Question

In: Statistics and Probability

A quality expert inspects 430 items to test whether the population proportion of defectives exceeds .02,...

A quality expert inspects 430 items to test whether the population proportion of defectives exceeds .02, using a right-tailed test at α = .10. (a) What is the power of this test if the true proportion of defectives is π = .03? (Round your intermediate calculations and final answer to 4 decimal places.) (b) What is the power of this test if the true proportion is π = .04? (Round your intermediate calculations and final answer to 4 decimal places.) (c) What is the power of this test if the true proportion of defectives is π = .05? (Round your intermediate calculations and final answer to 4 decimal places.)

Solutions

Expert Solution

a)

true proportion,   p=   0.03                      
                              
hypothesis proportion,   po=    0.02                      
significance level,   α =    0.100                      
sample size,   n =   430                      
                              
std error of sampling distribution,   σpo = √(po*(1-po)/n) = √ (   0.020   *   0.980   /   430   ) =   0.0068
std error of true proportion,   σp = √(p(1-p)/n) = √ (   0.03   *   0.97   /   430   ) =   0.0082

Zα =       1.282   (right tailed test)      
                  
We will fail to reject the null (commit a Type II error) if we get a Z statistic <                   1.282
this Z-critical value corresponds to X critical value( X critical), such that                  
                  
(p^ - po)/σpo ≤ Zα                  
p^ ≤ Zα*σpo + po                  
p^ ≤    1.282*0.0068+0.02       =   0.0287  
                  
now, type II error is ,ß =    P( p^ ≤    0.0287   given that p =   0.03  
                  
   = P ( Z < (p^ - p)/σp )=       P(Z < (0.0287-0.03) / 0.0082)      
   = P ( Z < (   -0.164   )      
ß   =   0.434933          
                  
                  
                  
power =    1 - ß =   0.5651          

b)

std error of true proportion,   σp = √(p(1-p)/n) = √ (   0.04   *   0.96   /   430   ) =   0.0094
now, type II error is ,ß =    P( p^ ≤    0.0287   given that p =   0.04
              
   = P ( Z < (p^ - p)/σp )=       P(Z < (0.0287-0.04) / 0.0094)  
   = P ( Z < (   -1.201   )  
ß   =   0.114911      
              
              
              
power =    1 - ß =   0.8851      

c)

std error of true proportion,   σp = √(p(1-p)/n) = √ (   0.05   *   0.95   /   430   ) =   0.0105
now, type II error is ,ß =    P( p^ ≤    0.0287   given that p =   0.05  
                  
   = P ( Z < (p^ - p)/σp )=       P(Z < (0.0287-0.05) / 0.0105)      
   = P ( Z < (   -2.031   )      
ß   =   0.021121          
                  
                  
                  
power =    1 - ß =   0.9789          


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