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

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.

Transcribed Image Text:

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