46 Lecture 10 - Goodness of Fit and Contingency Analysis

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

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Lecture 10: Chi-Squared (Non-Parametric) Tests Goodness of Fit Contingency Analysis

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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?

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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?

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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)?

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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?

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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

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12 The Steps of Goodness of Fit 1.
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