A random sample of size 256 is drawn from a population whose

A random sample of size 256 is drawn from a population whose distribution, mean, and standard deviation are all unknown. The summary statistics are x = 1011ands = 34.

a. Construct a 90% condence interval for the population mean .

b. Construct a 99% condence interval for the population mean .

c. 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)=101
Standard deviation( sd )=34
Sample Size(n)=256
Confidence Interval = [ 101 ± t a/2 ( 34/ Sqrt ( 256) ) ]
= [ 101 - 1.651 * (2.125) , 101 + 1.651 * (2.125) ]
= [ 97.492,104.508 ]

b)
AT 99% C.I
Confidence Interval = [ 101 ± t a/2 ( 34/ Sqrt ( 256) ) ]
= [ 101 - 2.595 * (2.125) , 101 + 2.595 * (2.125) ]
= [ 95.486,106.514 ]
c)
Value of t-table value is diffrent at 95%, 99% interval. Thus, it changes the result