Section 11 2 Goodness of Fit Copyright 2010

Key Concept Sample data consist of observed frequency counts arranged in a single row

Definition A goodness-of-fit test is used to test the hypothesis that an observed frequency

Goodness-of-Fit Test Notation O (letter, not number) represents the Observed frequency of an outcome

Goodness-of-Fit Test Requirements 1. The data have been randomly selected. 2. For each category,

Goodness-of-Fit Test Statistic x 2 is pronounced “chi-square” Copyright © 2010, 2007, 2004 Pearson

Goodness-of-Fit Critical Values 1. Found in Table A- 4 using k – 1 degrees

Goodness-of-Fit P-Values P-values are typically provided by computer software, or a range of Pvalues

Expected Frequencies If all expected frequencies are equal: the sum of all observed frequencies

Expected Frequencies If expected frequencies are not all equal: Each expected frequency is found

Goodness-of-Fit Test v A close agreement between observed and expected values will lead to

Goodness-of-Fit Test “If the P is low, the null must go. ” (If the

Relationships Among the Test Statistic, P-Value, and Goodness-of-Fit Copyright © 2010, 2007, 2004 Pearson

Example: Data Set 1 in Appendix B includes weights from 40 randomly selected adult

Example: When people report weights, they typically round to a whole number, so reported

Example: (For example, the weight of 201. 5 lb has a last digit of

Example: Copyright © 2010, 2007, 2004 Pearson Education, Inc. . 17

Example: Requirements are satisfied: randomly selected subjects, frequency counts, expected frequency is Step 1:

Example: Step 4: no significance specified, use Step 5: testing whether a uniform distribution

Example: Copyright © 2010, 2007, 2004 Pearson Education, Inc. . 20

Example: Step 7: Because the test statistic does not fall in the critical region,

Example: Step 8: There is not sufficient evidence to support the claim that the

Goodness-of-fit by TI-83/84 • Press STAT and select EDIT • Enter Observed frequencies into

Goodness-of-fit by TI-83/84 • In TI-83: Clear screen and type (L 1 - L

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Key Concept Sample data consist of observed frequency counts arranged in a single row (called a one-way frequency table). We will test the claim that the observed frequency counts agree with some claimed distribution. In other words, there is a good fit of the observed data with the claimed distribution. Copyright © 2010, 2007, 2004 Pearson Education, Inc. . 2

Goodness-of-Fit Test Notation O (letter, not number) represents the Observed frequency of an outcome E represents the Expected frequency of an outcome k represents the number of different categories or outcomes n represents the total number of trials Copyright © 2010, 2007, 2004 Pearson Education, Inc. . 4

Goodness-of-Fit Test Requirements 1. The data have been randomly selected. 2. For each category, the expected frequency is at least 5. (There is no requirement on the observed frequency for each category. ) Copyright © 2010, 2007, 2004 Pearson Education, Inc. . 5

Goodness-of-Fit 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. Copyright © 2010, 2007, 2004 Pearson Education, Inc. . 7

Expected Frequencies If expected frequencies are not all equal: Each expected frequency is found by multiplying the sum of all observed frequencies by the probability for the category. Copyright © 2010, 2007, 2004 Pearson Education, Inc. . 10

Goodness-of-Fit Test v A close agreement between observed and expected values will lead to a small value of and a large P-value. v A large disagreement between observed and expected values will lead to a large value of and a small P-value. v A significantly large value of will cause a rejection of the null hypothesis of no difference between the observed and the expected. Copyright © 2010, 2007, 2004 Pearson Education, Inc. . 11

Goodness-of-Fit Test “If the P is low, the null must go. ” (If the P-value is small, reject the null hypothesis that the distribution is as claimed. ) Copyright © 2010, 2007, 2004 Pearson Education, Inc. . 12

Example: Data Set 1 in Appendix B includes weights from 40 randomly selected adult males and 40 randomly selected adult females. Those weights were obtained as part of the National Health Examination Survey. When obtaining weights of subjects, it is extremely important to actually weigh individuals instead of asking them to report their weights. By analyzing the last digits of weights, researchers can verify that weights were obtained through actual measurements instead of being reported. Copyright © 2010, 2007, 2004 Pearson Education, Inc. . 14

Example: When people report weights, they typically round to a whole number, so reported weights tend to have many last digits consisting of 0. In contrast, if people are actually weighed with a scale having precision to the nearest 0. 1 pound, the weights tend to have last digits that are uniformly distributed, with 0, 1, 2, … , 9 all occurring with roughly the same frequencies. Table 11 -2 shows the frequency distribution of the last digits from 80 weights listed in Data Set 1 in Appendix B. Copyright © 2010, 2007, 2004 Pearson Education, Inc. . 15

Example: (For example, the weight of 201. 5 lb has a last digit of 5, and this is one of the data values included in Table 11 -2. ) Test the claim that the sample is from a population of weights in which the last digits do not occur with the same frequency. Based on the results, what can we conclude about the procedure used to obtain the weights? Copyright © 2010, 2007, 2004 Pearson Education, Inc. . 16

Example: Requirements are satisfied: randomly selected subjects, frequency counts, expected frequency is Step 1: at least one of the probabilities , is different from the others Step 2: at least one of the probabilities are the same: Step 3: null hypothesis contains equality : At least one probability is different Copyright © 2010, 2007, 2004 Pearson Education, Inc. . 18

Example: Step 4: no significance specified, use Step 5: testing whether a uniform distribution so use goodness-of-fit test: Step 6: see the next slide for the computation of the test statistic. The test statistic , using and degrees of freedom, the critical value is Copyright © 2010, 2007, 2004 Pearson Education, Inc. . 19

Example: Step 7: Because the test statistic does not fall in the critical region, there is not sufficient evidence to reject the null hypothesis. Copyright © 2010, 2007, 2004 Pearson Education, Inc. . 21

Example: Step 8: There is not sufficient evidence to support the claim that the last digits do not occur with the same relative frequency. This goodness-of-fit test suggests that the last digits provide a reasonably good fit with the claimed distribution of equally likely frequencies. Instead of asking the subjects how much they weigh, it appears that their weights were actually measured as they should have been. Copyright © 2010, 2007, 2004 Pearson Education, Inc. . 22

Goodness-of-fit by TI-83/84 • Press STAT and select EDIT • Enter Observed frequencies into the list L 1 • Enter Expected frequencies into the list L 2 then the procedure differs for TI-83 and TI-84: • In TI-84: Press STAT, select TESTS scroll down to c 2 GOF-Test , press ENTER Type in: Observed: L 1 Expected: L 2 df: number of degrees of freedom Press on Calculate read the test statistic c 2 =. . . and the P-value p=… Copyright © 2010, 2007, 2004 Pearson Education, Inc. 23

Goodness-of-fit by TI-83/84 • In TI-83: Clear screen and type (L 1 - L 2 )2 ÷ L 2 → L 3 (the key STO produces the arrow → ) Then press 2 nd and STAT, select MATH scroll down to sum( press ENTER at prompt sum( type L 3 then ) and press ENTER This gives you the c 2 test statistic For the P-value use the function c 2 CDF from DISTR menu and at prompt c 2 CDF( type test_statistic, 9999, df) Copyright © 2010, 2007, 2004 Pearson Education, Inc. 24