46 Lecture 10 - Goodness of Fit and Contingency Analysis

46 Lecture 10 - Goodness of Fit and Contingency Analysis -...

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Unformatted text preview: Lecture 10: Chi-Squared (Non-Parametric) Tests Goodness of Fit Contingency Analysis 2 Agenda 1. Goodness of fit tests Test of uniform distribution Test of normality Test of more than 2 proportions 2. Contingency analysis 3 Announcements Quiz next week? Goodness of Fit Test 5 Introduction Remember one of the key ANOVA assumptions: Each population is normally distributed This is also relevant to our use of the Central Limit Theorem How do we test for this? In general, how do we test whether a population fits a particular distribution? 6 The Goodness of Fit Test The title describes this test very well We begin with a categorical variable This could be the classes of a frequency distribution We will check whether the frequencies in the observed frequencies conform to what we would expect to see Here is a simple example 7 Example 12.2 (p.484) The data on the following slide reflect` the choices of majors by a sample of 200 incoming freshmen students. Using a significance level of α = 0.05, is there sufficient evidence to conclude that the distribution of major choices is not uniform (i.e. not the same frequency for all majors)? 8 Example 12.2 (cont’d) Just by looking at the data, do we think they follow a uniform distribution? Should we draw this conclusion right away? Major # of Students Liberal Arts 44 Engineering 30 Education 26 Business 45 Sciences 55 200 9 The Logic of Goodness of Fit At a first glance, the sample data do not indicate a uniform distribution Nearly all categories are different! But this may well be due to sampling error We need to check whether these differences are significant Once again we face the problem of inference : What sound judgment about the population can we make based on sample data only? 10 The Logic… (cont’d) What we have in the table are the observed values for each category We will compare these to the expected values for each category These are the values we would expect to see if the data really did follow a uniform distribution 11 The Logic… (cont’d) Our test statistic will reflect the total difference between observed and expected values If this difference is too large (relatively speaking), we will conclude that the data do not follow the said distribution 12 The Steps of Goodness of Fit 1. State the hypotheses 2. Calculate the expected values 3. Calculate the test statistic 4. Determine the rejection region 5. Reach a decision regarding H 6. Draw a conclusion in terms of the original question 13 Step 1. HypothesesStep 1....
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This note was uploaded on 09/25/2010 for the course OMIS OMIS 1000 taught by Professor Alexandershoumarov during the Fall '09 term at York University.

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46 Lecture 10 - Goodness of Fit and Contingency Analysis -...

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