A company with a fleet of 150 cars found that the emissions

A company with a fleet of 150 cars found that the emissions systems of 17 out of the 45 they tested failed to meet the pollution control guidelines. Is there strong evidence that more than 25% of the fleet might be out of compliance? Test at = 0.05. Use the p-value and the rejection region methods

Solution

Formulating the null and alternatuve hypotheses,          
          
Ho:   p   <=   0.25
Ha:   p   >   0.25
As we see, the hypothesized po =   0.25      
Getting the point estimate of p, p^,          
          
p^ = x / n =    0.377777778      
          
Getting the standard error of p^, sp,          
          
sp = sqrt[po (1 - po)/n] =    0.064549722      
          
Getting the z statistic,          
          
z = (p^ - po)/sp =    1.979524821      
          
As this is a    1   tailed test, then, getting the p value,  
          
p =    0.023878474
      

significance level =    0.05      

As p < 0.05, we   REJECT THE NULL HYPOTHESIS.      

Also, zcrit = 1.645.

As z > zcrit, we REJECT THE NULL HYPOTHESIS.

Thus, the is significant evidence that p > 0.25.