Question

(a)
The table below relates the ages and heights of a group of individuals participating in a study.





Part A

Complete the table by filling in the missing values.

Ages
Tall
Medium
Short
Total
Under 18 18 28 14
60

18 - 50
20

51
28

99
51+ 12 25

9

46
Total
50 104
51

205


(b)
Part B

Are the events of being under 18 and being short independent? Explain your reasoning.

Answer 1

Part A — completed table

Ages / Height | Tall | Medium | Short | Total
Under 18 | 18 | 28 | 14 | 60
18–50 | 20 | 51 | 28 | 99
51+ | 12 | 25 | 9 | 46
Total | 50 | 104 | 51 | 205

(Values found by using the given row/column totals.)

Part B — independence

Let U = event "under 18" and S = event "short".

P(U) = 60/205, P(S) = 51/205, P(U and S) = 14/205.

If independent, P(U and S) = P(U)·P(S) = (60/205)(51/205) ≈ 0.0728, but 14/205 ≈ 0.0683. Also P(S | U) = 14/60 ≈ 0.2333 ≠ P(S) ≈ 0.2488.

Since P(U and S) ≠ P(U)P(S) (equivalently P(S|U) ≠ P(S)), the events are not independent.