Question

For a given population with σ=10.5 lb. we want to test the null hypothesis μ=66.5 against the alternative hypothesis μ ≠66.5 on the basis of a random sample of size n=64. If the null hypothesis is rejected when x¯<64.6 lb. or x¯>68.8.

a) (3 points) What is the probability of a type I error?

b) (4 points) What is the probability of a type II error and the power of the test when in reality μ=67.0?

Answer 1

Solution ? o= 10.5, n =64 Ho: M = 66.5 4 l 66.5 Reject Ho, when X < 64.616 or X > 68.816. so acceptance region will be {64.6 < X < 68.8} a) P (Type - 1 Exxox)=(«) is obtained as P (Reject Ho l Ho true ) - P ( 64.6<x<68.8 ) = 1-a o - 1-PC 64.6 < X < 68.8) x = 1 - P 6406-Mo <x - lo <68.8 Mo DAAR Plse where lo=66.5 con since Under Hos M = 66.5) x = 1 - P 64. 6-66.5 <z < 68.8 - 66.5 - 66.5) X=1 - P (-1.9*8 < Z < 2.3x8) 10.5 ( (0.5/8 10.5/8 10.5 x=1- P (-1.4476<2<107524) as 1- [PC z <1.7524) - P(Z <-1.4476)] Q-1- P(Z1.7524) + P(Z <-14476) a = 1- 9.9601+ 0.0738 -0. 1137 Probability of type error 0.137. b) p (type 1error ) - B. ß = P(Accept Hol Ho false) B = P ( 64.6 < X < 68. 8) 8) 4 = 67 64.6 67 < Z < 68.8 - 67 567) 10.5/8 10.5/8 B = P( 1:848) - 2.448 10.5 < Z < 1.888

B = p 61.8286 < z < 1.3714) B = P(Z < 1.3714) - P(Z < -1.8286) B = 0.914% - 0.0337 ß = 0.8811 proba bility of type :D estor i.e. Bis 0.8811 Power of test = 1-ß = 1-0.8811 Power of test = 0.1189