Everything Maths Grade 11

CHAPTER 19. INDEPENDENT AND DEPENDENT EVENTS 19.2

For example we can draw and analyse a two-way contingency table to solve the following problem.

Example 3: Contingency Tables

QUESTION

A medical trial into theeffectiveness of a new medication was carried out. 120 males and

90 females responded. Out of these 50 males and 40 females responded positively to the

medication.

1. Was the medication’s success independent of gender? Explain.


2. Give a table for the independence of genderresults.

SOLUTION

Step 1 : Draw a contingency table

Male Female Totals

Positive result 50 40 90

No Positive result 70 50 120

Totals 120 90 210

Step 2 : Work out probabilities

P(male).P(positive result) =

120

210

= 0, 57

P(female).P(positive result) =

90

210

= 0, 43

P(male and positive result) =

50

210

= 0, 24

Step 3 : Draw conclusion

P(male and positive result) is the observed probability and P(male).P(positive

result) is the expected probability. These two are quite different. So there is no

evidence that the medication’s success is independent of gender.

Step 4 : Gender-independent results

To get gender independence we need the positive results in the same ratio as the

gender. The gender ratio is: 120 : 90, or 4 : 3, so the number in themale and

positive column wouldhave to be^47 of the total number of patients responding

positively which gives 51 , 4. This leads to the following table:

Male Female Totals

Positive result 51 , 4 38 , 6 90

No Positive result 68 , 6 51 , 4 120

Totals 120 90 210