Question

2. An electrical firm manufactures light bulbs that have a lifetime that is approximately normally distributed with a standard deviation of 35 hours. A lifetime test of n=25 samples resulted in the sample average of 1007 hours. Assume the significance level of 0.05.

(a) Test the hypothesis H0:μ=1000 versus H1:μ≠1000 using a p-value. (6 pts)

(b) Calculate the power of the test if the true mean lifetime is 1010. (8 pts)

(c) What sample size would be required to detect a true mean lifetime as low as 990 hours if we wanted the power of the test to be at least 0.95? (6 pts)

Answer 1

a)

p value =0.3174

since p value >0.05 , we fail to reject null hypothesis

we do not have sufficient evidence that mean is different from 1000

b)

rejection region: μ-Zα*σx <Xbar >μ+Zα*σx or 986.28<Xbar>1013.72

P(Power) =1-P(986.28<Xbar<1013.72|μ=1010)=1-P(986.28-1010)/7<z<(1013.72-1010)/7)=1-P(-3.39<z<0.53)=

0.2984

c)

Hypothesized mean μo=	1000
true mean μa=	990
std deviation σ=	21.3
0.05 level critical Z=	1.960
0.05 level critical Zβ=	1.640
n=(Zα/2+Zβ)2σ2/(μo-μa)2=	59