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The mean of the following frequency distribution is 28. If sum of all frequencies is 100, then find the values of $p$ and $q$:
Class Interval 0-10 10-20 20-30 30-40 40-50 50-60
Frequency 12 p 27 20 q 6
Class Interval 0-10 10-20 20-30 30-40 40-50 50-60
Frequency 12 p 27 20 q 6
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Sol. Class Interval $x_i$ $f_i$ $f_ix_i$
0-10 5 12 60
10-20 15 p 15p
20-30 25 27 675
30-40 35 20 700
40-50 45 q 45q
50-60 55 6 330
Total 65+p+q 15p+45q+1765
Correct table (I) (2)
$\sum f_i = 100 = 65 + p + q \Rightarrow p + q = 35$ (II) (1/2)
Mean = $28 = \frac{15p + 45q + 1765}{100}$ (III) (1)
$\Rightarrow p + 3q = 69$ (IV) (1/2)
On solving, we get $p = 18, q = 17$ (V) (1/2+1/2)
0-10 5 12 60
10-20 15 p 15p
20-30 25 27 675
30-40 35 20 700
40-50 45 q 45q
50-60 55 6 330
Total 65+p+q 15p+45q+1765
Correct table (I) (2)
$\sum f_i = 100 = 65 + p + q \Rightarrow p + q = 35$ (II) (1/2)
Mean = $28 = \frac{15p + 45q + 1765}{100}$ (III) (1)
$\Rightarrow p + 3q = 69$ (IV) (1/2)
On solving, we get $p = 18, q = 17$ (V) (1/2+1/2)