MAT 201 Statistics Lab #2 Part 1: Identify each of the following are discrete or continuous random variables: a) The speed of a race car in mph. b) The number of cups of coffee that Mrs. Lowery drinks each day. c) The number of people that play the SC Lottery each day. d) The weight of a rhinoceros. e) The time it takes to complete Mrs. Lowery’s midterm. f) The number of math majors at USC. g) The blood pressures of patients at Lexington Medical Center. Part 2: Discrete Probability Distributions From past experience, a company has found that in each production run of cartons of transistors, 92% contain no defective transistors, 3% contain one defective transistor, 3% contain two defective transistors, and 2% contain three defective transistors. 1. Construct a probability distribution displaying this data: x P(x) 2. Find each of the following: 1) P(X = 2) 2) P(X < 2) 5) P(X > 2) 6) 3) P(X ≤ 2) 4) P(X ≤ 1) P(X = 3 or X = 4) 3. Calculate the mean, variance, and standard deviation for the defective transistors. µ= σ2 = σ= Part 3: Binomial Probability Distributions The manufacturing sector contributes 17% of Canada’s gross domestic product. A customer orders 50 components from a factory that has a 99% quality production rate (99% of the products are defect-free). 1. Based on this information, find the probability that: a. none of the components in the order are defective b. There is at least one defective product in the order. c. There are at least two defective products in the order. 2. Calculate the mean, variance, and standard deviation for the defective transistors. µ= σ2 = σ= 3. Determine what outcomes would be considered unusual and explain why. Part 4: Normal Distributions and z scores Summary statistics for Batting Averages in 2013 MLB season: League Mean Std. Dev. Median Range Min Max Q1 Q3 AL 0.2556 0.0130 0.255 0.046 0.237 0.283 0.242 0.262 NL 0.2511 0.0115 0.249 0.039 0.231 0.245 0.27 0.26 a. Find and compare the Z-scores. The American League Leader for 2013 and MVP was Miguel Cabrera (Det.) with a .348 Batting Average. The z score is: The National League Leader for 2013 was Michael Cuddyer (Col) with a .331 Batting Average. The z score is: ________________ The MVP for the National League for 2013 was Andrew McCutchen (Pit) with a .317 Batting Average. The z score is: ______________ What do the z scores indicate? ________________________________________ b. A math professor gives two different tests to measure students' aptitude for math. Scores on the first test are normally distributed with a mean of 23 and a standard deviation of 4.2. Scores on the second test are normally distributed with a mean of 71 and a standard deviation of 10.8. Assume that the two tests use different scales to measure the same aptitude. If a student scores 29 on the first test, what would be his equivalent score on the second test? c) A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. If an applicant is randomly selected, find the probability of a rating that is between 170 and 220. Part 5: Sampling Distributions a. The amount of snowfall falling in a certain mountain range is normally distributed with a mean of 94 inches, and a standard deviation of 14 inches. What is the probability that the mean annual snowfall during 49 randomly picked years will exceed 96.8 inches? b. The weights of the fish in a certain lake are normally distributed with a mean of 19 lb and a standard deviation of 6. If 4 fish are randomly selected, what is the probability that the mean weight will be between 16.6 and 22.6 lb? c. Suppose that replacement times for washing machines are normally distributed with a mean of 9.3 years and a standard deviation of 1.1 years. Find the probability that 70 randomly selected washing machines will have a mean replacement time less than 9.1 years. Part 6: Normal as an Approximation to Binomial Distributions 1. It is estimated that 10% of the vehicles entering Canada from the United States carry undeclared goods. Use the normal approximation to calculate the probability that a search of 500 randomly selected vehicles will find: a. fewer than 50 with undeclared goods. b. more than 60 with undeclared goods. 2. A bank found that 25% of its loans to new small businesses become delinquent. If 500 small businesses are selected randomly from the bank’s files, what is the probability that at least 130 of them are delinquent? Compare the result from the normal approximation with that from a calculation using the binomial distribution. Explain whether the difference between the two results is significant.