Suppose that S^2 is the sample variance based on a sample of...

Suppose that $S^{2}$ is the sample variance based on a sample of size $n$ from a normal population with unknown mean and variance. Derive a $100(1-\alpha) \%$
a. upper confidence bound for $\sigma^{2}$.
b. lower confidence bound for $\sigma^{2}$.

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Key Concepts

Confidence Interval for Variance

A confidence interval for the population variance is derived by inverting the chi-square distribution relation. By finding the appropriate chi-square quantiles, one can establish upper and lower bounds that, with a specified level of confidence, contain the true variance of the underlying normal population.

Sample Variance

The sample variance is an unbiased estimator of the population variance, calculated as the sum of squared deviations from the sample mean divided by n-1 when the population mean is unknown. It quantifies the variability within a sample drawn from the population.

Chi-Square Distribution

When the underlying population is normally distributed, the statistic (n-1)S²/?² follows a chi-square distribution with n-1 degrees of freedom. This pivotal relationship is instrumental in setting up confidence intervals for the population variance.

Degrees of Freedom

Degrees of freedom, typically n-1 in the context of variance estimation, represent the number of independent values that can vary in the calculation of a statistic. It affects the shape of the chi-square distribution and is crucial in determining the critical values for confidence bounds.

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