Theory question 114
StatisticsThe table shows the distribution of ages of workers in a company
Age in (years): 17 − 21, 22 − 26, 27 − 31, 32 − 36, 37 − 41, 42 − 46, 47 − 51, 52 − 56
No of workers: 12, 24, 30, 37, 45, 25, 10, 7
(a) Using an assumed mean of 39, calculate the (i) mean; (ii) standard deviation; of the distribution.
(b) If a worker is selected at random from the company for an award, what is the probability that he is at most 36 years old?
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Midpoints 19, 24, 29, 34, 39, 44, 49, 54; d = x − 39: −20, −15, −10, −5, 0, 5, 10, 15
Σf = 190, Σfd = −755, Σfd² = 17325
(a)(i) Mean = 39 + (−755/190) = 39 − 3.97 ≈ 35.03 years
(ii) SD = √(17325/190 − (755/190)²) = √(91.18 − 15.79) = √75.39 ≈ 8.68
(b) Workers aged at most 36: 12 + 24 + 30 + 37 = 103
Probability = 103/190 ≈ 0.542
Work with deviations from the assumed mean: mean = A + Σfd/Σf, SD = √(Σfd²/Σf − (Σfd/Σf)²).