The table gives the distribution of heights in... | WAEC SSCE - Further Mathematics - 2012 (Essay)

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The table gives the distribution of heights in metres of 100 students.

Height	1.40-1.42	1.43-1.45	1.46-1.48	1.49-1.51	1.52-1.54	1.55-1.57	1.58-1.60	1.61-1.63
Freq	2	4	19	30	24	14	6	1

(a) Calculate the : (i) mean height ; (ii) mean deviation of the distribution.

(b) What is the probability that the height of a student selected at random is greater than the mean height of the distribution?

Answer

Each class has width 3; the mid-values are \(1.41,1.44,\dots,1.62\). Use a working-mean coding \(u=\dfrac{x-A}{c}\) with \(A=1.50\) and \(c=0.03\).

Height	Mid \(x\)	\(f\)	\(u\)	\(fu\)	\(fu^2\)
1.40-1.42	1.41	2	-3	-6	18
1.43-1.45	1.44	4	-2	-8	16
1.46-1.48	1.47	19	-1	-19	19
1.49-1.51	1.50	30	0	0	0
1.52-1.54	1.53	24	1	24	24
1.55-1.57	1.56	14	2	28	56
1.58-1.60	1.59	6	3	18	54
1.61-1.63	1.62	1	4	4	16
Total		100		41	203

(a)(i) Mean height.\[\bar{x}=A+\frac{\sum fu}{\sum f}\,c=1.50+\frac{41}{100}(0.03)=1.50+0.0123=1.5123\approx 1.51\text{ m}\]

(a)(ii) Mean deviation. \(\text{MD}=\dfrac{\sum f\,|x-\bar{x}|}{\sum f}\) with \(\bar{x}=1.5123\).

\(x\)	\(f\)	\(\|x-\bar{x}\|\)	\(f\|x-\bar{x}\|\)
1.41	2	0.1023	0.2046
1.44	4	0.0723	0.2892
1.47	19	0.0423	0.8037
1.50	30	0.0123	0.3690
1.53	24	0.0177	0.4248
1.56	14	0.0477	0.6678
1.59	6	0.0777	0.4662
1.62	1	0.1077	0.1077
Total	100		3.333

\[\text{MD}=\frac{3.333}{100}=0.0333\approx 0.03\text{ m}\]

(b) Probability of height greater than the mean. The mean \(1.5123\) m lies below the upper boundary \(1.515\) m of the class \(1.49-1.51\), so every student in the classes \(1.52-1.54\) and above exceeds the mean:\[24+14+6+1=45\]\[P(\text{height}>\bar{x})=\frac{45}{100}=0.45\]

Mean = 1.51 m, mean deviation \(\approx\) 0.03 m, and the required probability = 0.45.

Read lesson note on Probability (WAEC)

Answer Details

Each class has width 3; the mid-values are \(1.41,1.44,\dots,1.62\). Use a working-mean coding \(u=\dfrac{x-A}{c}\) with \(A=1.50\) and \(c=0.03\).

Height	Mid \(x\)	\(f\)	\(u\)	\(fu\)	\(fu^2\)
1.40-1.42	1.41	2	-3	-6	18
1.43-1.45	1.44	4	-2	-8	16
1.46-1.48	1.47	19	-1	-19	19
1.49-1.51	1.50	30	0	0	0
1.52-1.54	1.53	24	1	24	24
1.55-1.57	1.56	14	2	28	56
1.58-1.60	1.59	6	3	18	54
1.61-1.63	1.62	1	4	4	16
Total		100		41	203

(a)(i) Mean height.\[\bar{x}=A+\frac{\sum fu}{\sum f}\,c=1.50+\frac{41}{100}(0.03)=1.50+0.0123=1.5123\approx 1.51\text{ m}\]

(a)(ii) Mean deviation. \(\text{MD}=\dfrac{\sum f\,|x-\bar{x}|}{\sum f}\) with \(\bar{x}=1.5123\).

\(x\)	\(f\)	\(\|x-\bar{x}\|\)	\(f\|x-\bar{x}\|\)
1.41	2	0.1023	0.2046
1.44	4	0.0723	0.2892
1.47	19	0.0423	0.8037
1.50	30	0.0123	0.3690
1.53	24	0.0177	0.4248
1.56	14	0.0477	0.6678
1.59	6	0.0777	0.4662
1.62	1	0.1077	0.1077
Total	100		3.333

\[\text{MD}=\frac{3.333}{100}=0.0333\approx 0.03\text{ m}\]

Mean = 1.51 m, mean deviation \(\approx\) 0.03 m, and the required probability = 0.45.

Read lesson note on Probability (WAEC)

Probability

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The table gives the distribution of heights in metres of 100 students. Height 1.40-1.42 1.43-1.45 1.46-1.48 1.49-1.51 1.52-1.54 1.55-1.57 1.58-1.60 1.61-1.6...

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The table gives the distribution of heights in metres of 100 students. Height 1.40-1.42 1.43-1.45 1.46-1.48 1.49-1.51 1.52-1.54 1.55-1.57 1.58-1.60 1.61-1.6...

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