Hypothesis Tests and Estimation for Population Variances. Copyright 2014 Pearson Education, Inc.

Transcription

1 Hypothesis Tests and Estimation for Population Variances 11-1

2 Learning Outcomes Outcome 1. Formulate and carry out hypothesis tests for a single population variance. Outcome 2. Develop and interpret confidence interval estimates for a population variance. Outcome 3. Formulate and carry out hypothesis tests for the difference between two population variances. 11-2

3 11.1 Hypothesis Tests and Estimation for a Single Population Variance When measuring variation, the standard deviation is used as the measure because it is measured in the same units as the mean There is no statistical test directly tests the standard deviation The chi-square test that can be used to test the population variance 11-3

4 Hypothesis for a Single Population Variance 11-4

5 Hypothesis for a Single Population Variance 11-5

6 Chi-Square Test for a Single Population Variance 11-6

7 Finding the Critical Value The critical value, chi-square table 2 χ α, is found from the χ 2 Do not reject H 0 2 Reject H 0 χ α 11-7

8 Hypothesis Tests for a Single Population Variance Hypothesis Tests for Population Variance 2 χ 1 α Don t reject H 0 χ 2 Don t reject H 0 Reject H 0 χ2 Don t reject H χ α χ 1 α /2 χ α /2 Reject H 0 χ 2 Reject H 0 Reject H

9 Hypothesis Tests for a Single Population Variance 11-9

10 One-Tailed Hypotheses Tests for a Population Variance - Example The company does small moving jobs for businesses and private residential customers. Looking at past records, the operations manager has determined the mean job time for a properly trained moving crew is 2 hours, with a standard deviation not to exceed 0.5 hour. Suppose the operations manager took a random sample of 20 service calls and found a variance of 0.33 hours squared. Compute the test statistic Decision and conclusion 11-10

11 Two-Tailed Hypotheses Tests for a Population Variance - Example The company is experimenting with devices that can be used to store solar energy. Engineers have determined that one particular storage design will yield an average of 88 minutes per cell with a standard deviation of 6 minutes. They also have made some modifications to the design and are interested in determining whether this change has impacted the standard deviation either up or down. A random sample of 12 individual storage cells containing the modified design have been used

12 Two-Tailed Hypotheses Tests for a Population Variance - Example Compute the test statistic Decision and conclusion 11-12

13 Confidence Interval Estimate for a Population Variance Confidence Interval α/2 α/

14 11.2 Hypothesis Tests for Two Population Variances Hypothesis Tests for Two Population Variance F-Test Statistic for Testing whether Two Populations Have Equal Variances 11-14

15 The F-Distribution F-distribution is formed by the ratio of two independent chi-square variables F-distribution is determined by its degrees of freedom Assumptions: The populations are normally distributed The samples are randomly and independently selected 11-15

16 Formulating the F-Ratio To use the F-distribution table, for a two-tailed test, always place the larger sample variance in the numerator. This will make the calculated F- value greater than 1.0 and push the F-value toward the upper tail of the F-distribution. For the one-tailed test, examine the alternative hypothesis. For the population that is predicted (based on the alternative hypothesis) to have the larger variance, place that sample variance in the numerator

17 Hypothesis Tests for Two Population Variances α 0 F 0 F Do not Reject H 0 Do not reject H 0 reject H 0 F α F α/2 Reject H 0 Rejection region for one-tailed test: Rejection region for two-tailed test: The larger sample variance in the numerator 11-17

18 Hypothesis Tests for Two Population Variances Step 1: Specify the population parameter of interest Step 2: Formulate the appropriate null and alternative hypotheses Step 3: Specify the level of significance Step 4: Construct the rejection region, and specify the critical value(s). To determine the critical value(s) from the F-distribution, either Excel s F.INV.RT function or the F-table can be used. The degrees of freedom are: D 1 = Numerator sample size 1 D 2 = Denominator sample size

19 Hypothesis Tests for Two Population Variances Step 5: Compute the test statistic which is formed by the ratio of the two sample variances. Select random samples from each population of interest, determine whether the assumptions have been satisfied, and compute the test statistic Step 6: Reach the decision: Compare the test statistic to the critical value(s) and reach a conclusion with respect to the null hypothesis Step 7: Draw the conclusion 11-19

20 Hypothesis Tests for Two Population Variances - Example Transportation agency is concerned about the waiting times for passengers who use the downtown transit center. Of particular interest is whether there is a difference in the standard deviations in waiting times at concourses A and B. Compute the test statistic Decision and conclusion 11-20

21 How to Do It in Excel? 1. Open file. 2. Select Data > Data Analysis. 3. Select F-Test Two Sample Variances. 4. Define the data range for the two variables. 5. Specify Alpha. 6. Specify output location. 7. Click OK

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