Statistics 3N03/3J04 - Assignment #3

2006-10-22

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 those 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 (see ed4p. 145, ed3p. 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. Use the complete gamma function gamma() when computing the theoretical mean and variance with formula (ed4 4-21, ed3 4-24).

Question 5

Generate 1000 samples, each of size n = 20, from the Weibull distribution used in Question 3. Compute the mean of each sample. Display the 1000 sample means on a histogram and on a normal probability plot with fitted line. This is a sample of 1000 observations from the sampling distribution of the sample mean. Find the mean and standard deviation of the sampling distribution. Repeat the exercise with n = 200. Discuss your results. What Theorem have you demonstrated?

Hint: Use weibsams <- matrix(rweibull(1000*20, 30, 2), ncol=20) to fill a 1000 x 20 matrix with independent Weibull data. Each row is a sample of 20 independent Weibull observations. Then, use weibmeans <- apply(weibsams, 1, mean) to find the 1000 row means.

Question 6

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

Fourth Edition 2-155 (p. 63) 3-78, 79, 80 (p. 87) 3-118 (p. 104) 3-137 (p. 106) 4-152, 156 (p. 152 ) 5-24 (p. 178) 7-48 (p. 255)

Third Edition 2-115 (p. 56) 3-66, 67, 68 (p.77) 3-106 (p. 94) 3-114 (p. 94) 4-148, 152 (p. 139) 5-52 (p. 166) 7-49 & 50 (p. 244)