Question

Consider the following distribution:
 
The mean of the distribution is 8.84 years. The value of x is:

• A. 8
• B. 13
• C. 18
• D. 23

Hint:

To find unknown frequencies using the mean, apply the formula: \[ \bar{x} = \frac{\sum f_i x_i}{\sum f_i} \] where \( x_i \) are class midpoints and \( f_i \) are frequencies.

Answer 1

The Correct Option is B

Use the midpoint of each class: \[ \text{Midpoints: } 6, 8, 10, 12, 14 \] Let \( x \) be the unknown frequency in the 7–9 class. Then: \[ \text{Mean} = \frac{6(16) + 8(x) + 10(10) + 12(6) + 14(5)}{16 + x + 10 + 6 + 5} = 8.84 \] \[ \Rightarrow \frac{96 + 8x + 100 + 72 + 70}{37 + x} = 8.84 \Rightarrow \frac{338 + 8x}{37 + x} = 8.84 \] Solve this equation: \[ 338 + 8x = 8.84(37 + x) = 327.08 + 8.84x
\Rightarrow 338 - 327.08 = 8.84x - 8x = 0.84x \Rightarrow 10.92 = 0.84x \Rightarrow x = 13 \] Correct value: 13