Question

Lake Vostok is Antarctica's largest and deepest subsurface lake. The mean depth of the lake is believed to be 344 meters. 25 random samples of the depth of this lake were obtained. Assume the underlying distribution is normal and σ = 30 meters. Please use 3 decimal places in all calculations.

b) Find the probability of the Type II error if the true mean depth of the lake is Pa = 328 meters; that is, find B(328). Assume that a = 0.01. Hint: If your numeric answer is wrong, please confirm that your hypotheses are correct before you run out tries.

Answer 1

true mean ,    µ =    328              
                      
hypothesis mean,   µo =    344              
significance level,   α =    0.01              
sample size,   n =   25              
std dev,   σ =    30              
                      
δ=   µ - µo =    -16              
                      
std error of mean,   σx = σ/√n =    30.0000   / √    25   =   6.00000

Zα/2   = ±   2.576   (two tailed test)                      
We will fail to reject the null (commit a Type II error) if we get a Z statistic between                           -2.576   and   2.576
these Z-critical value corresponds to some X critical values ( X critical), such that                                  
                                  
-2.576   ≤(x̄ - µo)/σx≤   2.576                          
328.545   ≤ x̄ ≤   359.455                          
                                  
now, type II error is ,ß =        P (   328.545   ≤ x̄ ≤   359.455   )          
       Z =    (x̄-true mean)/σx                      
       Z1 = (   328.545   -   328   ) /   6.00000   =   0.091
       Z2 = (   359.455   -   328   ) /   6.00000   =   5.242
                                  
   so, P(   0.091   ≤ Z ≤   5.242   ) = P ( Z ≤   5.242   ) - P ( Z ≤   0.091   )
                                  
       =   1.000   -   0.536   = 0.464   [ Excel function: =NORMSDIST(z) ]  

ß = 0.464

Please revert back in case of any doubt.

Please upvote. Thanks in advance.