Suppose that average weight of men is 70 kg and the standard deviation is 10 kg, while for women the average is 60 kg and the standard deviation is 8 kg. If 10 men and 6 women get on an elevator, what is the probability that their combined weight will exceed 1 tonne? Will exceed 1.1 tonnes? State any additional assumptions you make.
Suppose that reportable accidents at a round-the-clock construction site occur independently of each other at an average rate of 0.1 accidents per day. Compute the probabilities of the following events: (a) no accidents in November; (b) more than four accidents in November; (c) no accidents next week; (d) five or more accident-free weeks over the next ten weeks; (e) seven or more accident-free weeks over the next ten weeks; (f) no accidents in the next seven weeks; (g) the time between accidents exceeds seven weeks. State any additional assumptions you make.
Graph the probability density function for a Poisson distribution with mean = 74. Superimpose a graph of the approximating normal probability density function. Use vertical bars to show the Poison probabilities and use a smooth line in a different colour for the normal curve. Compute the exact Poisson probability of getting 80 or more. Compare the exact calculation with the normal approximation, computed with and without the continuity correction. [Hint: This is very easy to do in R. Use plot(0:110, dpois(0:110, 74), type = "h") to draw vertical lines for the Poisson probabilities, then use lines() to add the normal pdf. As a finishing touch, you could highlight the probability of getting 80 or more with lines(80:110, dpois(80:110, 74), type = "h", col = "blue")]
Generate 500 samples, each with n = 25 independent observations from a normal distribution with mean = 10 and standard deviation = 2. Arrange the data in 500 rows, so that each row is one sample. Use these samples to demonstrate the Central Limit Theorem, which says that the distribution of the sample mean will be normal with mean = 10 and standard deviation = 2/5.
The attached data file gives 500 samples of n = 25 observations from a normal distribution with mean = 10 and standard deviation = 2. The data are arranged in 500 rows, so that each row is one sample. Is the distribution of the sample mean normal with mean = 10 and standard deviation = 2/5? How are these samples different from the samples you generated in Question 4? Use appropriate graphs to explain your answer.
4-58 (p 121), 4-108 (p 142), 4-110 (p 142);
5-44 (p 169), 5-130 (p 197).