# 11. Hypothesis Tests Concerning a Single Variance

## j. identify the appropriate test statistic and interpret the results for a hypothesis test concerning 1) the variance of a normally distributed population, and 2) the equality of the variances of two normally distributed populations based on two independent random samples;

How would you express a null and alternative hypothesis if you wanted to test whether the variation from a single population is statistically equal to some hypothesized value?

$$H_0: \sigma^2 = \sigma_0^2$$

versus

$$H_1: \sigma^2 \neq \sigma_0^2$$

How does a chi-square graph differ from a z-graph or t-graph?

The chi-square graph is positively skewed and truncated at zero (not defined for negative values).

How is the chi-square statistic defined in math notation?

$$X^2$$

What is the formula for the test-statistic for chi-square with `n - 1`

degrees of freedom?

$$X^2 = \frac{(n - 1)s^2}{\sigma_0^2}$$

Does the chi-square probability table look more like a z-square or t-square table? Why? What is the title of the table?

It's more like a t-square table because it uses degrees of freedom.

The title is "Probability in the right tail"

Unlike the t-test, the chi-square test is sensitive to violations of its assumptions. What are some of the ways a chi-square test can be faulty?

If a sample is not actually random, or if it does not come from a normally distributed population, inferences based on a chi-square test are likely to be faulty.

##### Source:

- CFA