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lecture14

# lecture14 - Psych 100A Winter 2010 Lecture 14 Statistical...

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Psych 100A Winter 2010 Lecture 14: Statistical Inference Categorical data Multinomial experiment Chi-square test ontingency table Contingency table

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Categorical data Previous inference methods applicable to quantitative data. Inference on qualitative data . umber of observations at each level of a Number of observations at each level of a qualitative variable = count or enumeration data . Population : individuals can be placed into various categories according to some characteristic. Sample : count of number of individuals who fall into each category. Data is characteristic of multinomial experiment .
1. Multinomial experiment. A natural extension of a binomial experiment ultinomial experiments re defined by the Multinomial experiments are defined by the following conditions: xperiment consists of d ials 1. Experiment consists of n iid trials 2. Polychotomous outcome on each trial: each ial results in one of utcomes trial results in one of k outcomes 3. Pr(outcome i) = i , i = 1,…,k; constant from trial to trial;  = 1 . I 4. Variables of interest are n i = the number of trials with outcome i observed during the n trials .

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Formula: Pr(outcomes) in a multinomial xperiment experiment Probability for the number of observations esulting in each of the k outcomes is given by resulting in each of the k outcomes is given by k n n n n 2 1 ! r   k k n n n n n n 3 2 1 2 1 2 1 ! ! ! , , , Pr where n = number of trials probability of i th utcome i = probability of i outcome n i = number of i th outcomes x! = x(x-1)(x-2)…(2)(1)
Multinomial experiment (cont.) ne use is to test specified probabilities ( for One use is to test specified probabilities ( iO ) for each outcome in a categorical study. In an study with k outcomes, the expected number of outcomes of type I in n trials equals: E i = n iO In 1900, Pearson devised a test statistic to test specified categorical probabilities called the 2 (chi-square) goodness-of-fit statistic :   k i E n 2 2 where n i ’s = observed cell counts and E i ’s = 1 expected cell counts .

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2. Chi-square distribution Pearson showed that if the n i ’s are sufficiently large, the test statistic 2
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lecture14 - Psych 100A Winter 2010 Lecture 14 Statistical...

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