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chapter14 - Introduction to Probability and Statistics and...

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Introduction to Probability Introduction to Probability and Statistics and Statistics Thirteenth Edition Thirteenth Edition Chapter 14 Analysis of Categorical Data
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Introduction Introduction Many experiments result in measurements that are qualitative qualitative or categorical categorical rather than quantitative. People classified by ethnic origin Cars classified by color M&M ® s classified by type (plain or peanut) These data sets have the characteristics of a multinomial experiment multinomial experiment . m m m m m m
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The Multinomial Experiment The Multinomial Experiment 1. The experiment consists of n n identical identical trials trials . . 2. Each trial results in one of one of k k categories categories . . 3. The probability that the outcome falls into a particular category i on a single trial is p i and remains constant remains constant from trial to trial. The sum of all k probabilities, p p 1 +p +p 2 +…+ +…+ p p k = 1 = 1 . 4. The trials are independent independent . 5. We are interested in the number of outcomes in each category, O 1, 1, O 2 , 2 , O k with m m m m m m
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The Binomial Experiment The Binomial Experiment A special case of the multinomial experiment with k = 2. Categories 1 and 2: success and failure p 1 and p 2 : p and q O 1 and O 2 : x and n-x We made inferences about p (and q = 1 - p) In the multinomial experiment, we make inferences about all the probabilities , p , p , p p . m m m m m m
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Pearson’s Chi-Square Pearson’s Chi-Square Statistic Statistic We have some preconceived idea about the values of the p i and want to use sample information to see if we are correct. The expected number expected number of times that outcome i will occur is E E i = = np np i . If the observed cell counts, observed cell counts, O O i , , are too far from what we hypothesize under H 0 , the more likely it is that H should be rejected. m m m m m m
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Pearson’s Chi-Square Pearson’s Chi-Square Statistic Statistic We use the Pearson chi-square statistic: i i i E E O X 2 2 ) ( - = m m m m m m
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Degrees of Freedom Degrees of Freedom These will be different depending on the application. 1. Start with the number of categories or cells in the experiment. 2. Subtract 1 df for each linear restriction on the cell probabilities. (You always lose 1 df since p p 1 +p +p 2 +…+ +…+ p p k = 1 = 1 . ) 3. Subtract 1 df for every population parameter you have to estimate to calculate or estimate E i . m m m m m m
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The Goodness of Fit Test The Goodness of Fit Test The simplest of the applications. A single categorical variable is measured, and exact numerical i i i E E O X 2 2 ) ( - = : statistic Test
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Example Example A multinomial experiment with k = 6 and O 1 to O 6 given in the table. We test: Upper Face 1 2 3 4 5 6 Number of times 50 39 45 62 61 43 H : p = 1/6; p = 1/6;… p = 1/6 (die is fair) Toss a die 300 times with the following results. Is the die fair or biased?
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Example Example Upper Face 1 2 3 4 5 6 50 39 45 62 61 43 E = np = 300(1/6) = 50 Calculate the expected cell counts:
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