Question

1. A sport utility vehicle design is suspected to have propensity for rollover. In a test, out of 600 vehicles, 17 are observed to rollover. At a significance level of 0.05, test the hypothesis that proportion of vehicles with rollover problem is 2%.

a. If the true fraction of defects is found to be 2.5%, find the type II error.

b. What is the minimum sample size needed to detect type II error within 5% accuracy if the true fraction of defects is 2.5%?

c. How would a. and b. change if the hypothesis was changed to test the proportion of vehicles with rollover problem is more than 2%?

Answer 1

Ho :   p =    0.02                  
H1 :   p ╪   0.02       (Two tail test)          
                          
Level of Significance,   α =    0.05                  
Number of Items of Interest,   x =   17                  
Sample Size,   n =    600                  
                          
Sample Proportion ,    p̂ = x/n =    0.0283                  
                          
Standard Error ,    SE = √( p(1-p)/n ) =    0.0057                  
Z Test Statistic = ( p̂-p)/SE = (   0.0283   -   0.02   ) /   0.0057   =   1.4580
                          
  
                          
p-Value   =   0.144832382   [excel formula =2*NORMSDIST(z)]              
Decision:   p value>α ,do not reject null hypothesis                       
There is not enough evidence that proportion of vehicles with rollover problem is different than 2%.

a)

true proportion,   p=   0.025                      
                              
hypothesis proportion,   po=    0.020                      
significance level,   α =    0.05                      
sample size,   n =   600                      
                              
std error of sampling distribution,   σpo = √(po*(1-po)/n) = √ (   0.020   *   0.980   /   600   ) =   0.0057
std error of true proportion,   σp = √(p(1-p)/n) = √ (   0.025   *   0.975   /   600   ) =   0.0064
Zα/2   = ±   1.960   (two tailed test)                  
We will fail to reject the null (commit a Type II error) if we get a Z statistic between                       -1.960   and   1.960
these Z-critical value corresponds to some X critical values ( X critical), such that                              
-1.960   ≤(p^ - po)/σpo≤   1.960                      
-1.960   *σpo + po≤ p^ ≤   1.960   *σpo + po                  
0.0088   ≤ p^ ≤   0.0312                      
                              
now, type II error is ,ß =        P(0.0088< p^ < 0.0312)       =P( (0.0088-p) /σp < Z < (0.0312-p)/σp )              
       =P( (0.0088-0.025)/0.0064) < Z < (0.0312-0.025)/0.0064 )                      
so, P(   -2.542   < Z <   0.973   ) = P ( Z ≤   0.973   ) - P ( Z ≤   -2.542   )
       =   0.835   -   0.006   =   0.8292  

b)

True mean,   p =    0.025
hypothesis mean,   po =    0.02
      
Level of Significance ,    α =    0.05
power =    1-ß =    0.95
ß=       0.05
      
      
Z (α/2)=       1.960
      
Z (ß) =        1.645
      
sample size needed =    n = po*(1-po)[Z(α/2) + Z(ß) ]² / (p-po)² =   10187.8526
      
so, sample size =        10188.000

C)

Probability of error increases.

Sample size will increase

Thanks in advance!

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