Download How I teach the interaction between contingency tables and tree

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
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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.
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