A machine makes drill bits When the machine is operating cor

A machine makes drill bits. When the machine is operating correctly, the variance of

the diameter of drill bits is no more than .04 inches. In a random sample of 81 drill

bits, the sample variance was .07. Test the hypothesis at the 1% level that the machine

is operating correctly.

Solution

Formulating the null and alternative hypotheses,              
              
Ho:   sigma   <=   0.2  
Ha:    sigma   >   0.2  
              
As we can see, this is a    left   tailed test.      
              
Thus, getting the critical chi^2, as alpha =    0.01   ,      
alpha =    0.01          
df = N - 1 =    80          
chi^2 (crit) =    53.54007729        
              
Getting the test statistic, as              

s = sample standard deviation =    0.264575131          
sigmao = hypothesized standard deviation =    0.2          
n = sample size =    81          
              
              
Thus, chi^2 = (N - 1)(s/sigmao)^2 =    140          
              
Comparing chi^2 > chi^2(crit), we REJECT THE NULL HYPOTHESIS.              

Thus, there is significant evidence that the machine is not operating correctly. [CONCLUSION]