University:
York University
Course:
MATH 1131 | Introduction to Statistics I
Academic year:

2022
Views:

316

Pages:

1
Author:

Niklas

LECTURE 30 Titanic Mutual Independence Recall that in a contingency table X and Y are mutually independent. Y is not associated with X if, for i = 1, ..., s, j = 1, ..., t, n_ij = ĥ_ij, that is, if n_ij = (n_i * n_j) / n. If the two variables are independent, the expected frequency for the categories i and j is ĥ_ij = (n_i * n_j) / n. We measure how much the observed frequencies differ from the theoretical ones with: χ² = Σ(Σ((n_ij - ĥ_ij)² / ĥ_ij)) The Pearson's χ² can be used also to make inference, indeed: - if the variables are mutually independent - n is large - the χ² is approximately distributed as χ²(s-1)(t-1) p-value and rejection region. Observations are far from the independence hypothesis if the χ² statistic is large, so the p-value is small. P(χ²(s-1)(t-1) > χ²) = 1 - F_χ²(s-1)(t-1)(χ²) where F_χ²(s-1)(t-1) is the cdf of a χ² with (s-1)(t-1) degrees of freedom. Moreover, if χ²(s-1)(t-1),1-α is the quantile 1-α of a distribution χ²(s-1)(t-1), then the region S = {χ² > χ²(s-1)(t-1),1-α} is a rejection region of level α.