Hypothesis Tests for a Standard Deviation Section 9.5

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

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

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.

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.

Example Find the critical value 𝜒 0.05 2 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 = 0.05. The critical value is 𝜒 0.05 2 = 18.307.

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

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.

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 0 2 . 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.

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 0.95 2 =7.261.

Solution Solution (continued): The sample size is 𝑛 = 16 and the sample variance is 𝑠 2 = (0.8) 2 = 0.64. 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 0.95 2 = 7.261. Since this is a left-tailed test, we reject 𝐻 0 if 𝜒 2 ≤ 𝜒 0.95 2 . 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 0.95 2 =7.261

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