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The median of the following data is $32.5$, find the missing frequencies $x$ and $y$ :
Class: $0-10$ $10-20$ $20-30$ $30-40$ $40-50$ $50-60$ $60-70$ Total
Frequency: $x$ $5$ $9$ $12$ $y$ $3$ $2$ $40$
Class: $0-10$ $10-20$ $20-30$ $30-40$ $40-50$ $50-60$ $60-70$ Total
Frequency: $x$ $5$ $9$ $12$ $y$ $3$ $2$ $40$
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Sol. Correct table (I) (1 Mark)
Median Class $= 30 - 40$ (II) ($\frac{1}{2}$ Mark)
$\therefore 32.5 = 30 + \frac{10}{12} (\frac{40}{2} - (x+14))$ (III) (1 Mark)
$\Rightarrow x = 3$ (IV) ($\frac{1}{2}$ Mark)
$x + y + 31 = 40$
$\Rightarrow 3 + y = 9$ (V) (1 Mark)
$\therefore y = 9-3 = 6$ (VI) ($\frac{1}{2}$ Mark)
Median Class $= 30 - 40$ (II) ($\frac{1}{2}$ Mark)
$\therefore 32.5 = 30 + \frac{10}{12} (\frac{40}{2} - (x+14))$ (III) (1 Mark)
$\Rightarrow x = 3$ (IV) ($\frac{1}{2}$ Mark)
$x + y + 31 = 40$
$\Rightarrow 3 + y = 9$ (V) (1 Mark)
$\therefore y = 9-3 = 6$ (VI) ($\frac{1}{2}$ Mark)