Theory question 114
StatisticsThe table below shows the distribution of hours spent at work by the employees of a factory in a week.
Time (hours): 20 − 29, 30 − 39, 40 − 49, 50 − 59, 60 − 69, 70 − 79
No. of Persons: 8, 11, 23, 25, 8, 5
(a) Draw an ogive for the distribution.
(b) Using your graph, estimate the (i) median; (ii) lower quartile; (iii) 40th percentile; (iv) number of employees that spent at least 50 hours 30 minutes.
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Cumulative frequencies at upper class boundaries 29.5, 39.5, 49.5, 59.5, 69.5, 79.5: 8, 19, 42, 67, 75, 80
(a) Plot these points (plus (19.5, 0)) and join with a smooth S-shaped curve
(b)(i) Median at 40th: 39.5 + (21/23) × 10 ≈ 48.6 hours
(ii) Lower quartile at 20th: 39.5 + (1/23) × 10 ≈ 39.9 hours
(iii) 40th percentile at 32nd: 39.5 + (13/23) × 10 ≈ 45.2 hours
(iv) Cumulative frequency at 50.5 ≈ 42 + 2.5 = 44.5, so about 80 − 44.5 ≈ 35 employees
(Readings from a graph are accepted within a small tolerance.)
Plot cumulative frequency against upper class boundaries; read positions N/2, N/4 and 0.4N.