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HOW I TEACH THE INTERACTION BETWEEN CONTINGENCY TABLES AND TREE DIAGRAMS Nina Scheepers Diamantveld High School, Kimberley We are often confronted with tables of summarised data by the media, census releases, etc. It is crucial to know how to draw meaningful conclusions from these tables, and knowledge of conditional probability can be very useful in this regard. We will investigate how the probability laws and concepts may be applied to contingency tables to draw meaningful conclusions in daily life. What is a two-way contingency table? A two-way contingency table shows the observed frequencies for two variables in various categories. Each variable has two categories (choices) which are mutually exclusive (each learner can only be a boy or girl; take mathematics or mathematics literacy) and exhaustive (the categories cover all possible answers for each variable). What is a tree diagram? Tree diagrams are graphical representations of the outcomes of a random experiment. This is particularly useful when evaluating the numerator and denominator in the classical definition of probability: P(E) = number of outcomes favourable to E total number of outcomes In order to show the interaction between a contingency table and a tree diagram a survey was done amongst all the grade 11 learners in my school, Diamantveld High School. A questionnaire was circulated amongst all grade 11 learners. They were asked to tick one option in each of the following two questions: Q1 Boy Q2 Take Girl maths Take maths literacy 169 When processing the recorded information the events were defined as follows: Let B: the learner is a boy G: the learner is a girl M: the learner takes maths as a learning area L: the learner takes maths literacy as a learning area. The response from the learners was tabled to give the following 2 x 2 contingency table. GENDER B G TOTAL LEARNING M 31 49 80 AREA L 33 49 82 TOTAL 64 98 162 Interpretation of contingency table: Amongst the 162 learners interviewed: Frequency of boys = 64 Relative frequency of boys = 64 162 P (learner is a boy) = 32 81 Frequency of girls = 98 Relative frequency of girls = 98 162 P (learner is a girl) = 49 81 Frequency of interviewing a learner who takes maths = 80 Relative frequency of interviewing a learner who takes maths = 170 80 162 P (learner takes maths) = 40 81 Frequency of interviewing a learner who takes maths literacy = 82 Relative frequency of interviewing a learner who takes maths literacy = 82 162 P (learner takes maths literacy) = 41 81 From this we may conclude that the two – way contingency table will lead to probability calculations as follows: GENDER LEARNING AREA B G TOTAL M 31 162 49 162 80 162 L 33 162 49 162 82 162 TOTAL 64 162 98 162 162 162 Instead of drawing up this 2 x 2 contingency table consisting of the different probabilities, the same information can be illustrated by a tree diagram. Tree diagrams show all possible outcomes of events and each path show a possible outcome and each branch of the tree has been assigned the frequency of the event. M 31 B 33 64 L M 98 49 G 49 L 171 Questions that can be asked are: 1. Estimate the probability that a randomly selected child would be a boy. 2. Estimate the probability that a randomly selected child would be a boy and take maths. 3. Given that a boy is chosen, what is the probability that he takes maths literacy? 4. Given that a child from the class takes maths literacy, what is the probability that she is a girl? Solutions: 1. P(Select boy) = 64 162 2. P(Select boy and he takes maths) = P(B ∩ M) = 31 162 3. P(Selected child takes maths literacy / boy) = P(L / B) = 33 64 4. P(Select girl / child takes maths literacy) = P (G / L) = P(G ∩ L) 49 49 = = P(L) 49 + 33 82 Notice that by ordering the way in which we label the branches of the tree we make it easier to see where to find the acceptable outcomes or probabilities. CONCLUSION The interaction between a contingency table and tree diagrams enables us to interpret the recorded information, determine frequencies, relative frequencies and estimate probabilities and handle conditional probabilities. BIBLIOGRAPHY Probability component of NCS Data Handling; Prof D. North, School of Statistics & Actuarial Sciences, University of Kwazulu – Natal. 172