Question

Consider the following hypothesis test:

H(0): mu ≥ 10

H(a): mu < 10

The sample size is 120 and the population standard deviation is assumed known with σ = 5. Use alpha = .05. If the population mean is 9, what is the probability of making a Type II error if the actual population mean is 8 (to 4 decimals)? Please provide the appropriate formula in excel for solving this problem.

Answer 1

true mean ,    µ =    8
      
hypothesis mean,   µo =    10
significance level,   α =    0.05
sample size,   n =   120
std dev,   σ =    9
      
δ=   µ - µo =    -2
      
std error of mean,   σx = σ/√n =    0.8216
Zα =       -1.6449   (left tail test)
[ excel formula for this =normsdist(0.05)]

We will fail to reject the null (commit a Type II error) if we get a Z statistic >               -1.6449
this Z-critical value corresponds to X critical value( X critical), such that              
              
       (x̄ - µo)/σx ≥ Zα      
       x̄ ≥ Zα*σx + µo      
       x̄ ≥ -1.6449*0.8216+10= 8.649   (acceptance region)

now, type II error is ,ß =    P( x̄ ≥    8.649   given that µ =   8
              
ß = P ( Z > (x̄-true mean)/σx )           
excel formula for this [=1 - norm.dist(8.649,8,0.8216,true]

hence, ß=0.2149

type II error = 0.2149