COMP 233, Winter 2016 CONCORDIA UNIVERSITY PROBABILITY AND STATISTICS FOR COMPUTER SCIENCE. Assignment 3. Due: March 25, 2016. 1. (a) [6 marks] A random variable X has the standard normal distribution. Use the standard normal Table to determine : P (X 0.5), P (| X | 0.5), P (X 0.5), P (1 X 1), P (| X | 0.5), P (1 X 0.5) . (b) [6 marks] Suppose X is normally distributed with mean = 1.5 and standard deviation = 2.5 . Use the standard normal Table to determine : P (X 0.5), P (X 0.5), P (| X | 0.5), P (| X | 0.5) . 2. (a) [6 marks] Use the Central Limit Theorem (CLT) to estimate P (2 24), 32 P (2 40), 32 P (| 2 32 | 8) . 32 (b) [6 marks] Suppose X1 , X2 , , X12 , are identical, independent, uniform random 1 variables on [0, 1] . Let X = 12 (X1 + X2 + + X12 ). Use the CLT to compute approximate values of 1 P (X ), 3 2 P (X ), 3 P (| X 1 1 | ) . 2 3 (c) [6 marks] Suppose X1 , X2 , , X9 , are identical, independent, exponential random 1 variables, with f (x) = ex . Let X = 9 (X1 + X2 + + X9 ) . Use the CLT to compute approximate values of P (X 0.4), P (X 1.6), P (| X 1 | 0.6) . 3. Mathematics test scores across a population of high school are normally distributed with mean 500 and standard deviation 100. Determine the following probabilities: (a) [4 marks] A randomly chosen student scored below 600. (b) [4 marks] Of ve randomly chosen students, all scored below 600. (c) [4 marks] Of ve randomly chosen students, exactly three scored above 640. Hint: Use binomial distribution for part (b) and (c). 4. [12 marks] According to a transportation safety board, the number of persons per car passing a certain intersection between 8:00 and 9:00am, is a random variable H with mean 4 and variance 2. For a random sample of 30 cars at this intersection during this time period, what is the probability that the average number of persons per car is at least 5? Hint: Use the CLT. COMP 233, Winter 2016 Assignment 3, Page 2 5. [12 marks] A small elevator has a maximum capacity C, which is normally distributed, with mean 400 kg., and standard deviation 4 kg. The weight of the boxes being loaded into the elevator is a random variable with mean 30 kg., and standard deviation 0.3 kg. Assume that the weights of the boxes and maximum capacity are independent random variables. How many boxes may be loaded into the elevator before the probability of disaster exceeds 20%? Hint: Let Xi denote the weight of the i-th box being loaded into the elevator, and let Pn be probability that the weight of n loaded boxes exceeds maximum capacity C. Then Pn = P ( n i=1 Xi > C) = P ( n i=1 Xi C > 0). Now use the CLT. 6. [12 marks] The following are the percentages of ash content in 12 samples of coal: 9.2 , 14.1 , 9.8 , 12.4 , 16.0 , 12.6 , 22.7 , 18.9 , 21.0 , 14.5 , 20.4 , 16.9 Compute the sample mean. the sample median. and the sample standard deviation. 7. The average particulate concentration, in micrograms per cubic meter, was measured in a petrochemical complex at 36 randomly chosen times, with the following results: 5, 18, 15, 7, 23, 220, 130, 85, 103, 25, 80, 7, 24, 6, 13, 65, 37, 25, 24, 65, 82, 95, 77, 15, 70, 110, 44, 28, 33, 81, 29, 14, 45, 92, 17, 53 (a) [2 marks] Represent the data in a histogram. (b) [2 marks] Is the histogram approximately normal? (c) [2 marks] Calculate the sample mean X. (d) [2 marks] Calculate the sample standard deviation S. (e) [2 marks] Determine the proportion of the data values that lies within X 1.5S and compare with the lower bound given by Chebyshev's inequality. (f) [2 marks] Determine the proportion of the data values that lies within X 2S and compare with the lower bound given by Chebyshev's inequality. 8. [12 marks] If 10 pairs of fair dice are rolled, approximate the probability that the sum of the values obtained (which ranges from 20 to 120) is between 30 and 40 inclusive. Hint: Use the CLT. 9. The age at which the fan assembly in a laptop fails is normally distributed with variance 2 . Let S 2 be the corresponding sample variance for seven laptops. Determine (a) [6 marks] P (S 2 / 2 1.5) (b) [6 marks] P (0.8 S 2 / 2 1.1) Hint: Use 2 distribution. 10. Consider the following data on the diameter of a rivet. Assume a normal population. 6.68 6.66 6.62 6.72 6.76 6.67 6.70 6.72 6.78 6.66 6.76 6.72 6.76 6.70 6.76 6.76 6.74 6.74 6.81 6.66 6.64 6.79 6.72 6.82 (a) [6 marks] Estimate the population variance 2 . (b) [6 marks] Compute a 95 percent two-sided condence interval for 2