12  One variance test

12.1 What is-it?

  • A one variance test, also known as a one-sample variance test or a test of a single variance, is a statistical method used to assess whether the variance of a single sample significantly differs from a hypothesized population variance.

  • The one variance test is commonly used when working with continuous data and aims to determine if the observed variance in a sample is significantly different from a predetermined value or a hypothesized variance.

12.2 Data

  • \(x_1,\cdots,x_n\in\mathbb{R}\) assmued to be realizations of iid random variables \(X_1,\cdots, X_n\)

  • Sum of squares > - Mean is knwon: \(SS=\sum_{i=1}^n\left(X_i-\mu\right)^2\) > - Mean is unknwon: \(SS=\sum_{i=1}^n\left(X_i-\overline{X}\right)^2\)

12.3 Hypothesis

12.3.1 Null hypothesis

  • \(\mathcal{H}_0\): \(\sigma^2=\sigma_0^2\)

12.3.2 Alternative \(\mathcal{H}_1\):

  • Two-sided: \(\sigma^2\neq\sigma_0^2\)
  • Left-sided: \(\sigma^2<\sigma_0^2\)
  • Right-sided: \(\sigma^2>\sigma_0^2\)

12.4 Test statistic

  • \(\mathbb{X}^2=\dfrac{SS}{\sigma_0^2}\overset{\mathcal{H}_0}{\rightarrow}\mathcal{X}_{dof}^2\)
  • Where > - \(dof=n\) if mean \(\mu=\mathbb{E}[X]\) is known > - \(dof=n-1\) if mean is unknwon

12.5 Critical region and P-value

12.5.1 Critical Region

  • Let \(q_{\alpha}=q_{\alpha}\left(\mathcal{X}_{dof}^2\right)\)

  • Two-sided: \(W=\left(0, q_{\alpha/2}\right)\cup\left(q_{1-\alpha/2}, +\infty\right)\)

  • Left-sided: \(W=\left(0, q_{\alpha}\right)\)

  • Right-sided: \(W=\left(q_{1-\alpha}, +\infty\right)\)

12.5.2 P-value

  • \(pValue=\mathbb{P}\left(\mathbb{X}^2>\mathcal{X}_{obs}^2\right)\)

12.6 Decision

12.6.1 Decision based on Critical Region

  • Reject \(\mathcal{H}_0\) if and only if \(\mathcal{X}_{obs}^2\in W\)

12.6.2 Decision based on P-value

  • Reject \(\mathcal{H}_0\) if and only if \(pValue<\alpha\)