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 (df1= n1-1, df2= n2-1). Since variance cannot be negative, the distribution is asymmetrical and starts from zero.
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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