Let X = ( X 1 , X n ) be a random sample from N( 1 , 2 ) and let Y = ( Y 1 , Y m ) be a random…

 Let X = (X1, Xn) be a random sample from N(1, 2) and let Y =

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(Y1, Ym) be a random sample from N(2, 2). Suppose that we
wish

to test H0: 1 = 2 versus Ha: 1 _= 2.

(a) Show that the likelihood function _(X,Y) is a monotone
function of T 2,

where T is the two-sample t-statistic that
uses the pooled estimator S2

p of the common variance 2.

(b) Let S2 1 be the sample variance for the Xi ’s. Let W =(¯Y X 0)/S1,

where_0 is a constant. Find a constant K such that  KWhas the noncentral

t-distribution. Give the degrees of freedom and
noncentrality parameter _.