Fortynine items are randomly selected from a population of 5

Forty-nine items are randomly selected from a population of 500 items. The sample mean is 40 and the sample standard deviation 9.

Develop a 99% confidence interval for the population mean. (Round your answers to 3 decimal places.)

Solution

Here,

fpc = sqrt[(N-n)/(N-1)] = sqrt((500-49)/(500-1)) = 0.950687969

Thus, the effective standard deviation is

s = sigma*fpc = 9*0.950687969 = 8.556191725

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 =    40          
z(alpha/2) = critical z for the confidence interval =    2.575829304          
s = sample standard deviation =    8.556191725          
n = sample size =    49          
              
Thus,              
Margin of Error E =    3.14846991          
Lower bound =    36.85153009          
Upper bound =    43.14846991          
              
Thus, the confidence interval is              
              
(   36.85153009   ,   43.14846991   ) [ANSWER]