Now assume that the population standard deviation of the ex

Now assume that the population standard deviation, , of the existing machine diameter measurements is not known (but still normally distributed). Using sample mean 5.00 mm, sample standard deviation 0.028 mm and sample size

100, compute the following and include units in your answers:

a) A 95% twosided confidence interval for the true standard deviation of bolt diameter, .

b) A 90% lower confidence bound for the true standard deviation of bolt diameter, .

Solution

a)

As              
              
df = n - 1 =    99          
alpha = (1 - confidence level)/2 =    0.025          
              
Then the critical values for chi^2 are              
              
chi^2(alpha/2) =    128.4219886          
chi^2(alpha/2) =    73.36108019          
              
Thus, as              
              
lower bound = (n - 1) s^2 / chi^2(alpha/2) =    0.000604382          
upper bound = (n - 1) s^2 / chi^2(1 - alpha/2) =    0.001058          
              
Thus, the confidence interval for the variance is              
              
(   0.000604382   ,   0.001058   )
              
Also, for the standard deviation, getting the square root of the bounds,              
              
(   0.024584192   ,   0.032526907   ) [answer]

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b)

As              
              
df = n - 1 =    99          
alpha = (1 - confidence level) =    0.1          
              
Then the critical values for chi^2 are              
              
chi^2(alpha) =    117.4068832          

              
Thus,              
              
lower bound variance= (n - 1) s^2 / chi^2(alpha) =    0.000661086

Therefore,

lower bound standard deviation = sqrt(0.000661086)

lower bound standard deviation = 0.025711593 [ANSWER]