Multiple choice

The mean of the following distribution is $50$. $x$ $10$ $30$ $50$ $70$ $90$ $f$ $17$ $5a+3$ $32$ $7a-11$ $19$ Find the value of $a$.

  1. $3$
  2. $4$
  3. $5$
  4. $6$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using the mean formula, the total frequency is 60 + 12a and the sum of xf is 2800 + 640a. Setting their ratio equal to 50 gives 2800 + 640a = 3000 + 600a, so a = 5.

AI explanation

The mean of 50 implies the total sum of observations is 50 times the total frequency. The total frequency is 17 + 5a + 3 + 32 + 7a - 11 + 19 = 60 + 12a. The sum of (frequency times value) is (17*10) + (5a+3)*30 + (32*50) + (7a-11)*70 + (19*90), which equals 170 + 150a + 90 + 1600 + 490a - 770 + 1710 = 2800 + 640a. Setting up the equation 2800 + 640a = 50 * (60 + 12a) gives 2800 + 640a = 3000 + 600a. Solving this yields 40a = 200, so a equals 5.

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