Let X1,...,Xn be a random sample from a n(μx, Ï2x), and let Y1,...,Ym be an independent random sample from a n(μY, Ï2Y). We are interested

Let X1,...,Xn be a random sample from a n(μx, σ2x), and let Y1,...,Ym be an independent random sample from a n(μY, σ2Y). We are interested in testing
H0: μx = μY versus H1: μx ‰  μY
with the assumption that σ2x = σ2Y = σ2.
(a) Derive the LRT for these hypotheses. Show that the LRT can be based on the statistic

where

(The quantity S2p is sometimes referred to as a pooled variance estimate. This type of estimate will be used extensively in Section 11.2.)
(b) Show that, under H0, T ~ tn+m-2. (This test is known as the two-sample t test.)
(c) Samples of wood were obtained from the core and periphery of a certain Byzantine church. The date of the wood was determined, giving the following data.

Use the two-sample t test to determine if the mean age of the core is the same as the mean age of the periphery.

ntm 2) Core Periphery 1294 1251 1284 1274 1279 1248 1272 1264 1274 1240 1256 1256 1264 1232 1254 1250 1263 1220 1242 1254 1218 1251 1210