Chi-square test for variance

## Testing variance or standard deviation

In this tutorial we will discuss a method for testing a claim made about the population variance $\sigma^2$ or population standard deviation $\sigma$. To test the claim about the population variance or population standard deviation we use chi-square test.

We will explain the six steps approach used in hypothesis testing to test hypothesis about the population variance or population standard deviation.

## Chi-square Test for Variance

Let $X_1, X_2, \cdots, X_n$ be a random sample from a normal population with mean $\mu$ and variance $\sigma^2$.

Let $\overline{x}=\frac{1}{n} \sum x_i$ be the sample mean and $s^2=\frac{1}{n-1} \sum (x_i-\overline{x})^2$ be the sample variance.

## Assumptions

a. The sample must be randomly selected from the population.

b. The population must be normally distribution for the variable under study.

c. The observations must be independent.

## Step by Step Procedure

We wish to test the null hypothesis $H_0 : \sigma^2 = \sigma^2_0$, where $\sigma^2_0$ is the specified value of the

population variance.

#### Step 1 State the hypothesis testing problem

The hypothesis testing problem can be structured in any one of the three situations as follows:

Situation | Hypothesis Testing Problem |
---|---|

Situation A : | $H_0: \sigma^2=\sigma^2_0$ against $H_a : \sigma^2 < \sigma^2_0$ (Left-tailed) |

Situation B : | $H_0: \sigma^2=\sigma^2_0$ against $H_a : \sigma^2 > \sigma^2_0$ (Right-tailed) |

Situation C : | $H_0: \sigma^2=\sigma^2_0$ against $H_a : \sigma^2 \neq \sigma^2_0$ (Two-tailed) |

#### Step 2 Define the test statistic

The test statistic for testing above hypothesis is

$$

\chi^2 =\frac{(n-1)s^2}{\sigma^2}

$$

The test statistic $\chi^2$ follows $\chi^2$ distribution with $n-1$ degrees of freedom.

#### Step 3 Specify the level of significance $\alpha$

#### Step 4 Determine the critical values

For the specified value of $\alpha$ determine the critical region depending upon the alternative hypothesis.

- For
**left-tailed**alternative hypothesis: Find the $\chi^2$-critical value using

` $$ P(\chi^2\leq \chi^2 _{1-\alpha,n-1}) = \alpha. $$ `

- For
**right-tailed**alternative hypothesis: $\chi^2_\alpha$.

` $$ P(\chi^2\geq\chi^2_{\alpha, n-1}) = \alpha. $$ `

- For
**two-tailed**alternative hypothesis: $\chi^2_{\alpha/2}$.

` $$ P(\chi^2\leq \chi^2 _{1-\alpha/2,n-1} \text{ or } \chi^2\geq \chi^2_{\alpha/2,n-1}) = \alpha. $$ `

#### Step 5 Computation

Compute the test statistic under the null hypothesis $H_0$ using

` $$ \chi^2_{obs} = \frac{(n-1)s^2}{\sigma^2_0} $$ `

#### Step 6 Decision (Traditional Approach)

Based on the critical values.

- For
**left-tailed**alternative hypothesis: Reject $H_0$ if`$\chi^2_{obs}\leq \chi^2_{1-\alpha,n-1}$`

. - For
**right-tailed**alternative hypothesis: Reject $H_0$ if`$\chi^2_{obs}\geq \chi^2_{\alpha,n-1}$`

. - For
**two-tailed**alternative hypothesis: Reject $H_0$ if`$\chi^2_{obs}\leq \chi^2_{1-\alpha/2, n-1}$`

or`$\chi^2_{obs}\geq \chi^2_{\alpha/2, n-1}$`

.

**OR**

#### Step 6 Decision ($p$-value Approach)

It is based on the $p$-value.

Alternative Hypothesis | Type of Hypothesis | $p$-value |
---|---|---|

$H_a: \sigma^2<\sigma^2_0$ | Left-tailed | $p$-value $= P(\chi^2\leq \chi^2_{obs})$ |

$H_a: \sigma^2>\sigma^2_0$ | Right-tailed | $p$-value $= P(\chi^2\geq \chi^2_{obs})$ |

$H_a: \sigma^2\neq \sigma^2_0$ | Two-tailed | $p$-value $= 2P(\chi^2\geq \chi^2_{obs})$ |

If $p$-value is less than $\alpha$, then reject the null hypothesis $H_0$ at $\alpha$ level of significance, otherwise fail to reject $H_0$ at $\alpha$ level of significance.

## Endnote

In this tutorial, you learned the $\chi^2$-test for testing population variance and the assumptions for $\chi^2$-test for testing population variance. You also learned about the step by step procedure to apply $\chi^2$-test for testing population variance.

To learn more about other hypothesis testing problems, hypothesis testing calculators and step by step procedure, please refer to the following tutorials:

Let me know in the comments if you have any questions on **$\chi^2$-test for population variance** and your thought on this article.