Reconsider Eddie's Bicycle Shop described in the preceding problem. Forty percent of the bicycles require only a minor repair. The repair time for these bicycles has a uniform distribution between zero and one hour. Sixty percent of the bicycles require a major repair. The repair time for these bicycles has a uniform distribution between one hour and two hours. You now need to estimate the mean of the overall probability distribution of the repair times for all bicycles by using the following alternative methods.
a. Use the random numbers 0.7256, 0.0817, and 0.4392 to simulate whether each of three bicycles requires minor repair or major repair. Then use the random numbers 0.2243, 0.9503, and 0.6104 to simulate the repair times of these bicycles. Calculate the average of these repair times to estimate the mean of the overall distribution of repair times.
b. Repeat part a with the complements of the random numbers used there, so the new random numbers are 0.2744, 0.9183, 0.5608, and then 0.7757, 0.0497, and 0.3896.
c. Combine the random observations from parts a and b and calculate the average of these six observations to estimate the mean of the overall distribution of repair times. (This is referred to as the method of complementary random numbers.)
d. The true mean of the overall probability distribution of repair times is 1.1. Compare the estimates of this mean obtained in parts a, b, and c. For the method that provides the closest estimate, give an intuitive explanation for why it performed so well.
e. Formulate a spreadsheet model to apply the method of complementary random numbers described in part c. Use 600 random numbers and their complements to generate 600 random observations of the repair times and calculate the average of these random observations. Compare this average with the true mean of the distribution?