The mean of the following distribution is $50$. Find the value of $a$ and hence the frequencies of $30$ and $70$.
$x$ $f$ $10$ $17$ $30$ $5a+3$ $50$ $32$ $70$ $7a-11$ $90$ $19$
| $x$ | $f$ |
| $10$ | $17$ |
| $30$ | $5a+3$ |
| $50$ | $32$ |
| $70$ | $7a-11$ |
| $90$ | $19$ |
(A) To find the mean,we use the formula $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$.
Constructing the frequency table:
Given $\bar{x} = 50$,we have:
$50 = \frac{640a + 2710}{12a + 60}$
$50(12a + 60) = 640a + 2710$
$600a + 3000 = 640a + 2710$
$3000 - 2710 = 640a - 600a$
$290 = 40a$
$a = \frac{290}{40} = 7.25$
Frequency of $30 = 5(7.25) + 3 = 36.25 + 3 = 39.25$
Frequency of $70 = 7(7.25) - 11 = 50.75 - 11 = 39.75$
Constructing the frequency table:
| $x_i$ | $f_i$ | $f_i x_i$ |
| $10$ | $17$ | $170$ |
| $30$ | $5a+3$ | $150a+90$ |
| $50$ | $32$ | $1600$ |
| $70$ | $7a-11$ | $490a-770$ |
| $90$ | $19$ | $1710$ |
| Total | $\sum f_i = 12a + 60$ | $\sum f_i x_i = 640a + 2710$ |
Given $\bar{x} = 50$,we have:
$50 = \frac{640a + 2710}{12a + 60}$
$50(12a + 60) = 640a + 2710$
$600a + 3000 = 640a + 2710$
$3000 - 2710 = 640a - 600a$
$290 = 40a$
$a = \frac{290}{40} = 7.25$
Frequency of $30 = 5(7.25) + 3 = 36.25 + 3 = 39.25$
Frequency of $70 = 7(7.25) - 11 = 50.75 - 11 = 39.75$
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