A random sample of 20 observations showed a standard deviati

A random sample of 20 observations showed a standard deviation of 8. At a 5% level of significance, test to see if the variance of the population is significantly less than 65.

Solution

Here, sqrt(65) = hypothesized standard deviation = 8.062257748.

Formulating the null and alternative hypotheses,              
              
Ho:   sigma   >=   8.062257748  
Ha:    sigma   <   8.062257748  
              
As we can see, this is a    left   tailed test.      
              
Thus, getting the critical chi^2, as alpha =    0.05   ,      
alpha =    0.05          
df = N - 1 =    19          
chi^2 (crit) =    10.11701306        
              
Getting the test statistic, as              
s = sample standard deviation =    8          
sigmao = hypothesized standard deviation =    8.062257748          
n = sample size =    20          
              
              
Thus, chi^2 = (N - 1)(s/sigmao)^2 =    18.70769231          
              
As chi^2 > chi^2(crit), then we FAIL TO REJECT THE NULL HYPOTHESIS.              

Thus, there is no significant evidence that the variance is less than 65. [CONCLUSION]