You randomly select and measure the contents of 10 bottles o

You randomly select and measure the contents of 10 bottles of cough syrup. The results are shown 4.218 4.282 4.292 4.265 4.258 4.249 4.241 4.229 4.185 4.234 Assume the sample is taken from a normally distributed population. Construct 80% confidence interals for (a)the population variance 2 and (b) the population standard deviation . What is the confidence interval for the population variance.

Solution

Getting the sample standard deviation,

s = 0.031418501

As              
              
df = n - 1 =    9          
alpha = (1 - confidence level)/2 =    0.1          
              
Then the critical values for chi^2 are              
              
chi^2(alpha/2) =    14.68365657          
chi^2(alpha/2) =    4.168159008          
              
Thus, as              
              
lower bound = (n - 1) s^2 / chi^2(alpha/2) =    0.000605033          
upper bound = (n - 1) s^2 / chi^2(1 - alpha/2) =    0.002131421          
              
Thus, the confidence interval for the variance is              
              
(   0.000605033   ,   0.002131421   ) [ANSWER, FOR VARIANCE, A]
              
Also, for the standard deviation, getting the square root of the bounds,              
              
(   0.024597423   ,   0.046167311   ) [ANSWER, FOR STANDARD DEVIATION, B]