A manufacturer produces piston rings for an automobile engin

A manufacturer produces piston rings for an automobile engine. It is known that ring diameter is normally distributed with sigma =0.001 millimeters. A random sample of 15 rings has a mean diameter of x = 74.036 millimeters. Construct a 99% two-sided confidence interval on the mean piston ring diameter. Construct a 95% lower-confidence bound on the mean piston ring diameter.

Solution

a)

Note that              
Margin of Error E = z(alpha/2) * s / sqrt(n)              
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 =    0.005          
X = sample mean =    74.036          
z(alpha/2) = critical z for the confidence interval =    2.575829304          
s = sample standard deviation =    0.001          
n = sample size =    15          
              
Thus,              
Margin of Error E =    0.000665076          
Lower bound =    74.03533492          
Upper bound =    74.03666508          
              
Thus, the confidence interval is              
              
(   74.03533492   ,   74.03666508   )

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b)

Note that              

      
Lower Bound = X - z(alpha) * s / sqrt(n)              
              
where              
alpha = (1 - confidence level) =    0.05          
X = sample mean =    74.036          
z(alpha) = critical z for the confidence interval =    1.644853627          
s = sample standard deviation =    0.001          
n = sample size =    15          
              
Thus,              

Lower bound =    74.0355753          

Thus, the confidence interval is u > 74.0356 [ANSWER]