CHI-SQUARE TEST OF GOODNESS OF FIT 1. A gambler ..., Lecture notes of Applied Statistics

Download CHI-SQUARE TEST OF GOODNESS OF FIT 1. A gambler ... and more Lecture notes Applied Statistics in PDF only on Docsity! CHI-SQUARE TEST OF GOODNESS OF FIT 1. A gambler plays a game that involves throwing 3 dice in a succession of trials. His winnings are directly proportional to the number of sixes recorded. If the dice are fair, what is the probability distribution that governs the outcome of each throw? The frequencies of the sixes observed in 100 trials are recorded, together with their expected values, in the following table: Number of sixes Expected Count Observed Count 0 58 47 1 34.5 35 2 7 15 3 0.5 3 You are asked to assess whether it is likely that the dice have been unfairly weighted, using a chi-square test of goodness of fit. Answer. The number of sixes x in a fair trial has a Binomial distribution: b(n = 3, p = 1/6) = n! (n − x)!x! px(1 − p)n−x = 3! (3 − x)!x! ( 1 5 )x ( 5 6 )n−x . We have b(x = 0) = ( 5 6 )3 = 125 216 = 0.579, b(x = 1) = 3 ( 1 6 ) ( 5 6 )2 = 75 216 = 0.347, b(x = 2) = 3 ( 1 6 )2 ( 5 6 ) = 5 215 = 0.069, b(x = 3) = ( 1 6 )3 = 1 216 = 0.0046. The Chi-square statistic takes the form of χ2 = ∑ (observed − expected)2 observed . We calculate that χ2 = (47 − 58)2 58 + (35 − 34.5)2 58 + (15 − 7)2 7 + (3 − 0.5)2 0.5 = 2.08 + 0.007 + 9.14 + 12.5 = 23.727. From the table, the 5 percent critical value of the χ2 of three degrees of freedom is 7.815. Therefore, there is evidence that the dice are unfairly weighted. 1