Suppose that we take a sample of size n1 from a normally distributed population with

Chapter 8, Problem 8.128

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Suppose that we take a sample of size n1 from a normally distributed population with mean and variance 1 and 2 1 and an independent of sample size n2 from a normally distributed population with mean and variance 2 and 2 2 . If it is reasonable to assume that 2 1 = 2 2 , then the results given in Section 8.8 apply. What can be done if we cannot assume that the unknown variances are equal but are fortunate enough to know that 2 2 = k2 1 for some known constant k = 1? Suppose, as previously, that the sample means are given by Y 1 and Y 2 and the sample variances by S2 1 and S2 2 , respectively. a Show that Z given below has a standard normal distribution. Z = (Y 1 Y 2) (1 2) 1 1 1 n1 + k n2 . b Show that W given below has a 2 distribution with n1 + n2 2 df. W = (n1 1)S2 1 + (n2 1)S2 2 /k 2 1 . c Notice that Z and W from parts (a) and (b) are independent. Finally, show that T = (Y 1 Y 2) (1 2) S p 1 1 n1 + k n2 , where S2 p = (n1 1)S2 1 + (n2 1)S2 2 /k n1 + n2 2 has a t distribution with n1 + n2 2 df. d Use the result in part (c) to give a 100(1 )% confidence interval for 1 2, assuming that 2 2 = k2 1 . e What happens if k = 1 in parts (a)(d)?