WAEC 2022 · Paper 2 · Q11

Marks	2	3	4	5	6
Frequency	$n - 2$	$n - 1$	$n - 3$	$2n - 6$	$8 - n$

The table shows the distribution of marks scored by students in a test.

(a)
If the mean mark is 3.75, find the value of $n$ .
(b)
Find the: (i) interquartile range; (ii) probability of selecting a student who scored at least 3 marks.
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

(a)

$\sum f = (n - 2) + (n - 1) + (n - 3) + (2n - 6) + (8 - n)$
$= 4n - 4$ and $\sum fx = 2(n - 2) + 3(n - 1) + 4(n - 3) + 5(2n - 6) + 6(8 - n)$
$= 13n - 1$ .
The mean is 3.75: $\frac{13n - 1}{4n - 4} = 3.75$ , so $13n - 1 = 15n - 15$ , $2n = 14$ and $n = 7$ .

(b)(i)

With $n = 7$ the frequencies are $5, 6, 4, 8, 1$ , a total of 24.
Running totals: $5, 11, 15, 23, 24$ .
$Q_1$ is at position $\frac{24}{4} = 6$ : the 6th mark is 3.
$Q_3$ is at position $\frac{3 \times 24}{4} = 18$ : the 18th mark is 5.
Interquartile range $= 5 - 3 = 2$ .

(ii)

"At least 3 marks" is everyone except the 5 who scored 2: $24 - 5 = 19$ .
The probability is $\frac{19}{24}$ .

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