# Hypothesis Tests for Variance | CFA Level I Quantitative Methods

## Understanding the Chi-Square Test

The chi-square test is used for tests concerning the variance of a normally distributed population. Like in the case of testing on the mean, we have a hypothesized value of the population variance. Denoting this as **σ²₀**, we have three structures for stating the hypotheses.

This test concerning variance requires the use of the chi-square distribution and its test statistic. The distribution is characterized by n-1 degrees of freedom. As variance cannot be negative, the distribution is asymmetrical and starts from zero. Its shape approaches the normal distribution as the degrees of freedom increases.

## Reading a Chi-Square Table

Let’s illustrate this with a two-tailed test with a 10% significance level and 10 degrees of freedom.

To read a chi-square table, note that the probability to the right of the chi-square value is denoted in the header row. To find the chi-square value where 5% is to the right of it, we look under this column. Match it with the degrees of freedom, and we get the critical value of **18.307** for the right tail.

For the critical value on the left tail, we want 5% to the left of it, which is 95% to the right of it. So likewise, looking under the column where the probability to the right is 0.95, we get the critical value of **3.94** for the left tail.

So, for this particular two-tail test, the rejection regions are to the left and to the right of the critical values. The decision rule is therefore to reject H₀ if the chi-square value is less than 3.94 or if the chi-square is greater than **18.307**.

The test statistic is the degrees of freedom multiplied by the sample variance, divided by the hypothesized value for the population variance.

**EXAMPLE**

**Suzie Li would like to investigate if a fund manager is under-reporting the standard deviation of a fund. The monthly standard deviation of the fund is advertised at 6%. Suzie obtained a random sample of 15 monthly standard deviations of the fund and measured a standard deviation of monthly returns of 8.2%. Is this sufficient evidence, at a 5% significance level, to prove that the standard deviation of the fund is greater than the advertised rate of 6%?**

Let’s break down the solution step by step:

**State the hypotheses**: We denote σ² as the variance for returns for the fund and s² as the variance of returns for the random sample. Since what we would like to prove is that the standard deviation is greater than the advertised value of 6%, our alternative hypothesis is that σ² is greater than 0.06². Remember to square the value as we are dealing with standard deviation here. The null hypothesis is therefore σ² ≤ 0.0036.**Select an appropriate test statistic**: Since this is a test concerning a single variance, we use the chi-square statistic.**Specify the significance level**: Given in the problem, the significance level is 5%.**State the decision rule**: The sample size is 15, so there are 14 degrees of freedom. Looking up the chi-square table where the right tail probability is 5%, we get the critical value of**23.685**. The decision rule is therefore to reject H₀ when the chi-square is more than 23.685.**Calculate the test statistic**: Using the formula, we calculate the chi-square value as**26.149**.**Make the statistical decision**: Since the test statistic is greater than 23.685, we reject H₀ at the 5% significance level. We conclude that the actual standard deviation is higher than the advertised rate of 6%.

## Testing Equality of Variance Between Two Populations

In this lesson, we’ll explore another common type of hypothesis test regarding variance – the test of equality of variance between two populations. We’ll use the F-distribution for this test, also known as the F-test.

## Understanding the F-test

The F-test is a test concerning the equality or inequality of variances between two populations. It assumes that both populations are normally distributed, and the samples are independent. Unlike the t-test, which tests the difference between means, the F-test focuses on variances.

The F-distribution is characterized by the degrees of freedom of the two sample sizes (**df _{1}= n_{1}-1**,

**df**). Since variance cannot be negative, the distribution is asymmetrical and starts from zero.

_{2}= n_{2}-1**EXAMPLE**

Now you’ve learned how to use the F-test to compare the variances of two populations. With this knowledge, you’ll be better prepared to tackle problems related to variance in the CFA Level I Quantitative Methods section.

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