Chi-squared test for a variance
Section of Hypothesis tests, part of Statistical inference
Chi-squared test for a variance
| Type | Section |
| Taught in | Hypothesis tests |
| Part | Statistical inference |
For the variance, (n-1)s^2/\sigma_0^2 is \chi^2_{n-1} when \sigma^2=\sigma_0^2. Reject for large values against \sigma^2>\sigma_0^2, using \chi^2_{n-1;1-\alpha}; for small values against \sigma^2<\sigma_0^2, using \chi^2_{n-1;\alpha}; and in either tail, at \alpha/2 each, against \sigma^2\neq\sigma_0^2.
Example 1 (Tuning-fork variability) Tuning-fork frequencies have been \operatorname{N}(\mu,1.1). A new process is meant to reduce the variance. A batch of 7 new forks has sample variance 0.488.
Question. Is there evidence that the new process has smaller variance?
Solution. Test H_0:\sigma^2=1.1 against H_1:\sigma^2<1.1 with 6s^2/1.1, which is \chi^2_6 under H_0. Its value is 6\times0.488/1.1=2.66, and the p-value is \mathbb{P}(\chi^2_6\le2.66)=0.15. At level 0.05 the critical region is (0,\ 1.635], and 2.66 is not in it. There is not enough evidence that the new process is less variable. With only 7 forks, the sample variance would have to be below 0.30 to reject.
Used in
See also
Null and alternative hypotheses · Critical region · p-value · Type I and type II errors · Neyman–Pearson approach · Power function · One-sample t test · Two-sample t test