Chi-square statistic for testing single variance:

The Chi-square statistic is used to test the population variance of a single sample.

The necessary assumptions for Chi-square test:

The sample should be collected using simple random sampling.

The population from which the sample is drawn should follow normal distribution.

The data should be continuous.

The chi-square test statistic is obtained as given below:

\(\displaystyle{x}^{2}=\frac{{{\left({n}-{1}\right)}{s}^{2}}}{{\sigma^{2}}}\)

\(\displaystyle{n}=\) Sample size

\(\displaystyle{s}^{2}=\) Sample variance

\(\displaystyle\sigma^{2}=\) Population variance

Decision rule based on P-value approach for both directional and non-directional tests:

The level of significance is \(\alpha.\)

If P-value \(\displaystyle\le\alpha\), then reject the null hypothesis \(\displaystyle{H}_{{0}}\).

If P-value \(\displaystyle>\alpha\), then fail to reject the null hypothesis \(\displaystyle{H}_{{0}}\).

P-value:

The P-value will be obtained from the chi-square distribution table based on the value of test statistic and the degrees of freedom \(\displaystyle{\left({n}–{1}\right)}\) and the type of hypothesis test (Two tailed, right tailed or left tailed).

Chi-square statistic for testing distribution:

The Chi-square goodness of fit test is used to test whether the sample data are consistent with a hypothesized distribution or not.

The chi-square goodness-of-fit test is used to test whether a sample of data comes from a population with a specific distribution. The chi-square goodness-of-fit can also be applied to discrete distributions

The necessary assumptions for Chi-square test for goodness of fit are given below:

The sample should be collected using simple random sampling.

The variable of interest must be categorical.

The expected value of each cell should not be less than 5.

Evidently, the test is to determine whether a sample of data comes from a population with a specific distribution.

Chi-square goodness of fit is a right tailed test. Therefore, it is a directional test.

The chi-square test statistic is obtained as given below:

\(\displaystyle{x}^{2}=\frac{{{\sum_{{{i}={1}}}^{{n}}}{\left({O}_{{i}}-{E}_{{i}}\right)}^{2}}}{{{E}_{{i}}}}\)

\(\displaystyle{O}_{{i}}=\) Observed frenquency

\(\displaystyle{E}_{{i}}=\) Expected frequency

Decision rule based on P-value approach:

The level of significance is \(\alpha.\)

If P-value \(\displaystyle\le\alpha\), then reject the null hypothesis \(\displaystyle{H}_{{0}}\).

If P-value \(\displaystyle>\alpha\), then fail to reject the null hypothesis \({H}_{{0}}\).

The P-value will be obtained from the chi-square distribution table based on the value of test statistic and the degrees of freedom \(\displaystyle{\left({n}–{1}\right)}\) for the right tailed test.