Let X1, X2, …, Xn be a random sample from N(μ, σ^2), where both parameters μand σ^2 are unknown.

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Let X1, X2, …, Xn be a random sample from N(μ, σ^2), where...

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 $P\left((n-1) S^{2} / \sigma^{2}<b\right)=0.975$ and $P(a<$ $\left.(n-1) S^{2} / \sigma^{2}<b\right)=0.95 .$ In $\mathrm{R}, \mathrm{b}=$ qchisq $(0.975, \mathrm{n}-1)$, while $\mathrm{a}=$ qchisq $(0.025, \mathrm{n}-1)$ (a) Show that this second probability statement can be written as $$ P\left((n-1) S^{2} / b<\sigma^{2}<(n-1) S^{2} / a\right)=0.95 $$ (b) If $n=9$ and $s^{2}=7.93$, find a $95 \%$ confidence interval for $\sigma^{2}$. (c) If $\mu$ is known, how would you modify the preceding procedure for finding a confidence interval for $\sigma^{2} ?$

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 $P\left((n-1) S^{2} / \sigma^{2}<b\right)=0.975$ and $P(a<$ $\left.(n-1) S^{2} / \sigma^{2}<b\right)=0.95 .$ In $\mathrm{R}, \mathrm{b}=$ qchisq $(0.975, \mathrm{n}-1)$, while $\mathrm{a}=$ qchisq $(0.025, \mathrm{n}-1)$
(a) Show that this second probability statement can be written as
$$
P\left((n-1) S^{2} / b<\sigma^{2}<(n-1) S^{2} / a\right)=0.95
$$
(b) If $n=9$ and $s^{2}=7.93$, find a $95 \%$ confidence interval for $\sigma^{2}$.
(c) If $\mu$ is known, how would you modify the preceding procedure for finding a confidence interval for $\sigma^{2} ?$

Key Concepts

Known vs. Unknown Mean

When the mean of the population is known, the estimation of the variance changes because the sample variance can be computed using the known mean, leading to a chi-squared distribution with n degrees of freedom instead of n-1. This adjustment provides more accurate bounds in the confidence interval for the variance since there is no loss of a degree of freedom for estimating the mean.

Sample Variance

The sample variance is a measure of the dispersion of data points around their mean in a sample. When sampling from a normal population, the scaled sample variance, specifically (n-1) times the sample variance divided by the true variance, follows a chi-squared distribution, which is key for constructing a confidence interval for the variance.

Chi-Squared Distribution

The chi-squared distribution is a probability distribution that arises when a sum of squared standard normal variables is considered. In the variance estimation of normally distributed data, the statistic formed by scaling the sample variance is distributed as a chi-squared variable with degrees of freedom corresponding to the sample size minus one.

Degrees of Freedom

Degrees of freedom refer to the number of independent values or quantities which can vary in an analysis without breaking any constraints. For the sample variance of a normally distributed population, n-1 degrees of freedom are used because one degree of freedom is lost in estimating the mean from the data.

Confidence Interval

A confidence interval is a range of values used to estimate an unknown parameter of a population. In the context of variance estimation, it provides a lower and upper bound such that there is a specified probability (confidence level) that the interval contains the true variance of the underlying normal distribution.

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