Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from \(N\left(\mu, \sigma^{2}\right)\), where both parameters \(\mu\) and \(\sigma^{2}\) are unknown. A confidence interval for \(\sigma^{2}\) can be found as follows. We know that \((n-1) S^{2} / \sigma^{2}\) is a random variable with a \(\chi^{2}(n-1)\) distribution. Thus we can find constants \(a\) and \(b\) so that \(P\left((n-1) S^{2} / \sigma^{2}

Short answer

a) Proven using properties of distributions and inequalities. b) Confidence interval is calculated by first determining \(a\) and \(b\) from chi-square distribution with \(n-1\) degrees of freedom, then substituting these, \(n=9\) and \(s^2 = 7.93\) into the formula. c) If known, \(\mu\) would cause the degrees of freedom to be \(n\) instead of \(n-1\), thus changing the calculations of a, b and altering the formula of the confidence interval.

Step-by-step solution

Proof of the Probability Statement

The given inequality is: \((n-1) S^{2} / b<\sigma^{2}<(n-1) S^{2} / a\). Now to prove this, remember the properties of probability and inequalities. Properties of distributions can be used here. Since \(a\) and \(b\) are values on the \(\chi^{2}\) distribution, which is strictly positive and \(a 1/b\). Thus, the statement is proven.

Calculation of Confidence Interval

Given, \(n=9\) and \(s^{2}=7.93\). Use these values to calculate \(a\) and \(b\) first. These are determined by calculating the \(\chi^{2}\) distribution quantile function in R; \(a=qchisq(0.025, n-1)\) and \(b=qchisq(0.975, n-1)\). Then substitute these along with the given \(n\) and \(s\) into the confidence interval formula, \((n-1) S^{2} / b<\sigma^{2}<(n-1) S^{2} / a\). This will provide the 95% confidence interval.

Modification with Known Mean

If the population mean, \(\mu\), is known, then we wouldn't estimate it from the data and degrees of freedom would be \(n\) instead of \(n-1\). Accordingly, the denominator in the \(\chi^{2}\) distribution would be \(n\) and the estimation of \(s^{2}\) would be different. . Thus the constants would be calculated as : \(a=qchisq(0.025, n)\) and \(b=qchisq(0.975, n)\). The confidence interval formula will then be altered.