Question

2. (10pts) The following 16 random samples; 5.33, 4.25, 3.15, 3.70, 1.61, 6.40, 3.12, 6.59,3.53, 4.74, 0.11, 1.60, 5.49, 1.72, 4.15, 2.30, came from normal distribution with mean μ and variance σ2, i.e., Xi, X2' .. . , X16 ~ N(μ, σ*), with the density function (a) (4pts) Find the maximum likelihood estimates of μ and σ2, denoted with μ and σ2. (b) (4pts) Based on above μ and σ2, construct 95% confidence intervals for μ and σ2 separately. (c) (2pts) Based on part (b) and the duality of confidence intervals and the hypothesis tests, carry out a hypothesis test of Ho : μ-: 4 versus H1: μ 4 at a-0.05.

Answer 1

luko) Cien -16 +2.30 5.33+4.25+--·· I6 X : 3.612 (5.33-3.6)2) (4.25-3612): = 51- 1198 15 3418 3.412 1.847 lese hle need to be test te fellouoing rypotheis

oD Stardoad deriation coil be osed 3.612- - 0 388 о, 46; -o ‘ 8 403 esikol voke of 005 I.96 0 400 p-value > : О.o5 Ho is not ซุejected

Confidence İnterval

3.612 1.96 x 1.847 3.612 96x o 46115 :3 G12 o 90503 3-612 090sa3, 8-612t o 40508 2.70694 ʼ 4.51703