Let X denote the distance (m) between to potholes on I75. Suppose that X has an exponential distribution with = 0.0238. We want to

“Let X denote the distance (m) between to potholes on I75. Suppose that X has an exponential distribution with λ = 0.0238. We want to see this in action by running some simulation experiments. We will use Excel to find random samples from a Gamma distribution (the exponential distribution is merely a gamma distribution with α = 1 and β = 1/λ). We will then examine the values we get for X from the various samples. a. The statistic of interest, which, for us, will be X. b. The population distribution, which in this case is the exponential distribution with λ = .0238. c. The sample size, which for us will vary (n = 8, 40). d. The number of replications, which for us will be k = 500. You should use a different sheet for each simulation. Create a sample of size n = 8 in the following way: 1. Randomly generate a number that comes from the specified gamma distribution. This may be accomplished by entering the command “=GAMMAINV(RAND(),α, β)”. Stretch this entry across eight columns. You’ve just generated a random sample of size n = 8. In this same row, two columns over, compute the sample mean of the cells to the left. You have now found E(X) for this particular random sample of size n = 8. Thus, the entry in the last column corresponds to the average distance between potholes on 8 different observations. Now highlight this row and drag down to row 500 to replicate the sample taking process 499 more times. 2. You should have a value of E(X) for each sample at this point, 500 of them, all in column E. Now construct a histogram of these values of E(X). Comment on its shape. Does it look Normal? 3. What should E(X) be? What is the standard deviation? Discuss what the mean and standard deviation of X should be (using formulas for exponential distribution) Now repeat the process with samples of size n = 40, using a new Excel file. Again, comment on the shape of the histogram. What shape should you expect and why?