this section.

9.7.5.Using the results of Section 4.5, show thatR, (9.7.1), is a consistent estimate
ofρ.

9.7.6.By doing the following steps, determine a (1−α)100% approximate confi-
dence interval forρ.

(a)For 0<α<1, in the usual way, start with 1−α=P(−zα/ 2 <Z<zα/ 2 ),

whereZis given by expression (9.7.5). Then isolateh(ρ)=(1/2) log [(1 +

ρ)/(1−ρ)] in the middle part of the inequality. Findh′(ρ) and show that it is

strictly positive on− 1 <ρ<1; hence,his strictly increasing and its inverse

function exists.

(b)Show that this inverse function is the hyperbolic tangent function given by

tanh(y)=(ey−e−y)/(ey+e−y).

(c)Obtain a (1−α)100% confidence interval forρ.

9.7.7.The intrinsic R functioncor.test(x,y)computes the estimate ofρand the
confidence interval in Exercise 9.7.6. Recall the baseball data which is in the file
bb.rda.