It is crucial that the variance of a production process be l

It is crucial that the variance of a production process be less than or equal to 28. A sample of 30 is taken. The sample variance equaled 29.

a) Construct a 90% confidence interval for the population variance

b) Construct a 90% confidence interval for the population standard deviation

c) test at 10% level of significance that whether the variance of the production process exceeds its standard (use the critical value approach)

Solution

a)

As              
              
df = n - 1 =    29          
alpha = (1 - confidence level)/2 =    0.05          
              
Then the critical values for chi^2 are              
              
chi^2(alpha/2) =    42.5569678          
chi^2(alpha/2) =    17.70836618          
              
Thus, as              
              
lower bound = (n - 1) s^2 / chi^2(alpha/2) =    573.090642          
upper bound = (n - 1) s^2 / chi^2(1 - alpha/2) =    1377.258622          
              
Thus, the confidence interval for the variance is              
              
(   573.090642   ,   1377.258622   ) [ANSWER, PART A]

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b)
              
Also, for the standard deviation, getting the square root of the bounds,              
              
(   23.93931164   ,   37.11143519   ) [ANSWER]

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c)

Formulating the null and alternative hypotheses,              
              
Ho:   sigma   <=   28  
Ha:    sigma   >   28  
              
As we can see, this is a    right   tailed test.      
              
Thus, getting the critical chi^2, as alpha =    0.1   ,      
alpha =    0.1          
df = N - 1 =    29          
chi^2 (crit) =    39.08746977        
              
Getting the test statistic, as              
s = sample standard deviation =    29          
sigmao = hypothesized standard deviation =    28          
n = sample size =    30          
              
              
Thus, chi^2 = (N - 1)(s/sigmao)^2 =    31.10841837          
As chi^2 < chi^2(crit), then we FAIL TO REJECT THE NULL HYPOTHESIS.              

There is no significant evidence that the production process exceeds its standard. [CONCLUSION]