Question

Consider n weeks, we denote X₀ as the random variable recording number of weeks with zero accidents, X₁ being the random variable recording the number of weeks with one accident, X₂ being the random variable recording number of accidents per week. We would like to test H₀: Y~Poisson(λ) against H_a: not H₀, where λ is unknown. Sample size n = 50, observed number of weeks with zero accidents, one accident, as well as two or more accidents are x₀: = 32, x₁ = 12, and x₂ = 6 respectively, observed average number of accidents per week ȳ = 0.48. We would like to apply chi-square test in goodness-of-fit test. (1) Translate the interested testing problem into the testing problem of chi-square test by applying the principle of Mean Likelihood Error; (2) Calculate chi-square test statistic and make a decision.

Answer 1

The r.vs X0,X1,X2 denote the no of weeks with zero, one and two or more accidents, and Y denotes the the no accidents per week. We have to perform a test for the goodness of fit of Poisson distribution to the given set of frequencies. The ch-square goodness of test is performed where the test statistic is 2= (Oi-Ei)2/Ei23-1-1, since the parameter is estimated by sample mean, df decreases by one. The detail explanations are given in the images...