Let
/math-3ac6004d77c0cc0055e95c99b9dfd7e0.png?v=0 "\sigma\,\!")
be the variance of sample/math-6373accf16c083723e8abae2f5401af2.png?v=0 "x\,\!")
and /math-dd695fbfdff84fbbf7d7fd4da890ad37.png?v=0 "\sigma_0\,\!")
is the hypothetical variance, this function tests the hypotheses:
Let be the variance of sample and is the hypothetical variance, this function tests the hypotheses:
/math-b62036202bb3c073a0f36d26144e465f.png?v=0 "H_0:\sigma = \sigma_0\,\!")
vs /math-fe006680bc392bd1d6a3eb90690a53f4.png?v=0 "H_1:\sigma \ne \sigma_0\,\!")
/math-ab76f24b6b8359ea08c9e3b4f35636ce.png?v=0 "H_0:\sigma\le\sigma_0")
vs /math-5f7e685d4c40faee1093470f026af541.png?v=0 "H_1:\sigma > \sigma_0\,\!")
/math-dc64eb17546349239d1776a3fe745e68.png?v=0 "H_0:\sigma\ge\sigma_0")
vs /math-d1a196c09895cc49fc4e06147097beb0.png?v=0 "H_1:\sigma < \sigma_0\,\!")
To compare the variance, we firstly compute the Chi-square value by:
/math-802a5896ddc6b75b6313ca8885a0949b.png?v=0 "x^2=\frac{(n-1)s^2}{\sigma_0^2}")
where /math-215891908ca120202db0e329cbccedcf.png?v=0 "s^2\,\!")
is the sample variance. For a given significance level, /math-06f7895cd704b1cb0921cf98aec71926.png?v=0 "\alpha\,\!")
, we will reject the null hypothesis /math-806277203dedea2ed8321f6cbd465a54.png?v=0 "H_0\,\!")
when:
/math-996bafe12348ca565dcd9ebb5c77136a.png?v=0 "|x^2| \ne \chi_{\alpha/2}^2\,\!")
, for two tailed test
/math-2cfd4ccaa5e40e5f0afdf28485c227be.png?v=0 "x^2>\chi_\alpha^2")
, for upper tailed test
/math-b9d2be7cfd6da04e4afda8d352e0b2e0.png?v=0 "x^2<\chi_{1-\alpha}^2")
, for lower tailed test
And the confidence interval for the sample variance can be generated by:
| Null Hypothesis | Confidence Interval |
|---|---|
|
|
/math-671806157726c91b7677f32b423d92cf.png?v=0 "\sqrt{\frac{(n-1)s^2}{\chi_{(\alpha/2,n-1)}^2}} \le \sigma \le \sqrt{\frac{(n-1)s^2}{\chi_{(1-\alpha/2,n-1)}^2}}")
|
|
|
/math-5ad36c1dfa487fc8d86ea6ae38a664df.png?v=0 "\sqrt{\frac{(n-1)s^2}{\chi_{(\alpha,n-1)}^2}} \le \sigma \le \infty")
|
|
|
/math-06dbb894cfaf3aaa3bc222f615531abf.png?v=0 "0 \le \sigma \le \sqrt{\frac{(n-1)s^2}{\chi_{(1-\alpha,n-1)}^2}}")
|