Question

QUESTION 4 a) What is meant by critical region and significance level in hypothesis testing? (2 marks) b) A random sample X1, X2...., X, of size n is obtained from a normal distribution with mean, u and variance, 81. We are interested to test Ho : 4 = 100 against H, : x = 104. 1) Show that a best critical region according to Neyman-Pearson lemma is 82c. (8 marks) ii) Find the sample size, n given that the significance level is approximately 0.05 and 1- B =0.6. (10 marks)

Answer 1

a A critical region is known as rejection region statistie This region is a set of values of test base which the null hypothesis would be rejected, observe that the test statistic is in crítical region, we can reject the Ho (null I b we the hypothesis) significance level: The significance level is nothing but the probability of rejecting to (null hypothesis When it's true it's de noted by a. a = P( rejecting Hol Ho is true) { Type I error n] Now we do the No test of level & bor testing with r =81, Hoi M=100 H,: M =104, (M, > Mo). Let. X, ,X₂ - - Xn be random sample of size n drawn likelihood bunction is given from Nm, ry The n as b(ul M) é par (x; -H) e & ar n [st (n-M 204 e ar sha The Ne lemma size a critical most powerbul is region given as c = { x: b (x) z, chulw}

the above ea. we need to find the value of c. 6CxIM) en [5+ (n - H.) f(x/Mo) é 2 [8+ (n -%o) ry > ¿ which gives n in [ [n-H) - (n-1). I} 202 >ť e n [ In - M.) - (n - Mo)"] , 40 ) = loge 200 [ taking loy both side ob ()] = anñ (M-Mo) >, zot laget nC t-Ho) n > 20-logé tn (Mr-Mon 2 (M, - Mo) 11 The best critical region is n Zee (Hence proved Here H =104 Mo - 100 or=81 as you need it

NOW we need to find n. Let x = 0.05 I- ß = Orf we know that P (xref Ho) = x PHO X - Me Tl in 7 e- Ho r/in PHO (ره )N T e - Ho orn ㅁ Here e-Mo za 5/10 [one tailed test -> c= Motta 8/10 za = Zoos 1.645. 100 +1.645 x 9 in I- B - 2 ß = 0.4 PHI C X-M, L olm - 0.4 from normal table 027 e-Mi =-0.206. (approx) r/in 104 & 0.206 x solve 104 -0. 206x9 n 100 tl.645 t. 2) 4 = t. 645X 9 + 0.206x9 m [complete this.)
I have taken approximate z value. Please find it from z table. Tell me if you have any doubt. Beta is nothing but type two error and it's complement is power of the test.