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 mean bolt diameter, .

b) A 99% upper confidence bound for the true mean bolt diameter, .

Solution

a)

Note that              
              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    5          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    0.028          
n = sample size =    100          
              
Thus,              
              
Lower bound =    4.994512101          
Upper bound =    5.005487899          
              
Thus, the confidence interval is              
              
(   4.994512101 mm   ,   5.005487899 mm   ) [ANSWER]

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B)

Note that              
              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.005          
X = sample mean =    5          
z(alpha/2) = critical z for the confidence interval =    2.575829304          
s = sample standard deviation =    0.028          
n = sample size =    100          
              
Thus,              
              
Upper bound =    5.007212322 mm [ANSWER]