The commuting time of a population of employees of the Post

The commuting time of a population of employees of the Post Office in Jacksonville has a mean or average of 45 minutes and a standard deviation of 8 minutes. (a) At least 75% of the employees will have commuting times between what two values? (b) At least what percentage of the employees will have commuting times 33 and 57 minutes.

Solution

A)

By Chebyshev\'s theorem, at least 75% of data are within 2 standard deviations from the mean.

Thus,

u - 2s = 45 - 2*8 = 29
u + 2s = 45 + 2*8 = 61

Thus, it is between 29 and 61. [ANSWER]

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b)

By the theorem, at least 1-1/k^2 lie within k standard deviations. Here, for 57,

k = (x-u)/s = (57-45)/8 = 1.5

Thus,

1-1/k^2 = 1-1/1.5^2 = 0.555555556 or 55.5556% [ANSWER]