## 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.

In this tutorial 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 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.