S3N03/3J04 Assignment #2

Statistics 3N03/3J04 - Assignment #2

2004-10-17

Due: 2004-11-02 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 = 10. 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 5 or less. 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 = 10 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 n = 40, 100, 1000. 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 an exponential distribution with mean = 10. 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 a chi-square distribution on 1 degree of freedom. Look up its mean and variance in the text. Demonstrate the Central Limit Theorem by generating 1000 samples, each of size n = 20, from a chi-square distribution on 1 degree of freedom. 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? [Hint: Use matrix(rchisq(1000*20, 1), ncol=20) to fill a 1000 x 20 matrix with independent chi-square data, then use apply() to find the 1000 row means.]

Part B

2-123 (p 58) Also: If there is a read error, what is the probability that the head is properly aligned?
3-66 (p 77)
3-68 (p 77)
4-134 (p 138 )
4-138 (p 138)
4-148 (p 139) Hint: The maximum diameter won't exceed 1.6 mm if and only if none of the 10 exceed 1.6 mm.