Multiple choice

The mean of the followlng frequency table is $53$. But the frequencies $f_1$ and $f_2$ in the classes $20-40 $ and $60-80 $ are missing. Find the missinq frequencies. Age (in years) 0-20 20-40 40-60 60-80 80-100 Total No. of people 15 $f_1$ 21 $f_2$ 17 100

  1. $f_1=20, f_2=28$
  2. $f_1=17, f_2=19$
  3. $f_1=19, f_2=25$
  4. $f_1=18, f_2=29$
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D Correct answer
Explanation

Total frequency = 15 + f1 + 21 + f2 + 17 = 100, so f1 + f2 = 47. Mean = (sum of f*x)/100 = 53. Using midpoints (10, 30, 50, 70, 90): (150 + 30f1 + 1050 + 70f2 + 1530) / 100 = 53. 30f1 + 70f2 = 2730 - 2700 = 30. This implies 3f1 + 7f2 = 270. Solving the system: f1 = 18, f2 = 29.

AI explanation

Using the step-deviation method for finding the mean, the class marks for the given intervals (0-20, 20-40, 40-60, 60-80, 80-100) are 10, 30, 50, 70, and 90, with a common class width of 20. Assuming the mean A as 50, the deviations u_i equal (x_i - 50) / 20, resulting in values of -2, -1, 0, 1, and 2 for the five classes. The equation for the mean becomes 53 = 50 + (20 / 100) multiplied by the sum of f_i and u_i, which simplifies to the sum of f_i and u_i equaling 15. We also know from the total frequency that f_1 plus f_2 equals 47, so solving the two equations gives the missing frequencies as f_1 equals 18 and f_2 equals 29.

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