Chi-Squared Distribution and Hypothesis Testing for Standard Deviation

Critical Values from the Chi-Squared Distribution

• When the population is normal, it is possible to construct confidence intervals for the standard deviation or variance.

• These confidence intervals are based on the chi-squared distribution.

• There are many different chi-squared distributions, each with a different number of degrees of freedom.

Characteristics of the Chi-Squared Distribution

• The chi-squared distributions are not symmetric; they are skewed to the right.

• Values of the chi-squared test statistic are always greater than or equal to zero.

Critical Values

• Critical values for a level $100(1 - \alpha)\%$ confidence interval are the values that contain the middle $100(1 - \alpha)\%$ of the area under the curve between them.

• The notation for the critical values tells how much area is to the right of the critical value.

• For a level $1 - \alpha$ confidence interval, the critical values are denoted:

  • $\chi^2_{1 - \alpha/2}$ which has an area of $1 - \alpha/2$ to its right.

  • $\chi^2_{\alpha/2}$ which has an area of $\alpha/2$ to its right.

Example

• Find the critical values for a 95% confidence interval using the chi-squared distribution with 10 degrees of freedom.

• The confidence level is 95%, so the critical values are the values that contain the middle 95% of the area under the curve between them.

• The lower critical value, denoted $\chi^2_{0.975}$, has an area of 0.975 to its right.

• The upper critical value, denoted $\chi^2_{0.025}$, has an area of 0.025 to its right.

• Using table A4, the critical values are found at the intersection of the row corresponding to 10 degrees of freedom and the columns corresponding to 0.975 and 0.025.

• Thus, the critical values are 3.247 and 20.483.

Hypothesis Tests for a Standard Deviation

• Hypothesis tests for a standard deviation are based on the chi-squared distribution.

• There are many different chi-squared distributions, each with a different number of degrees of freedom.

• Recall that the chi-squared distributions are skewed to the right, and the values of the chi-squared statistic are always greater than or equal to zero.

• The notation $\chi^2_{\alpha}$ represents the value that has an area of $\alpha$ to its right.

• Table A4 is consulted to find critical values associated with the distribution.

Finding Critical Values

• To find the critical value $\chi^2_{0.05}$ for a chi-squared distribution with 10 degrees of freedom, consult table A4.

• The critical value is located at the intersection of the row corresponding to 10 degrees of freedom and the column corresponding to $\alpha = 0.05$. The critical value is 18.307.

Hypothesis Testing

• The null hypothesis for a standard deviation $\sigma$ is of the form $\sigma = \sigma_0$.

• The test is based on the fact that if the null hypothesis is true, then the test statistic: $\chi^2 = \frac{(n - 1)s^2}{\sigma_0^2}$

  • has a chi-squared distribution with $n - 1$ degrees of freedom.

• Caution: The methods of this section apply only for samples drawn from a normal distribution.

  • If the distribution differs even slightly from normal, these methods should not be used.

Steps for Performing a Hypothesis Test for a Standard Deviation

1. Check to be sure that the assumptions are satisfied.

2. State the null and alternate hypotheses.

3. Choose a significance level $\alpha$ and find the critical value based on whether it is a left-tailed, a right-tailed, or a two-tailed test.

4. Compute the test statistic: $\chi^2 = \frac{(n - 1)s^2}{\sigma_0^2}$

5. Determine whether to reject the null hypothesis based on whether it is a left-tailed, a right-tailed, or a two-tailed test.

6. State a conclusion.

Example

• To check the reliability of a scale in a butcher shop, a test weight known to weigh 400 grams was weighed 16 times.

• For the scale to be considered reliable, the standard deviation of repeated measurements must be less than 1 gram.

• The standard deviation of the 16 measured weights was $s = 0.8$ grams.

• Assume that the measured weights are independent and follow a normal distribution.

• Can we conclude that the population standard deviation of the measurements is less than 1 gram?

• Use the $\alpha = 0.05$ level of significance.

Solution

1. Check the assumptions.

   • We have a random sample from a normal population, so the assumptions are satisfied.

2. State the null and alternate hypotheses.

   • Null hypothesis: $\sigma = 1$

   • Alternate hypothesis: \sigma < 1

3. Determine the degrees of freedom.

   • $16 - 1 = 15$ degrees of freedom.

4. Determine the type of test and critical value.

   • This is a left-tailed test, so the critical value is $\chi^2<em>{1 - \alpha} = \chi^2</em>{0.95} = 7.261$.

5. Compute the test statistic.

   • Sample size: $n = 16$

   • Sample variance: $s^2 = 0.8^2 = 0.64$

   • Value specified by the null hypothesis: $\sigma_0 = 1$

   • $\chi^2 = \frac{(n - 1)s^2}{\sigma_0^2} = \frac{(16 - 1)(0.64)}{1^2} = 9.6$

6. Determine whether to reject the null hypothesis.

   • The value of the test statistic is $\chi^2 = 9.6$.

   • The critical value is 7.261.

   • Since this is a left-tailed test, we reject the null hypothesis if $\chi^2$ is less than or equal to the critical value.

   • Since 9.6 is greater than 7.261, we do not reject the null hypothesis at the $\alpha = 0.05$ level.

7. State the conclusion.

   • There is not enough evidence to conclude that the population standard deviation $\sigma$ is less than 1 gram.

   • We cannot consider the scale to be reliable.