WAEC 2008 · Paper 2 · Q15

(a)
Five (5) female and seven (7) male teachers applied for 4 vacancies in a Junior High School. The teachers are equally qualified. Find the number of ways of employing 4 teachers if (i) there is no restriction; (ii) at least 2 of them are females.
Separate values with commas, e.g. 3, −2
(b)
The table shows the positions awarded to 7 contestants by judges $X$ and $Y$ in a competition.

Contestant P Q R S T U V

Judge X 2 7 1 3 6 5 4

Judge Y 4 6 2 3 7 1 5

(i) Calculate, correct to one decimal place, the Spearman's rank correlation coefficient. (ii) Interpret your answer in (b)(i).

Contestant	P	Q	R	S	T	U	V
Judge X	2	7	1	3	6	5	4
Judge Y	4	6	2	3	7	1	5

Worked solution (try it first)

Choose any 4 of the 12 teachers: $^{12}C_4 = \dfrac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1}$
$= 495$ .

"At least 2 females" means 2, 3 or 4 females.
2 females and 2 males: $^5C_2 \times {^7C_2} = 10 \times 21 = 210$ .
3 females and 1 male: $^5C_3 \times {^7C_1} = 10 \times 7 = 70$ . 4 females: $^5C_4 = 5$ .
Add: $210 + 70 + 5 = 285$ ways.

Differences $d = X - Y$ : $-2, 1, -1, 0, -1, 4, -1$ .
Squares: $4, 1, 1, 0, 1, 16, 1$ , so $\sum d^2 = 24$ .
Use $\rho = 1 - \dfrac{6\sum d^2}{n(n^2 - 1)}$ with $n = 7$ : $\rho = 1 - \dfrac{6 \times 24}{7 \times 48}$
$= 1 - \dfrac{144}{336}$ .
$\rho \approx 1 - 0.429 = 0.571$ , which is $0.6$ to one decimal place.

$\rho = 0.6$ is positive and fairly close to 1: there is a fairly strong positive correlation, so the two judges broadly agree in their rankings.