Hypothesis Testing Module 3.3: Parametric Tests c University of New South Wales School of Risk and Actuarial Studies 1/63
Hypothesis Testing Introduction Fisher’s exact test Fisher’s exact test Contingency tables One sample Multinomial r -Sample Multinomial Contingency tables One sample Multinomial Contingency tables Examples Contingency tables Examples Non-parametric tests Two sample test Non-parametric tests Test on quantiles Goodness of fit tests An overview Goodness of fit tests Exercises Summary 2/63
Hypothesis Testing Introduction Introduction I Last two weeks we have seen hypothesis testing; One part of it was the test statistic; I To do so, we assumed a distribution for X . I For example, normal sample and testing mean with unknown variance, ¯ X - μ S / √ n ∼ t n - 1 . However, there are cases where we don’t know the distribution of X . 3/63
Hypothesis Testing Introduction Introduction I Use non-parametric methods : I The non-parametric methods do not rely on the estimation of the parameters describing the distribution. I We introduce the following tests: I Fisher exact test; I Quantile test; I Signed rank test; I Wilcoxon paired sample test. 4/63
Hypothesis Testing Introduction Introduction I We also introduce an important concept: Chi-squared tests. I Contingency tables; I Test of independence; I Test of homogeneity. I Verify distributional assumption—Goodness-of-fit tests: I Chi-squared test; I Anderson-Darling & Cramér-von Mises tests; I Kolmogorov-Smirnoff & Kuiper tests. 5/63
Hypothesis Testing Fisher’s exact test Fisher’s exact test Example: Fisher’s exact test I By random selection, 24 male and female employees are selected. Their promotion records are tabulated below, Number Male Female Promote x = 21 14 M = 35 Hold file 3 10 13 n = 24 24 N = 48 I We want to test the following statement Are promotion decisions made independent of gender? 6/63
Hypothesis Testing Fisher’s exact test Fisher’s exact test Hypergeometric distribution I Take n draws from a sample without replacement; Let the total number of objects be N ( ≥ n ) ; Have two possible different outcomes (e.g. “ blue ball ” and “ red ball ”), one of them (say blue balls ) has M ( ≤ N ) objects. I Let X be the random variable of blue balls selected. This provides us the Hypergeometric distribution : X ∼ HYP ( n , M , N ) with n = 1 , 2 , . . . , N , M = 0 , 1 , . . . , N . The probability mass function of a Hypergeometric distribution is given by: p X ( x ) = ( M x ) · ( N - M n - x ) ( N n ) . 7/63
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- Normal Distribution, Statistical tests, Chi-square distribution, Pearson's chi-square test, Non-parametric statistics