Question

We have seen that the standard deviation σ measures the spread of a data set about the mean μ. Chebyshev's inequality gives an estimate of how well the standard deviation measures that spread. One consequence of this inequality is that for every data set at least 75% of the data points lie within two standard deviations of the mean, that is, between μ − 2σ and μ + 2σ (inclusive). For example, if μ = 20 and σ = 5, then at least 75% of the data are at least 20 − 2 × 5 = 10 and at most 20 + 2 × 5 = 30.
We have 1200 lightbulbs in our building. Over a 10-month period, we record the number of bulbs that burn out each month. The result is the data list

19, 31, 35, 32, 29, 37, 39, 27, 24, 15.

(a) What is the average number of bulbs that burn out each month? (Round your answer to the nearest whole number.)
  bulbs

(b) What is the standard deviation of these data? Round the standard deviation to one decimal.

(c) Use Chebyshev's inequality and your answers to parts (a) and (b) to estimate how many replacement bulbs you should keep on hand so that for at least 75% of the months you don't have to acquire additional replacement bulbs. (Round your answer up to the nearest whole number.)
bulbs

Answer 1

a) Mean ( ) = = 288 / 10 = 28.8

Mean ( ) = 29 bulbs

b) Standard deviation ( ) =   =  

Standard deviation ( ) = 7.3

c)  μ − 2σ = 29 - (2*7.3 ) = 14.4 ~ 14

μ + 2σ = 29 + (2*7.3 ) = 43.6 ~ 44

You should keep on hand 14 to 44 bulbs