The questions that follow are based on the following information. Following is the frequency distribution of heights of students in 7th standard of a certain school.Height in cm Number of boys Number of girls 145 2 5 150 3 6 155 5 9 160 15 10 165 5 0 Total 30 30
Which of the following statements is true ?
The arithmetic mean of height of boys is greater than that of girls by 4 cm.
For a discrete frequency distribution, the arithmetic mean is found by multiplying each height by its frequency, adding these products and dividing by the total frequency: Mean = \(\frac{\Sigma fx}{\Sigma f}\). Here both the boys and the girls have a total frequency of 30, so the two means can be computed separately and then compared directly.
Boys:
\(\Sigma fx = 145 \times 2 + 150 \times 3 + 155 \times 5 + 160 \times 15 + 165 \times 5\)
\(= 290 + 450 + 775 + 2400 + 825\)
\(= 4740\)
Mean of boys = \(\frac{4740}{30} = 158\) cm
Girls:
\(\Sigma fx = 145 \times 5 + 150 \times 6 + 155 \times 9 + 160 \times 10 + 165 \times 0\)
\(= 725 + 900 + 1395 + 1600 + 0\)
\(= 4620\)
Mean of girls = \(\frac{4620}{30} = 154\) cm
Difference = \(158 - 154 = 4\) cm
The boys' mean is larger than the girls' mean by 4 cm. A difference of 0.8 cm does not match the computed values, the boys' mean is not smaller than the girls' mean, and the two means are clearly not equal, so those statements are false.
Hence, The arithmetic mean of height of boys is greater than that of girls by 4 cm.
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