Consider the problem of testing the equality of two normal means when the variances are unequal. This problem was introduced on page 593 in Sec. 9.6. The data are two independent samples X1, . . . ,Xm and Y1, . . . , Yn. The Xi’s are i.i.d. having the normal distribution with mean μ1 and variance σ21, while the Yj’s are i.i.d. having the normal distribution with mean μ2 and variance σ22.
a. Assume that μ1 = μ2. Prove that the random variable V in Eq. (9.6.14) has a distribution that depends on the parameters only through the ratio σ2/σ1.
b. Let ν be the approximate degrees of freedom for Welch’s procedure from Eq. (9.6.17). Prove that the distribution of ν depends on the parameters only through the ratio σ2/σ1.
c. Use simulation to assess the approximation in Welch’s procedure. In particular, set the ratio σ2/σ1 equal to each of the numbers 1, 1.5, 2, 3, 5, and 10 in succession. For each value of the ratio, simulate 10,000 samples of sizes n = 11 and m = 10 (or the appropriate summary statistics). For each simulated sample, compute the test statistic V and the 0.9, 0.95, and 0.99 quantiles of the approximate t distribution that corresponds to the data in that simulation. Keep track of the proportion of simulations in which V is greater than each of the three quantiles. How do these proportions compare to the nominal values 0.1, 0.05, and 0.01?