Statistic Distribution for Normal Population

The question asks for the distribution of the statistic $Y = \frac{(n-1)s^2}{\sigma^2}$, where a sample is drawn from a normally distributed population. Here:

• $n$ = sample size
• $s^2$ = sample variance
• $\sigma^2$ = population variance

Deriving the Chi-square Distribution

A fundamental result in statistical theory states that if a sample of size $n$ is taken from a normally distributed population with variance $\sigma^2$, then the quantity $\frac{(n-1)s^2}{\sigma^2}$ follows a specific probability distribution.

This statistic, $Y$, is known to follow the Chi-square distribution.

The degrees of freedom associated with this Chi-square distribution are determined by the sample size, specifically $(n-1)$.

Therefore, the statistic $Y = \frac{(n-1)s^2}{\sigma^2}$ follows a Chi-square distribution with $(n-1)$ degrees of freedom.