Question

A simple random sample from a population with a normal distribution of 109 body temperatures has x = 98.50 degrees Upper F and s =0.69degrees Upper F. Construct a 90​% confidence interval estimate of the standard deviation of body temperature of all healthy humans.

____ degree F < o <____ degree F

Answer 1

CONFIDENCE INTERVAL FOR STANDARD DEVIATION
ci = (n-1) s^2 / ᴪ^2 right < σ^2 < (n-1) s^2 / ᴪ^2 left
where,
s = standard deviation
ᴪ^2 right = (1 - confidence level)/2
ᴪ^2 left = 1 - ᴪ^2 right
n = sample size
since alpha =0.1
ᴪ^2 right = (1 - confidence level)/2 = (1 - 0.9)/2 = 0.1/2 = 0.05
ᴪ^2 left = 1 - ᴪ^2 right = 1 - 0.05 = 0.95
the two critical values ᴪ^2 left, ᴪ^2 right at 108 df are 133.2569 , 85.015
s.d( s )=0.69
sample size(n)=109
confidence interval for σ^2= [ 108 * 0.4761/133.2569 < σ^2 < 108 * 0.4761/85.015 ]
= [ 51.4188/133.2569 < σ^2 < 51.4188/85.0149 ]
[ 0.3859 < σ^2 < 0.6048 ]
and confidence interval for σ = sqrt(lower) < σ < sqrt(upper)
= [ sqrt (0.3859) < σ < sqrt(0.6048), ]
= [ 0.6212 < σ < 0.7777 ]
90​% confidence interval estimate of the standard deviation of body temperature of all healthy humans.
[ 0.6212 < σ < 0.7777 ]