The length of 10 carrots that are believed to be distributed

  The length of 10 carrots that are believed to be distributed N(, ² ) with an unknown ², had a sample mean of 7.2 inches and a sample standard deviation s = .25 inches. Find 98% Confidence Interval for ².

Solution

CI = (n-1) S^2 / ^2 right < ^2 < (n-1) S^2 / ^2 left
Where,
S = Standard Deviation
^2 right = (1 - Confidence Level)/2
^2 left = 1 - ^2 right
n = Sample Size

Since aplha =0.02
^2 right = (1 - Confidence Level)/2 = (1 - 0.98)/2 = 0.02/2 = 0.01
^2 left = 1 - ^2 right = 1 - 0.01 = 0.99
the two critical values ^2 left, ^2 right at 9 df are 21.666 , 2.088
S.D( S^2 )=0.25
Sample Size(n)=10
Confidence Interval = [ 9 * 0.0625/21.666 < ^2 < 9 * 0.0625/2.088 ]
= [ 0.5625/21.666 < ^2 < 0.5625/2.0879 ]
= [ 0.026 , 0.2694 ]