The Bureau of Labor Statistics’ American Time Use Survey showed that the amount

Question

The Bureau of Labor Statistics’ American Time Use Survey showed that the amount of time spent using a computer for leisure varied greatly by age. Individuals age 75 and over averaged .20 hour (12 minutes) per day using a computer for leisure. Individuals ages 15 to 19 spend .8 hour per day using a computer for leisure. If these times follow an exponential distribution, find the proportion of each group that spends: (a) Less than 10 minutes per day using a computer for leisure.

(Round your answers to 4 decimal places.)

Proportion ______ and _________

Explanation / Answer

Solution

Back-up Theory

If X ~ Exponential with parameter (average inter-event time), the pdf (probability density function) of X is given by f(x) = (1/)e-x/, 0 x < ………………................................………………(1)

CDF (cumulative distribution function), F(t) = P(X t) = 1- e-t/ ……………….…(2)

From (2), P(X > t) = e-t/ ……………….………………………………..……………(3)

Proportion = probability……………………………………………………....………..(4)

Now, to work out the solution,

Let X = the amount of time spent per day using a computer for leisure by Individuals aged 75 and over

Y = the amount of time spent per day using a computer for leisure by Individuals aged 15 to 19. Then,

X ~ Exp(12) and Y ~ Exp(48)

And we want P(X < 10) and P(Y < 10)

[vide (2) under Back-up Theory ], P(X < 10) = 1 – e-10/12 = 1 – 0.4346 = 0.5654 [using Excel Function]

Similarly,

[vide (2) under Back-up Theory ], P(Y < 10) = 1 – e-10/48 = 1 – 0.8119 = 0.1881 [using Excel Function]

Thus,

Proportion that spends less than 10 minutes per day using a computer for leisure is:

0.5654 for Individuals aged 75 and over and

0.1881 Individuals aged 15 to 19.

DONE

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