11.6 Test of a Single Variance

Test of a Single Variance

The chi-square test for a single variance lets you determine whether the spread of data in a population matches a hypothesized value. Instead of testing a mean, you're testing how variable the data is. This matters in fields like manufacturing, where too much or too little variation in a product can signal a problem.

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Test Statistic for Single Variance

The test statistic compares your observed sample variance to the variance you'd expect under the null hypothesis. The formula is:

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

• $n$ = sample size (number of observations)
• $s^2$ = sample variance (calculated from your data)
• $\sigma_0^2$ = hypothesized population variance (the value stated in $H_0$)

The result follows a chi-square distribution with $n - 1$ degrees of freedom. That distribution is what you'll use to find your p-value.

How to calculate it, step by step:

1. Identify your sample size $n$ from the problem (e.g., $n = 50$)
2. Calculate or find the sample variance $s^2$ from the data (e.g., $s^2 = 25.6$)
3. Identify the hypothesized population variance $\sigma_0^2$ from the null hypothesis (e.g., $\sigma_0^2 = 20$)
4. Plug into the formula: $\chi^2 = \frac{(50-1)(25.6)}{20} = \frac{49 \times 25.6}{20} = \frac{1254.4}{20} = 62.72$

A large $\chi^2$ value suggests the sample variance is much bigger than the hypothesized variance. A small value suggests it's much smaller. Whether that difference is statistically significant depends on the tail direction and your significance level.

Hypotheses for Variance Testing

The null hypothesis always claims the population variance equals a specific value:

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

The alternative hypothesis depends on what the research question is asking. There are three options:

• Right-tailed: $H_a: \sigma^2 > \sigma_0^2$ (you suspect variance is greater than the hypothesized value)
• Left-tailed: $H_a: \sigma^2 < \sigma_0^2$ (you suspect variance is less than the hypothesized value)
• Two-tailed: $H_a: \sigma^2 \neq \sigma_0^2$ (you suspect variance is different in either direction)

Setting up your hypotheses:

1. Read the problem carefully. Look for keywords like "exceeds," "less than," or "different from" to determine the direction.
2. Write $H_0: \sigma^2 = \sigma_0^2$ using the specific value given (e.g., $H_0: \sigma^2 = 40$).
3. Write $H_a$ with the appropriate inequality (e.g., if the question asks whether variance has decreased, use $H_a: \sigma^2 < 40$).

Tail Direction in Variance Tests

The tail direction controls where you look on the chi-square distribution to find your p-value.

• Right-tailed test: You reject $H_0$ if your $\chi^2$ statistic falls in the upper tail. This tests whether the true variance is larger than claimed. For example, a quality inspector might test whether the variance in bottle fill amounts exceeds the acceptable standard.
• Left-tailed test: You reject $H_0$ if your $\chi^2$ statistic falls in the lower tail. This tests whether the true variance is smaller than claimed. For example, you might test whether a new training program reduced variability in employee performance scores.
• Two-tailed test: You reject $H_0$ if your $\chi^2$ statistic falls in either tail. This tests whether the variance is simply different from the hypothesized value, regardless of direction. You split your significance level $\alpha$ between both tails ($\alpha/2$ in each).

Statistical Inference and Decision Making

After computing your $\chi^2$ statistic, you compare it to a critical value from the chi-square table (using $n - 1$ degrees of freedom and your chosen $\alpha$), or you find the p-value directly.

• The significance level ($\alpha$) is the threshold you set before testing, typically 0.05. It represents the probability of rejecting $H_0$ when it's actually true (a Type I error).
• If your p-value $\leq \alpha$, you reject $H_0$. If your p-value $> \alpha$, you fail to reject $H_0$.

One thing to watch: the chi-square distribution is not symmetric, so critical values for left-tailed and right-tailed tests are not mirror images of each other. Always use the correct tail when looking up values in a chi-square table.