Statistics 3N03/3J04 - Assignment #2

2005-10-23

Due: 2005-11-02 18:00

The numbered problems and data sets are taken from Montgomery & Runger, Applied Statistics and Probability for Engineers, 3rd edition. You will find it easy to do your graphs and calculations in R but, where appropriate, try the calculations on your calculator to check your results.

Part A

Question 1

Graph the probability density function of a Poisson distribution with mean = 8. Superimpose a graph of the approximating normal probability density function. Use vertical bars to show the Poisson probabilities and use a smooth line in a different colour for the normal curve. Compute the exact Poisson probability of getting 6 or more. Indicate this tail of the distribution by using a different colour for the vertical bars. Compare the exact calculation with the normal approximation, computed with and without the continuity correction.

Question 2

Generate a sample of n = 10 pseudorandom observations from a normal distribution with mean = 30 and variance = 100. Test for normality graphically by plotting a histogram with the true normal density superimposed and by plotting a probability plot with fitted line, using qqnorm() and qqline(). Repeat for some larger values of n. How many observations do you need before you can say with any confidence that the data came from a normal distribution?

Question 3

Repeat Question 2, this time with pseudorandom observations from a Weibull distribution (p. 133) with shape parameter = 30 and scale parameter = 2. How many observations do you need before you can say with any confidence that the data did not come from a normal distribution?

Question 4

Plot the probability density function for Weibull distribution used in Question 3. Compute its mean and variance using the formulas given in the text. Demonstrate the Central Limit Theorem by generating 1000 samples, each of size n = 20, from this Weibull distribution. Compute the mean of each sample. Display the 1000 sample means on a histogram and on a normal probability plot with fitted line. Find the mean and standard deviation of the distribution of the means. Repeat the exercise with n = 200. Are your results consistent with the Central Limit Theorem? [Hints: Use matrix(rweibull(1000*20, 30, 2), ncol=20) to fill a 1000 x 20 matrix with independent Weibull data, then use apply() to find the 1000 row means. Use the complete gamma function gamma() when computing the mean and variance.]

Question 5

Suppose that you have apples with a mean weight of 100 g and standard deviation 10 g. You are you are packing them randomly, 10 to a bag, and you pack 1000 bags. What is the probability that you will get more than 60 bags each weighing less than 950 g? How does this probability change if the standard deviation of an apple is 9 g? 11 g?

Part B

2-74 (p. 45)
3-106 (p. 94)
3-114 (p. 94)
4-152 (p. 139 )
5-52 (p. 166)
7-49 & 50 (p. 244)

Statistics 3N03/3J04