Question

We are to test H0 : µ = 1200 versus H1 : µ ≠ 1200

Assume that the random variable of interest Y~N( µ, σ2), where both mean µ and standard deviation were unknown. And the sample mean Ῡ=1422.2, sample standard deviation s=246.86, obtained from a random sample of size n=10 taken from this population.

Compute the following:

a) P- value of the test

b) Probability of making Type II error of this test at µ= 1300

Answer 1

a)

Ho :   µ =   1200  
Ha :   µ ╪   1200   (Two tail test)
          
Level of Significance ,    α =    0.010  
sample std dev ,    s =    246.8600  
Sample Size ,   n =    10  
Sample Mean,    x̅ =   1422.2000  
          
degree of freedom=   DF=n-1=   9  
          
Standard Error , SE = s/√n =   246.86/√10=   78.0640  
t-test statistic= (x̅ - µ )/SE =    (1422.2-1200)/78.064=   2.8464  
          

          
p-Value   =   0.0192   [Excel formula =t.dist.2t(t-stat,df) ]

b)

true mean ,    µ =    1300
      
hypothesis mean,   µo =    1200
significance level,   α =    0.05
sample size,   n =   10
std dev,   σ =    246.8600
      
δ=   µ - µo =    100
      
std error of mean=σx = σ/√n =    246.86/√10=   78.0640
(two tailed test) Zα/2   = ±   1.9600  
type II error is          
ß = P(Z < Zα/2 - δ/σx) - P(Z < -Zα/2-δ/σx)          
P(Z<1.96-100/78.064) - P(Z<-1.96-100/78.064)=          
P(Z<0.679)-P(Z<-3.241)=          
=   0.751419523   -   0.0006
          
=   0.750819523   (answer)