A personnel manager at a large firm needs to estimate the mean length of stay, m

Question

A personnel manager at a large firm needs to estimate the mean length of stay, mu of its employees. She takes a random sample of employees and finds the sample mean (x bar) to be 25 years. Find a 90% Confidence Interval for mu if the employee sample size is 64, and the standard deviation is known to be 16. (sigma = 16) What is the appropriate distribution (z or t)?_________ What is the numerical value of (z or t)?__________ What is the Confidence Interval? (Show equation with filled in numerical values. Calculate answer.)___________ What is the appropriate interpretation of the Confidence Interval? There is a 90% probability that the true population mean is in the Confidence Interval. There is a 90% probability that the Confidence Interval includes the true population mean. Both of the above choices are correct. None of the above choices are correct.

Explanation / Answer

1.

a)

sigma is known, and n = 64, which is large enough. Hence, we use Z DISTRIBUTION. [ANSWER]

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b)

As confidence level = 0.90,

alpha/2 = (1 - confidence level)/2 =    0.05      
Thus, by table/technology,
  
z(alpha/2) = critical z for the confidence interval =    1.644853627   [ANSWER]

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c)      

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 =    0.05          
X = sample mean =    25          
z(alpha/2) = critical z for the confidence interval =    1.644853627          
s = sample standard deviation =    16          
n = sample size =    64          
              
Thus,              
Margin of Error E =    3.289707254          
Lower bound =    21.71029275          
Upper bound =    28.28970725          
              
Thus, the confidence interval is              
              
(   21.71029275   ,   28.28970725   ) [ANSWER]

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D)

Strictly speaking, none of these choices are correct, but some classes do consider

OPTION II: There is a 90% probability that the confidence interval includes the true population mean.   

[The strictly correct answer is that "we are 90% confident that our interval includes the true population mean."]

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