Constructing a 95% Confidence Interval for Population Variance
The image shows a statistics problem that needs to be solved. The problem states:
"A simple random sample of size n is drawn from a population that is known to be normally distributed. The sample variance, s^2, is determined to be 13.0. Complete part (a) through (c).
(a) Construct a 95% confidence interval for σ^2. The sample size is n = 20."
To construct a 95% confidence interval for the population variance σ^2, we use the chi-squared (χ^2) distribution because the sample comes from a normally distributed population. The confidence interval is given by:
[(n-1)s^2/χ^2_(1-α/2), (n-1)s^2/χ^2_(α/2)]
Here, n is the sample size, s^2 is the sample variance, α is the significance level (1 - confidence level), and χ^2_(1-α/2) and χ^2_(α/2) are the critical values from the chi-squared distribution for degrees of freedom (df = n - 1) and significance levels (α/2 and 1 - α/2), respectively.
Given:
n = 20 (sample size)
s^2 = 13.0 (sample variance)
α = 1 - 0.95 = 0.05 (significance level since we are looking for a 95% confidence interval)
Degrees of freedom (df) = n - 1 = 20 - 1 = 19
The critical values for χ^2 distribution at the 95% confidence level for 19 degrees of freedom can be found in a χ^2 distribution table or using statistical software.
Let's assume we have determined the critical values:
χ^2_(0.025, 19) (lower critical value for α/2 = 0.025)
χ^2_(0.975, 19) (upper critical value for 1 - α/2 = 0.975)
Using the chi-squared distribution table or a calculator with chi-squared functionality, we get:
χ^2_(0.025, 19) ≈ 32.85 (rounded to two decimal places as needed)
χ^2_(0.975, 19) ≈ 8.91 (rounded to two decimal places as needed)
Now we can substitute these into the confidence interval formula:
Lower bound = (n-1) * s^2 / χ^2_(1-α/2) = 19 * 13.0 / 32.85 ≈ 7.55 (rounded to two decimal places)
Upper bound = (n-1) * s^2 / χ^2_(α/2) = 19 * 13.0 / 8.91 ≈ 27.90 (rounded to two decimal places)
Thus, the 95% confidence interval for σ^2 is (7.55, 27.90). Please note that the critical values I used were assumed for the explanation, and you need to check the actual χ^2 values from a chi-squared distribution table or use statistical software to find the accurate critical points.
