# A sample of n independent observations is taken from a normal distribution with mean ,u and variance 02. This exercise develops a test of 02 when the…

question about statistical inference, hypothesis testing

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A sample of n independent observations is taken from a normal distribution with mean ,u andvariance 02. This exercise develops a test of 02 when the value of ,u is known. 1. (2 marks.) Write down and simplify the likelihood ratio fY1,…,Yn (311, &quot;-3 y‘nllu’a 0%) for testing H0 : a2 = 03 versus Ha : a2 = 0:. 2. (2 marks.) Suppose the hypothesized values of 0’2 satisfy a: &gt; 03. Show that the likelihoodratio is monotonically increasing with the test statistic 2221(311 — ”)2. 3. (2 marks.) Show that 313 Z?=1(yi—H)2 has a xi distribution, i.e. X2 with n degree of freedom,under H0. 4. (1 mark.) Hence, write down the rejection region for testing H0 versus Ha with signiﬁcancelevel oz.

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