Question

We are given that n=15, the sample mean Ῡ=2.5, the sample standard deviation s=1.5 and random variable Y distributed Normal with mean µ and variance σ2, where both µ and σ2 are unknown and we are being concentrated on testing the following set of hypothesis about the mean parameter of the population of interest. We are to test: H0 : µ ≥ 3.0 versus H1 : µ < 3.0. Compute the following:

a) P- value of the test

b) Probability of making Type II error and the power of this test at µ= 2.0

Answer 1

a)

Ho :   µ ≥ 3
Ha :   µ <   3
      
Level of Significance ,    α =    0.050
sample std dev ,    s =    1.5000
Sample Size ,   n =    15
Sample Mean,    x̅ =   2.5000
      
degree of freedom=   DF=n-1=   14
      
Standard Error , SE = s/√n =   1.5/√15=   0.3873
t-test statistic= (x̅ - µ )/SE =    (2.5-3)/0.3873=   -1.291
      

      
p-Value   =   0.1088

b)

true mean ,    µ =    2
      
hypothesis mean,   µo =    3
significance level,   α =    0.05
sample size,   n =   15
std dev,   σ =    1.5000
      
δ=   µ - µo =    -1
      
std error of mean=σx = σ/√n =    1.5/√15=   0.3873

Zα =   -1.6449   (left tail test)
P(type II error) , ß =   P(Z > Zα - δ/σx)  
= P(Z >    -1.6449-(-1)/0.3873)  
=P(Z>    0.937   ) =
type II error, ß =   0.1743