The data shows the ordered marks scored by... | WAEC SSCE - Further Mathematics - 2024 (Essay)

Question 1 Report

The data shows the ordered marks scored by students in a test: 11, 12, (2x + y), (x + 2y), 14, and ((y\(^2\) - 2x). Given that the median is 13\(\frac{1}{2}\) and y is greater than x by 1, find:

(a) the values of x and y

(b) correct to three significant figures, the standard deviation of the distribution.

Answer

Given: 11, 12, (2x + y), (x + 2y), 14, and ((y\(^2\) - 2x). median = 13\(\frac{1}{2}\) = \(\frac{27}{2}\)

Median = \(\frac{(2x + y) + (x + 2y)}{2}\) = \(\frac{27}{2}\)

\(\frac{3x + 3y}{2}\) = \(\frac{27}{2}\)

3x + 3y = 27

x + y = 9 - - - - - - - -(i)

also, y = x + 1 - - - - -(ii)

put y = x + 1 into eqn (ii)

x + x + 1 = 9

2x = 9 - 1 = 8

x = \(\frac{8}{2}\) = 4

y = x + 1 = 4 + 1 = 5

(b) Since x = 4 and y = 5

11, 12, (2(3) + 5), (4 + (2(5)), 14, (5\(^2\) - 2(4))

11, 12, 13, 14, 14, 17

Mean(\(\overline{x}\)) = \(\frac{\sum{x}}{n}\) = \(\frac{11 + 12 + 13 + 14 + 14 + 17}{6}\) = 13.5

x	f	fx	(x - \(\overline{x}\))	(x - \(\overline{x}\))\(^2\)	f(x - \(\overline{x}\))\(^2\)
11	1	11	- 2.5	6.25	6.25
12	1	12	- 1.5	2.25	2.25
13	1	13	- 0.5	0.25	0.25
14	2	28	0.5	0.25	0.50
17	1	17	3.5	12.5	12.25

\(\sum f(x - \overline{x})^2\) = 21.50

S.D = \(\sqrt{\frac{\sum f(x - \overline{x})^2}{\sum {f}}}\) = \(\sqrt{\frac{21.50}{6}}\) = 1.89

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