Chapter 7, Problem 19RP

Random Numbers. A variable is said to be uniformly distributed or to have a uniform distribution with parameters a and b if its distribution has the shape of the horizontal line segment with equation y = 1/(b − a), for a < x < b. The mean and standard deviation of such a variable are $(a+b)/2\text{ and }(b-a)/\sqrt{12}$, respectively. The basic random-number generator on a computer or calculator, which returns a number between 0 and 1, simulates a variable having a uniform distribution with parameters 0 and 1.

1. a. Sketch the distribution of a uniformly distributed variable with parameters 0 and 1. Observe from your sketch that such a variable is far from being normally distributed.
2. b. Use the technology of your choice to simulate 2000 samples of two random numbers between 0 and 1.
3. c. Find the sample mean of each of the 2000 samples obtained in part (b).
4. d. Determine the mean and standard deviation of the 2000 sample means.
5. e. Theoretically, what are the mean and the standard deviation of all possible sample means for samples of size 2? Compare your answers to those you obtained in part (d).
6. f. Obtain a histogram of the 2000 sample means. Is the histogram bell shaped? Would you expect it to be?
7. g. Repeat parts (b)–(f) for a sample size of 35.