STA113: Probability and Statistics in Engineering

Homework #5

Due: Friday, Mar 6

Problem 1

Generate 1600 random samples, each of size 16, from the Poisson distribution with mean 2. Draw histograms of the sample means and sample variances of your 1600 samples. Hint: Here is a Minitab routine to help; ignore the "* NOTE * No data in column" warning.

MTB > ERASE c1, c2
MTB > NAME C1='Means', C2='Vars'
MTB > STORE 'RANSAM'
STOR> RANDOM 16 INTO C3;
STOR> POISS 2.
STOR> LET C4 = STDEV (C3)
STOR> LET C4 = C4 * C4
STOR> LET C3 = MEAN (C3)
STOR> STACK C1 C3 C1
STOR> STACK C2 C4 C2
END > END.
MTB > EXECUTE 'RANSAM' 1600
MTB > HISTOGRAM 'Means'
MTB > HISTOGRAM 'Vars'

a)	What are the mean and variance of your sample means?
b)	How many of your sample means are at least 2.5?
	(HINT: CODE (0:2.5) 0 (2.5:1599) 1 C1, C5 will fill column C5 with zero's and one's for the sample means that are less than or greater than 2.5, respectively; now use SUM)
c)	Use the normal approximation to calculate about how many sample means you should EXPECT to be greater than 2.5.
d)	Plot a histogram of the values of 1600 simulated Poisson random variables (all with mean two). Give the mean and variance. Does this look similar to the histogram of sample means? How do they differ?

Problem 2

Make the few changes needed to the Minitab program above needed to produce histograms from simulated distributions of the sample variance S^2 from samples of size n=100 observations from a

a)	Uniform distribution on (0,1)
b)	Normal distribution, with mean 0 and variance 1
c)	Exponential distribution with mean 1

Problem 3

Problem 7.18, on page 316 of the textbook (Mendenhall & Sincich).

Problem 4

Problem 7.29, on page 320 of the textbook (Mendenhall & Sincich).

No Optional Exercise this week.