Lecture 2_Chi-square test

Lecture 2_Chi-square test - ECON1320 Quantitative Economic...

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1 ECON1320 Quantitative Economic and Business Analysis B LECTURE 2 Chi Square tests Chapter 12.1 – 12.2 2 Topics Use the Chi-square ( χ 2 ) statistic to test 1. whether a given variable follows a hypothesized population distribution ( χ 2 Goodness-of-Fit Test) 2. hypotheses on population proportions as an alternative to the z test 3. whether two or more categorical variables are independent ( χ 2 Test of Independence) 3 Chi-Square distribution r Statistical experiment: select a random sample of size n from a normal population, having a SD = σ. We find that the sample SD = S . r Compute a statistic, called chi-square , using the following equation: r If this experiment is repeated an infinite number of times, we could obtain a sampling distribution for the chi-square statistic. 2 2 2 * ) 1 ( σ χ S n - = 4 Chi-Square distribution r always positive – first quadrant r there is a family of χ 2 distributions r a particular member is specified by the degrees of freedom r right skewed for low d.f. r more symmetrical as d.f. increase χ 2 0 df = 1 5 5 Chi-square statistic r used for qualitative (categorical) data r uses count data – frequency (not proportion or %) of success r contingency table (shows success/failure for two or more groups) r χ 2 statistic compares observed frequency (f o ) with expected frequency (f e ) in each cell 6 Chi-square statistic r the test statistic is given by χ 2 calc = r d.f. = k-m-1, where k = the number of categories, m = the number of parameters estimated from the sample data (used to determine expected frequencies) NB: if expected distribution is given, m=0, df=k-1 r all expected frequencies must be 5, if not, cells must be combined to have more than 5 - cells all e e o f f f 2 ) (
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2 7 Topic1 :Chi-Square Goodness-of-Fit Test r is used to determine whether a given variable follows a hypothesised population distribution (i.e. normal, uniform, binomial or multinomial distribution) r determines how well the theoretical distribution fits the observed data (how good is the fit?) r compares an observed frequency distribution with a theoretically expected distribution. 8 The five-step test r Step 1: state hypotheses H 0 : The variable follows the … distribution H 1 : The variable does not follow the … distribution Or H 0 : No change to the distribution H 1 : The distribution has been changed 9 The five-step test r Step 2: decision rule: reject H 0 if χ 2 cal > χ 2 α (k-1) Non Rejection region Rejection region χ 2 α (k-m-1) α 10 The five-step test r Step 3: compute test statistic & compare with the decision rule r Step 4: make a decision r Step 5: state your conclusion 2 2 all cells ( ) o e calc e f f f χ - = 11 Example 1 r A wholesale distributor buys goblets from a supplier. The supplier claims that they are producing goblets at a 10% defective rate. On every delivery, the distribution company check randomly 10 goblets and record this statistic. r
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Lecture 2_Chi-square test - ECON1320 Quantitative Economic...

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