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Answer the following : The mean height of 200 students is 65 inches. The mean heights of boys and girls are 70 inches and 62 inches respectively and the standard deviations are 8 and 10 respectively. - Mathematics and Statistics

Sum

Answer the following :

The mean height of 200 students is 65 inches. The mean heights of boys and girls are 70 inches and 62 inches respectively and the standard deviations are 8 and 10 respectively. Find the number of boys and the combined S.D.

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Solution

Let suffix 1 denote quantities for boys and suffix 2 for girls.

Given : `bar(x)_1` = 70, `sigma_1` = 8, `bar(x)_2` = 62, `sigma_2` = 10, n1 + n2 = 200

∴ n2 = 200 – n1 

Combined mean = `bar(x)` = 65, where

`bar(x) = ("n"_1bar(x)_1 + "n"_2bar(x)_2)/("n"_1 + "n"_2)`

∴ 65 = `(70"n"_1 + 62(200 - "n"_1))/200`

∴ 13000 = 8n1 + 12400

∴ 600 = 8n1

∴ n1 = 75

∴ n2 = 200 – 75 = 125

d1 = `bar(x)_1 - bar(x)` = 70 – 65 = 5

d2 = `bar(x)_2 - bar(x)` = 62 – 65 = – 3

If combined S.D. is `sigma`, then

`sigma = sqrt(("n"_1("d"_1^2 + sigma_1^2) + "n"_2("d"_2^2 + sigma_2^2))/("n"_1 + "n"_2)`

= `sqrt((75(25 + 64) + 125(9 + 100))/200`

= `sqrt((6675 + 13625)/200`

= `sqrt(101.5)`

= 10.07

Hence, the number of boys = 75 and combined S.D. = 10.07.

Concept: Standard Deviation for Combined Data
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APPEARS IN

Balbharati Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 8 Measures of Dispersion
Miscellaneous Exercise 8 | Q II. (13) | Page 191
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