(a) The "quantile part" for the confidence interval is:
χ²(α/2) = 7.779 and χ²(1 - α/2) = 0.773.
(b) The "standard deviation" part for the confidence interval is: √49,954.62 ≈ 223.80 for the lower bound and √4,690.26 ≈ 68.49 for the upper bound.
Confidence Interval for the Standard Deviation:
We're trying to estimate the population standard deviation σ, with a sample standard deviation s = 188, a sample size n = 5, and a confidence level of 80%. Since the population variance is unknown and the sample size is small, we'll use the chi-square distribution to construct the confidence interval.
Degrees of freedom: Calculate the degrees of freedom
ν = n - 1 = 5 - 1 = 4.
Chi-square quantiles:
Find the chi-square quantiles χ²(α/2) and χ²(1 - α/2) for the desired confidence level α = 1 - 0.8 = 0.2.
Using a chi-square table or calculator, we get χ²(α/2) = 7.779 and χ²(1 - α/2) = 0.773.
Confidence interval: Calculate the lower and upper bounds of the confidence interval:
Lower bound: s² / χ²(1 - α/2) = 188² / 0.773 ≈ 49,954.62
Upper bound: s² / χ²(α/2) = 188² / 7.779 ≈ 4,690.26
Take the square root of both bounds to get the interval in terms of standard deviation:
Lower bound: √49,954.62 ≈ 223.80
Upper bound: √4,690.26 ≈ 68.49