Based on a random sample of 40 values from a normal populati

Based on a random sample of 40 values from a normal population with mean and variance 2 , you calculated that the sample mean is 87 and the sample standard deviation is 4. Establish a 95% confidence interval for the population mean. Which probability distribution did you use? Why?

Solution

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 =    87          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    4          
n = sample size =    40          
              
Thus,              
              
Lower bound =    85.76040994          
Upper bound =    88.23959006          
              
Thus, the confidence interval is              
              
(   85.76040994   ,   88.23959006   ) [ANSWER]

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I used the z distribution because n > 30, which is many enough to approximate it as a normal distribution.