1. Hypothesis Tests for a Standard Deviation
Section 9.5

2. Objectives Find critical values of the chi-square distribution
Test hypotheses about the standard deviation of a normal distribution

3. Find critical values of the chi-square distribution
Objective 1
Find critical values of the chi-square distribution

4. The Chi-Square Distribution
Hypothesis tests for a standard deviation are based on the chi-square distribution, denoted χ2. There are actually many different chi-square distributions, each with a different number of degrees of freedom. The figure below shows several examples of chi-square distributions.
Recall that the chi-square distributions are skewed to the right, and the values of the χ2 statistic are always greater than or equal to 0. They are never negative.

5. Critical Values
The notation 𝜒 𝛼 2 represents the value that has an area of 𝛼 to its right. We consult Table A. 4 to find critical values associated with the chi-square distribution.

6. Example
Find the critical value 𝜒 for a chi-square distribution with 10 degrees of freedom. Solution: We consult Table A.4. The critical value is located at the intersection of the row corresponding to 10 degrees of freedom and the column corresponding to α = The critical value is 𝜒 =

7. Test hypotheses about the standard deviation of a normal distribution
Objective 2
Test hypotheses about the standard deviation of a normal distribution

8. Test Statistic for the Hypothesis Test
The null hypothesis for a standard deviation 𝜎 is of the form 𝐻 0 :𝜎= 𝜎 0 . The test is based on the fact that if 𝐻 0 is true, then the test statistic χ 2 = 𝑛 −1 ∙ 𝑠 2 𝜎 0 2 has a chi-square 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.

9. Hypothesis Test for a Standard Deviation
Check to be sure that the assumptions are satisfied. We must have a simple random sample from a normal population. Step 1. State the null and alternate hypotheses. Step 2. Choose a significance level 𝛼 , and find the critical value, using Table A.4 with 𝑛 −1 degrees of freedom. Left-tailed: The critical value is 𝜒 1−𝛼 2 Right-tailed: The critical value is 𝜒 𝛼 2 Two-tailed: The critical value is 𝜒 𝛼 2 2 and 𝜒 1−𝛼 2 2 Step 3. Compute the test statistic χ 2 = 𝑛 −1 ∙ 𝑠 2 𝜎 Step 4. Determine whether to reject 𝐻 0 , as follows: Left-tailed: Reject if χ 2 ≤ 𝜒 1−𝛼 2 Right-tailed: Reject if χ 2 ≥ 𝜒 𝛼 2 Two-tailed: Reject if χ 2 ≥ 𝜒 𝛼 2 2 or χ 2 ≤ 𝜒 1−𝛼 2 2 Step 5. State a conclusion.

10. 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 𝛼 = 0.05 level of significance. Solution: We have a random sample from a normal population, so the assumptions are satisfied. The null and alternate hypotheses are: 𝐻 0 : 𝜎=1 versus 𝐻 1 : 𝜎<1. We have 16 − 1 = 15 degrees of freedom. This is a left-tailed test, so the critical value is 𝜒 1−𝛼 2 = 𝜒 =7.261.

11. Solution
Solution (continued): The sample size is 𝑛 = 16 and the sample variance is 𝑠 2 = (0.8) 2 = The value specified by 𝐻 0 is 𝜎 0 = 1. We compute the value of the test statistic as 𝜒 2 = 𝑛 −1 𝑠 2 𝜎 0 = (16 −1)(0.64) 1 2 =9.6. The value of the test statistic is 𝜒 2 = 9.6. The critical value is 𝜒 = Since this is a left-tailed test, we reject 𝐻 0 if 𝜒 2 ≤ 𝜒 Since 9.6 > 7.261, we do not reject 𝐻 0 at the 𝛼 = 0.05 level. There is not enough evidence to conclude that the population standard deviation 𝜎 is less than 1 gram. We cannot consider the scale to be reliable.
Remember 𝐻 0 :𝜎=1
𝐻 1 :𝜎<1
𝑛=16
𝜒 =7.261

12. You Should Know…
How to find the critical values for a chi-square distribution
The assumptions for performing a hypothesis test for a standard deviation of a normal distribution
How to perform a hypothesis test for a standard deviation of a normal distribution