5 Use the following data to construct confidence intervals f

5. Use the following data to construct confidence intervals for the population mean and population var o2 both at 95% confidence limits. Data: 90 88 67 70 93 87 60 100 82 83

Solution

The sample mean and standard deviations are

X = 82
s = 12.57864151

*************

For the 95% confidence interval for population mean:

Note that              
Margin of Error E = t(alpha/2) * s / sqrt(n)              
Lower Bound = X - t(alpha/2) * s / sqrt(n)              
Upper Bound = X + t(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    82          
t(alpha/2) = critical t for the confidence interval =    2.262157163          
s = sample standard deviation =    12.57864151          
n = sample size =    10          
df = n - 1 =    9          
Thus,              
Margin of Error E =    8.998218072          
Lower bound =    73.00178193          
Upper bound =    90.99821807          
              
Thus, the confidence interval is              
              
(   73.00178193   ,   90.99821807   ) [ANSWER]

******************
For the 95% confidence interval for population variance:

As              
              
df = n - 1 =    9          
alpha = (1 - confidence level)/2 =    0.025          
              
Then the critical values for chi^2 are              
              
chi^2(alpha/2) =    19.0227678          
chi^2(alpha/2) =    2.7003895          
              
Thus, as              
              
lower bound = (n - 1) s^2 / chi^2(alpha/2) =    74.8576661          
upper bound = (n - 1) s^2 / chi^2(1 - alpha/2) =    527.331335          
              
Thus, the confidence interval for the variance is              
              
(   74.8576661   ,   527.331335   ) [ANSWER]