Compute the test statistic and report your conclusion

28.6 Exercises 425

Table 28.3. Salaries in two kinds of occupations.

Source: D.J. Hand, F. Daly, A.D. Lunn, K.J. McConway, and E. Ostrowski.

b. Do the same without the assumption of equal variances.

c. As a comparison, one carries out an empirical bootstrap simulation for the nonpooled studentized mean difference. The bootstrap approximations for the critical values are c∗about the salaries on the basis of the bootstrap results. l= −2.004 and c∗u= 2.133. Report your conclusion

Suppose we view the datasets as realizations of random samples from normal

distributions with expectations µ1, µ2, and µ3 and variances σ2 1, σ2 2, and σ2 3,

28.3 ⊡ In a seven-day study on the effect of ozone, a group of 23 rats was

kept in an ozone-free environment and a group of 22 rats in an ozone-rich

are

Ozone-free: ¯x23 = 22.40

13.1 27.3 28.5 −9.9 6.8 28.2

−16.9 17.4 21.8 17.9 −12.9 14.0

26.0 26.6 −9.0

equal variances, i.e., compute the test statistic and report your conclusion.

b. One also carries out a bootstrap simulation for the test statistic used in

c. Also perform the test at level 0.05 without the assumption of equal vari-

ances, where you may use the normal approximation for the distribution

simulation?

28.4 Show that in the case when n = m, the random variables Tp and Td are

�S2 X�= 2σ4

n − 1

a. Show that a and b must satisfy a + b = 1.

b. Show that Var

b. Show that the pooled variance S2 Var� ¯Xn − ¯Ym

p, as defined on page 417, is a biased � =σ2n+ σ2 m .

unbiased estimator for Var

X= σ2 Y= σ2. Show that S2� ¯Xn − ¯Ym�of the form aS2

What about when n = m?

dalso an unbiased estimator for Var� ¯Xn − ¯Ym�= σ2(1/n + 1/m).

Summary of distributions

Discrete distributions

P(X = k) =�n�pk(1 − p)n−k for k = 0, 1, . . ., n.

E[X] = np and Var(X) = np(1 − p).

P(X = k) =µk k! e−µ for k = 0, 1, . . . .

E[X] = µ and Var(X) = µ.

π (β2+ (x − α)2) for −∞ < x < ∞.

F(x) = 1