Stat104_Lecture26v4_1up

# Stat104_Lecture26v4_1up - Stat 104: Quantitative Methods...

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Stat 104: Quantitative Methods for Economists Class 26: Chi-Square Tests 1

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Two Techniques… The first is a goodness-of-fit test applied to data produced by a multinomial experiment , a generalization of a binomial experiment and is used to describe one population of data. The second uses data arranged in a contingency table to determine whether two classifications of a population of nominal data are statistically independent ; this test can also be interpreted as a comparison of two or more populations. In both cases, we use the chi-squared ( ) distribution. 2
Goodness of Fit b A goodness-of-fit test is used to test the hypothesis that an observed frequency distribution fits (or conforms to) some claimed distribution. b It can be used to test for normality, or if some observed data follow some other distribution. b We will use it simply in the multinomial setting. 3

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The Multinomial Experiment… Unlike a binomial experiment which only has two possible outcomes (e.g. heads or tails), a multinomial experiment : • Consists of a fixed number, n , of trials. • Each trial can have one of k outcomes, called cells. • Each probability p i remains constant. • Our usual notion of probabilities holds, namely: p 1 + p 2 + … + p k = 1, and • Each trial is independent of the other trials. 4
Chi-squared Goodness-of-Fit Test We test whether there is sufficient evidence to reject a specified set of values for p i . To illustrate, our null hypothesis is: p a p a …, H 0 : p 1 = a 1 , p 2 = a 2 , …, p k = a k where a 1 , a 2 , …, a k are the values we want to test. Our research hypothesis is: H a : At least one p i is not equal to its specified value 5

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O represents the observed frequency of an outcome. E represents the expected frequency of an outcome. Goodness-of-Fit Notation . k represents the number of different categories or outcomes. n represents the total number of trials . 6
Example b Two companies, A and B, have recently conducted aggressive advertising campaigns to maintain and possibly increase their respective shares of the market for fabric softener. b These two companies enjoy a dominant position in e market. the market. b Before the advertising campaigns began, the market share of company A was 45%, whereas company B had 40% of the market. Other competitors accounted for the remaining 15%. b Has their market share changed after the campaign? 7

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Example b A marketing analyst solicited the preferences of a random sample of 200 customers of fabric softener. b Of the 200 customers, 102 indicated a preference for company A's product, 82 preferred company B's fabric softener, and the remaining 16 preferred the products of ne of the competitors. one of the competitors. b
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## This note was uploaded on 03/27/2012 for the course STATS 104 taught by Professor Michaelparzen during the Fall '11 term at Harvard.

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Stat104_Lecture26v4_1up - Stat 104: Quantitative Methods...

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