The table below shows the distribution of ages of... | WAEC SSCE - Further Mathematics - 2007 (Essay)

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The table below shows the distribution of ages of workers in a company.

Age/ yr	17 - 21	22 - 26	27 - 31	32 - 36	37 - 41	42 - 46	47 - 51	52 - 56
Workers	12	24	30	37	45	25	10	7

(a) Using an assumed mean of 39, calculate the (i) mean (ii) standard deviation; of the distribution.

(b) If a worker is selected at random from the company for an award, what is the probability that he is at most 36 years old?

Answer

Class width \(=5\); mid-values \(19,24,\dots,54\). Code with \(u=\dfrac{x-39}{5}\), \(A=39\).

Age	Mid \(x\)	\(f\)	\(u\)	\(fu\)	\(fu^2\)
17-21	19	12	-4	-48	192
22-26	24	24	-3	-72	216
27-31	29	30	-2	-60	120
32-36	34	37	-1	-37	37
37-41	39	45	0	0	0
42-46	44	25	1	25	25
47-51	49	10	2	20	40
52-56	54	7	3	21	63
Total		190		-151	693

(a)(i) Mean.\[\bar{x}=39+\frac{\sum fu}{\sum f}\times 5=39+\frac{-151}{190}\times 5=39-3.97=35.0\text{ years}\]

(a)(ii) Standard deviation.\[\sigma=c\sqrt{\frac{\sum fu^2}{\sum f}-\left(\frac{\sum fu}{\sum f}\right)^2}=5\sqrt{\frac{693}{190}-\left(\frac{-151}{190}\right)^2}\]\[=5\sqrt{3.647-0.632}=5\sqrt{3.015}=5(1.737)=8.68\text{ years}\]

(b) Probability at most 36 years old. Ages \(\le 36\) cover the classes \(17-21,\,22-26,\,27-31,\,32-36\), giving \(12+24+30+37=103\) workers out of 190.\[P(\text{at most }36)=\frac{103}{190}=0.542\]

Mean \(\approx\) 35.0 years, standard deviation \(\approx\) 8.68 years, and \(P=0.542\).

Read lesson note on Statistics (WAEC)

Answer Details

Class width \(=5\); mid-values \(19,24,\dots,54\). Code with \(u=\dfrac{x-39}{5}\), \(A=39\).

Age	Mid \(x\)	\(f\)	\(u\)	\(fu\)	\(fu^2\)
17-21	19	12	-4	-48	192
22-26	24	24	-3	-72	216
27-31	29	30	-2	-60	120
32-36	34	37	-1	-37	37
37-41	39	45	0	0	0
42-46	44	25	1	25	25
47-51	49	10	2	20	40
52-56	54	7	3	21	63
Total		190		-151	693

(a)(i) Mean.\[\bar{x}=39+\frac{\sum fu}{\sum f}\times 5=39+\frac{-151}{190}\times 5=39-3.97=35.0\text{ years}\]

Mean \(\approx\) 35.0 years, standard deviation \(\approx\) 8.68 years, and \(P=0.542\).

Read lesson note on Statistics (WAEC)

Statistics

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The table below shows the distribution of ages of workers in a company. Age/ yr 17 - 21 22 - 26 27 - 31 32 - 36 37 - 41 42 - 46 47 - 51 52 - 56 Workers 12 2...

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The table below shows the distribution of ages of workers in a company. Age/ yr 17 - 21 22 - 26 27 - 31 32 - 36 37 - 41 42 - 46 47 - 51 52 - 56 Workers 12 2...

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