Question

In a particular factory, workers produce various types of output units.
The following distribution was obtained.

Output units Produced	No. of workers
70 – 74	40
75 – 79	45
80 – 84	50
85 – 89	60
90 – 94	70
95 – 99	80
100 – 104	100

Find the percentage of workers who have produced less than 82 output units.

Answer 1

Since the given data is not continuous, we have to convert it in the continuous form by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit of every class interval.

∴ The class intervals will be 69.5 – 74.5, 74.5 – 79.5, etc.

We construct the less than cumulative frequency table as given below:

Output units produced	No. of workers (f)	Less than Cumulative frequency (c.f.)
69.5 – 74.5	40	40
74.5 – 79.5	45	85
79.5 – 84.5	50	135
84.5 – 89.5	60	195
89.5 – 94.5	70	265
94.5 – 99.5	80	345
99.5 – 104.5	100	445
Total	445

Here, N = 445
Let P_x = 82
The value 82 lies in the class 79.5 – 84.5.
∴ L = 79.5, f = 50, c.f. = 85, h = 5

∴ Px = `"L"+"h"/"f"(("xN")/100-"c.f.")`

∴ 82 = `79.5 + 5/50(("x"xx445)/100-85)`

∴ 82 − 79.5 = `1/10(4.45"x" - 85)`

∴ 2.5 × 10 = `4.45"x" - 85`

∴ 25 + 85 = 4.45x

∴ `110/4.45` = x

∴ x = `11000/445`

∴ x = `(2200)/89`

∴ x = 24.72

∴ 24.72% of workers produced less than 82 output units.