Question

For a sample drawn from a normally distributed population, the statistic \[ Y = \frac{(n-1)s^2}{\sigma^2} \] where \( n \) = sample size, \( \sigma \) = population standard deviation, and \( s \) = sample standard deviation, has

• A. \(\chi^2 \text{ distribution with } (n - 1) \text{ degrees of freedom}\)
• B. \(\chi^2 \text{ distribution with } n \text{ degrees of freedom}\)
• C. \(\chi^2 \text{ distribution with } (n + 1) \text{ degrees of freedom}\)
• D. \(\text{Gaussian distribution with } n \text{ degrees of freedom}\)

Hint:

Whenever you're dealing with the ratio of sample variance to population variance for normally distributed populations, recall that it follows a Chi-square distribution with \( n-1 \) degrees of freedom.

Answer 1

The Correct Option is A

In inferential statistics, when we draw a random sample of size \( n \) from a normal population with known population variance \( \sigma^2 \), the statistic \[ Y = \frac{(n-1)s^2}{\sigma^2} \] follows a Chi-square distribution with \( (n - 1) \) degrees of freedom. This result comes from the fundamental property of the sampling distribution of variance from a normal population, where the scaled sample variance follows the chi-square distribution with degrees of freedom equal to one less than the sample size.