A variable is said to have an exponential distribution or to be exponentially distributed if its distribution has the shape of an exponential curve, that is, a curve of the form y = e−x/μ/μ for x > 0, where μ is the mean of the variable. The standard deviation of such a variable also equals μ. At the emergency room at Desert Samaritan Hospital in Mesa, Arizona, the time from the arrival of one patient to the next, called an interarrival time, has an exponential distribution with a mean of 8.7 minutes.
a. Sketch the exponential curve for the distribution of the variable "interarrival time." Note that this variable is far from being normally distributed. What shape does its distribution have?
b. Use the technology of your choice to simulate 1000 samples of four interarrival times each.
c. Find the sample mean of each of the 1000 samples.
d. Determine the mean and standard deviation of the 1000 sample means.
e. Theoretically, what are the mean and the standard deviation of all possible sample means for samples of size 4? Compare your answers to those you obtained in part (d).
f. Obtain a histogram of the 1000 sample means. Is the histogram bell shaped? Would you necessarily expect it to be?
g. Repeat parts (b)-(f) for a sample size of 40.