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Question
The mean of the following distribution is 50.
| x | f |
| 10 | 17 |
| 30 | 5a + 3 |
| 50 | 32 |
| 70 | 7a – 11 |
| 90 | 19 |
Find the value of a and hence the frequencies of 30 and 70.
Solution
The formula of the mean `(barx)` is:
`barx = (sum_(i = 1)^n f_ix_i)/(sum_(i = 1)^n f_i)`
So, the value of a wall be calculated as follows:
`50 = (17 xx 10 + (5a + 3) xx 30 + 32 xx 50 + (7a - 11) xx 70 + 19 xx 90)/(17 + 5a + 3 + 32 + 7a - 11 + 19)`
`50 = (170 + 150a + 90 + 1600 + 490a - 770 + 1710)/(12a + 60)`
`50 = (3570 - 770 + 640a)/(12a + 60)`
2800 + 640a = 600a + 3000
40a = 200
a = 5
So, frequency of 30 will be = 5a + 3 = 5 × 5 + 3 = 25 + 3 = 28
Similarly, frequency of 70 will be = 7a – 11 = 7 × 5 – 11 = 35 – 11 = 24
Therefore, the frequency of 30 and 70 are 28 and 24 respectively.
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