The Information Systems Audit and Control Association surveyed office workers to learn about the anticipated usage of office computers for personal holiday shopping (USA Today, November 11, 2009). Assume that the number of hours a worker spends doing holiday shopping on an office computer follows an exponential distribution.
a. The study reported that there is a .53 probability that a worker uses an office computer for holiday shopping 5 hours or less. Is the mean time spent using an office computer for holiday shopping closest to 5.8, 6.2, 6.6, or 7 hours?
b. Using the mean time from part (a), what is the probability that a worker uses an office computer for holiday shopping more than 10 hours (to 4 decimals)?
c. What is the probability that a worker uses an office computer for holiday shopping between 4 and 8 hours (to 4 decimals)?
To answer these questions, we need to use the properties of exponential distribution. The exponential distribution is often used to model the time between consecutive events in a Poisson process, where events occur randomly and independently over time.
The probability density function (pdf) of an exponential distribution is given by:
f(x) = λ * e^(-λx)
where λ is the rate parameter and x is the time variable. The mean (μ) and standard deviation (σ) of an exponential distribution are both equal to 1/λ.
a. We are given that there is a 0.53 probability that a worker uses an office computer for holiday shopping 5 hours or less. From the properties of exponential distribution, we know that this corresponds to the cumulative distribution function (CDF) at x = 5, which can be calculated as:
CDF(5) = ∫[0 to 5] f(x) dx = 0.53
To find the mean time spent using an office computer for holiday shopping, we need to find the λ parameter. Solving the equation CDF(5) = ∫[0 to 5] λ * e^(-λx) dx = 0.53 gives us the λ value. Once we have λ, we can compute the mean as 1/λ.
b. Once we have the mean time spent on holiday shopping, we can calculate the probability of spending more than 10 hours using the survival function (1 - CDF) of the exponential distribution. The survival function at x = 10 is given by:
Survival(10) = 1 - CDF(10)
c. To find the probability of using an office computer for holiday shopping between 4 and 8 hours, we can calculate the difference between the CDFs at x = 8 and x = 4:
P(4 < x < 8) = CDF(8) - CDF(4)
Now, let's solve these separately:
a. Calculate the mean time spent using an office computer for holiday shopping:
1. Find λ: Solve the equation CDF(5) = ∫[0 to 5] λ * e^(-λx) dx = 0.53 numerically or using a statistical software.
2. Calculate the mean as 1/λ.
b. Calculate the probability of using an office computer for holiday shopping more than 10 hours:
1. Use the mean time calculated in part a.
2. Calculate the survival function at x = 10 using the formula Survival(10) = 1 - CDF(10).
c. Calculate the probability of using an office computer for holiday shopping between 4 and 8 hours:
1. Use the mean time calculated in part a.
2. Calculate the CDF at x = 8 and x = 4 using the formula CDF(x) = ∫[0 to x] λ * e^(-λt) dt.
3. Calculate the difference between the CDFs at x = 8 and x = 4: P(4 < x < 8) = CDF(8) - CDF(4).
Once you have the mean time and the respective formulas for each part, plug in the values into the formulas to calculate the probabilities.