1. Section 11.2-1 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series by Mario F. Triola

2. Section 11.2-2 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Chapter 11 Goodness-of-Fit and Contingency Tables 11-1 Review and Preview 11-2 Goodness-of-Fit 11-3 Contingency Tables

3. Section 11.2-3 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Key Concept In this section, we consider sample data consisting of observed frequency counts arranged in a single row or column (called a one-way frequency table). We will use a hypothesis test for the claim that the observed frequency counts agree with some claimed distribution, so that there is a good fit of the observed data with the claimed distribution.

4. Section 11.2-4 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Definition A goodness-of-fit test is used to test the hypothesis that an observed frequency distribution fits (or conforms to) some claimed distribution.

5. Section 11.2-5 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Notation O represents the observed frequency of an outcome, found from the sample data. E represents the expected frequency of an outcome, found by assuming that the distribution is as claimed. k represents the number of different categories or cells. n represents the total number of trials. Goodness-of-Fit Test

6. Section 11.2-6 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Goodness-of-Fit Test 1.The data have been randomly selected. 2.The sample data consist of frequency counts for each of the different categories. 3.For each category, the expected frequency is at least 5. (The expected frequency for a category is the frequency that would occur if the data actually have the distribution that is being claimed. There is no requirement that the observed frequency for each category must be at least 5.) Requirements

7. Section 11.2-7 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Goodness-of-Fit Hypotheses and Test Statistic

8. Section 11.2-8 Copyright © 2014, 2012, 2010 Pearson Education, Inc. P-Values and Critical Values P-Values P-values are typically provided by technology, or a range of P-values can be found from Table A-4. Critical Values 1. Found in Table A-4 using k – 1 degrees of freedom, where k = number of categories. 2. Goodness-of-fit hypothesis tests are always right-tailed.

9. Section 11.2-9 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Finding Expected Frequencies If all expected frequencies are assumed equal: If all expected frequencies are assumed not equal:

10. Section 11.2-10 Copyright © 2014, 2012, 2010 Pearson Education, Inc. A close agreement between observed and expected values will lead to a small value of χ 2 and a large P-value. A large disagreement between observed and expected values will lead to a large value of χ 2 and a small P-value. A significantly large value of χ 2 will cause a rejection of the null hypothesis of no difference between the observed and the expected. Goodness-of-Fit Test

11. Section 11.2-11 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Goodness- Of-Fit Tests

12. Section 11.2-12 Copyright © 2014, 2012, 2010 Pearson Education, Inc... Example A random sample of 100 weights of Californians is obtained, and the last digit of those weights are summarized on the next slide. When obtaining weights, it is extremely important to actually measure the weights rather than ask people to self-report them. By analyzing the last digit, we can verify the weights were actually measured since reported weights tend to be rounded to something ending with a 0 or a 5. Test the claim that the sample is from a population of weights in which the last digits do not occur with the same frequency.

13. Section 11.2-13 Copyright © 2014, 2012, 2010 Pearson Education, Inc... Example - Continued

14. Section 11.2-14 Copyright © 2014, 2012, 2010 Pearson Education, Inc... Example - Continued Requirement Check: 1.The data come from randomly selected subjects. 2.The data do consist of counts. 3.With 100 sample values and 10 categories that are claimed to be equally likely, each expected frequency is 10, which is greater than 5. All requirements are met to proceed.

15. Section 11.2-15 Copyright © 2014, 2012, 2010 Pearson Education, Inc... Example - Continued Step 1: The original claim is that the digits do not occur with the same frequency. That is: Step 2: If the original claim is false, then all the probabilities are the same:

16. Section 11.2-16 Copyright © 2014, 2012, 2010 Pearson Education, Inc... Example - Continued Step 3: The hypotheses can be written as: Step 4: No significance level was specified, so we select α = 0.05. Step 5: We use the goodness-of-fit test with a χ 2 distribution.

17. Section 11.2-17 Copyright © 2014, 2012, 2010 Pearson Education, Inc... Example - Continued Step 6: The calculation of the test statistic is given:

18. Section 11.2-18 Copyright © 2014, 2012, 2010 Pearson Education, Inc... Example - Continued Step 6: The test statistic is χ 2 = 212.800 and the critical value is χ 2 = 16.919 (Table A-4). The P-value was found to be less than 0.0001 using technology.

19. Section 11.2-19 Copyright © 2014, 2012, 2010 Pearson Education, Inc... Example - Continued Step 7: Reject the null hypothesis, since the P-value is small and the test statistic is in the critical region. Step 8: We conclude there is sufficient evidence to support the claim that the last digits do not occur with the same relative frequency. In other words, we have evidence that the weights were self-reported by the subjects, and the subjects were not actually weighed.