Consider the following two way tables illustrating two possible samples designed to study the prevalence of a certain trait among men and women in the population of the U.S adults. The rows represent the gender of the individuals in the sample and the columns represent whether or not the individuals have the trait of interest.

Your task is to determine in which of the two samples being female and having a trait are independent. For each sample, perform two test for independence - one using the joint probability P(A and B) and one using a conditional probability P(A | B) or P (B | A) with the appropriate comparison ( you should thus have four comparisons in total). These comparisons may require you to compute additional marginal probabilities. Show your work for each computation. IN which sample are being female and having the trait independent.

This is the work I have started for the question, is this the correct way to answer it?

Joint Test - Sample ! A = female B = has trait Total females = 315+185= 500 total yes trait = 693 + 315 = 1008 Total sample = 1600 total Ves and female = 315 P (A ) = total female / total sample = 500/ 1600 = 0.3125 P CB ) = total yes / total sample = 1008/ 1600 = 0,63 P(AnB ) = total yest female / Jutul sample = 315/ 1600- 0, 196875 P CA ) P ( B ) = ( 500 / 16 00 ) . ( 1 608 / 1600 ) = 0. 0196 87 Because P (A n. B. ) = P (A ] X DCB ) , A and B are incle pendel Conditional Test : PCA |B ) or PCBIA) P (BI A ) = tolal yes + female / total female = 315/500= 0.63 P (B ) = 6.63 There fore because "PCBIA ) = PCB) they are inclependent