A random sample of size 28 is drawn from a normal population

A random sample of size 28 is drawn from a normal population. The summary statistics are x=68.6 and s = 1.28.

Construct a 95% confidence interval for the population mean .

Construct a 99.5% confidence interval for the population mean .

Comment on why one interval is longer than the other.

Solution

a.
CI = x ± t a/2 * (sd/ Sqrt(n))
Where,
x = Mean
sd = Standard Deviation
a = 1 - (Confidence Level/100)
ta/2 = t-table value
CI = Confidence Interval
Mean(x)=68.6
Standard deviation( sd )=1.28
Sample Size(n)=28
Confidence Interval = [ 68.6 ± t a/2 ( 1.28/ Sqrt ( 28) ) ]
= [ 68.6 - 2.0518 * (0.242) , 68.6 + 2.0518 * (0.242) ]
= [ 68.104,69.096 ]

b.
AT 99.5 confidenec level
Confidence Interval = [ 68.6 ± t a/2 ( 1.28/ Sqrt ( 28) ) ]
= [ 68.6 - 3.0565 * (0.242) , 68.6 + 3.0565 * (0.242) ]
= [ 67.861,69.339 ]
c.
value of t a/2 is different at each of the confidence level