📂Statistical Test

Estimation of Population Variance for Normally Distributed Groups

Hypothesis Testing ¹

Assume the distribution of a population with a sample size of $n$ follows a normal distribution $N \left( \mu , \sigma^{2} \right)$. The hypothesis test for the candidate $\sigma_{0}$ of the population variance is as follows.

• $H_{0}$: $\sigma^{2} = \sigma_{0}^{2}$
• $H_{1}$: $\sigma^{2} \neq \sigma_{0}^{2}$

Test Statistic

The test statistic for the sample variance $S^{2}$ is as follows. $$\mathcal{X}^{2} = \frac{ \left( n - 1 \right) S^{2} }{ \sigma_{0}^{2} }$$ Under the assumption that the null hypothesis is true, this test statistic follows a chi-squared distribution with degrees of freedom of $(n-1)$.

Explanation

The hypothesis is performed through a two-tailed test to check whether $\mathcal{X}^{2}$ falls between $\chi^{2}_{1 - \alpha} (n-1)$ and $\chi^{2}_{\alpha} (n-1)$ at a significance level of $\alpha$. If $\mathcal{X}^{2}$ falls within this range, the null hypothesis cannot be rejected, leading to the conclusion that the population variance is $\sigma_{0}^{2}$. $$\chi^{2}_{\alpha/2} (n-1) \le \mathcal{X}^{2} \le \chi^{2}_{1 - \alpha/2} (n-1)$$