Suppose that Y is normally distributed with mean 0 and...

Suppose that $Y$ is normally distributed with mean 0 and unknown variance $\sigma^{2} .$ Then $Y^{2} / \sigma^{2}$ has a $x^{2}$ distribution with 1 df. Use the pivotal quantity $Y^{2} / \sigma^{2}$ to find a a. $95 \%$ confidence interval for $\sigma^{2}$ b. $95 \%$ upper confidence limit for $\sigma^{2}$. c. $95 \%$ lower confidence limit for $\sigma^{2}$.

Key Concepts

Normal Distribution

A normal distribution is a probability distribution that is symmetric about its mean, with its spread determined by its variance. In inference problems, such as this one, the assumption of normality allows for the exact derivation of the sampling distribution of functions of the data (like Y²), which is key for constructing confidence intervals.

Pivotal Quantity

A pivotal quantity is a function of both the observed data and the unknown parameter that has a probability distribution independent of the parameter. In this context, the ratio Y²/?² is a pivotal quantity because it follows a chi-square distribution with 1 degree of freedom regardless of the value of ?². This property enables us to form confidence intervals for ?².

Chi-Square Distribution

The chi-square distribution is a family of distributions that arise naturally when summing the squares of independent standard normal variables. Here, since Y is normally distributed, Y²/?² has a chi-square distribution with 1 degree of freedom. The known quantiles of the chi-square distribution are used to determine the endpoints of the confidence intervals for ?².

Confidence Interval Construction

Confidence intervals provide a range of values within which the true value of a parameter is believed to lie with a certain level of confidence (e.g., 95%). By inverting the probability statements associated with the pivotal quantity Y²/?² and utilizing the quantiles from its chi-square distribution, one can derive a two-sided confidence interval for the unknown variance ?².

One-sided Confidence Limits

One-sided confidence limits are derived when only an upper or lower bound is required for a parameter, rather than both bounds simultaneously. In this problem, the 95% upper and lower confidence limits for ?² are calculated by identifying the appropriate quantiles from the chi-square distribution, which yields bounds that guarantee, with 95% confidence, that ?² lies below or above the computed limit, respectively.

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