MEMORIAL UNIVERSITY OF NEWFOUNDLAND DEPARTMENT OF MATHEMATICS AND STATISTICS MIDTERM EXAM - STATISTICS 2550 - FALL 2014, SECTION 005 Instructor: A. Oyet Date: October 16, 2014 Name(Surname First): Student Number Time Allowed: 75mins INSTRUCTIONS 1. Make sure that your examination paper has 13 multiple choice questions in Section 1 and 3 QUES- TIONS in Section 2. 2. No books or notes are allowed. You may use your calculator. 3. Statistical tables can be found on the last page. Section Question # Mark Score 1 1-13 30 1 20 2 2 20 3 20 Total Section 1: Multiple Choice - 30 points Instruction: Answer all questions in Section 1. Circle the correct answer. 1. As part of an economics class project, students were asked to randomly select 500 New York Stock Exchange (NYSE) stocks from the Wall Street Journal. As part of the project, students were asked to summarize the current prices (also referred to as the closing price of the stock for a particular trading date) of the collected stocks using graphical and numerical techniques. a) Identify the variable of interest for this study. A) the entire set of stocks that are traded on the NYSE B) the 500 NYSE stocks that current prices were collected from C) the current price (or closing price) of a NYSE stock D) a single stock traded on the NYSE b) What type of variable is being collected? A) Qualitative B) Quantitative 1
2. A recent survey was conducted to compare the cost of solar energy to the cost of gas or electric energy. Results of the survey revealed that the distribution of the amount of the monthly utility bill of a 3-bedroom house using gas or electric energy had a mean of $112 and a standard deviation of $15. Three solar homes reported monthly utility bills of $59, $63, and $64. a) Using the mean and standard deviation of the gas or electric energy bills, compute the z-score of the bills for the three solar homes. A) 3.53, 3.17, and 3.3 B) -3.53, 3.27, and 3.2 C) -3.53, -3.17, and -3.3 D) -3.53, -3.27, and -3.2 b) Based on your z-score, which of the following statement is true? A) Homes using solar power always have lower utility bills than homes using only gas and electricity. B) The utility bills for homes using solar power are about the same as those for homes using only gas and electricity. C) Homes using solar power may actually have higher utility bills than homes using only gas and electricity. D) Homes using solar power may have lower utility bills than homes using only gas and electricity. 3. The Fresh Oven Bakery knows that the number of pies it can sell varies from day to day. The owner believes that on 50% of the days she sells 100 pies. On another 25% of the days she sells 150 pies, and she sells 200 pies on the remaining 25% of the days. To make sure she has enough product, the owner bakes 200 pies each day at a cost of $2.50 each. Assume any pies that go unsold are thrown out at the end of the day. If she sells the pies for $3 each, find the probability distribution for her daily profit. A) -$200 0.5 -$50 0.25 $100 0.25 B) $300 0.5 $450 0.25 $600 0.25 C) $100 0.5 $250 0.25 $400 0.25 4. Consider the given discrete probability distribution. Find P (X = 5). x 3 5 7 9 p(x) 0.33? 0.27 0.31 A) 0.09 B) 0.91 C) 0.45 D) 4.55 D) $50 0.5 $75 0.25 $100 0.25 5. The random variable X represents the number of boys in a family with three children. Assuming that births of boys and girls are equally likely, find the mean and standard deviation for the random variable X. A) mean: 1.5; standard deviation: 0.75 B) mean: 2.25; standard deviation: 0.87 C) mean: 1.5; standard deviation: 0.87 D) mean: 2.25; standard deviation: 0.75 6. The university police department must write, on average five tickets per day to keep department revenues at budgeted levels. Suppose the number of tickets actually written follows a Poisson distribution with a mean of 9. What is the probability that exactly four tickets are written on a randomly selected day? A) 0.9643 B) 0.055 C) 0.0338 D) 0.0212 2
7. After a particularly heavy snowstorm, the depth of snow reported in a mountain village followed a uniform distribution over the interval from 15 to 22 inches of snow. What is the standard deviation of the snowfall amounts? A) 18.5 inches B) 4.08 inches C) 2.02 inches D) 1.42 inches 8. Let Z be a standard normal random variable. The mean and standard deviation of Z are A) mean: µ, standard deviaion: σ B) mean: 0, standard deviaion: σ C) mean: 0, standard deviaion: 1 B) mean: µ, standard deviaion: σ 2 9. Let Z be a standard normal random variable. What is the value of z 0 such that P (Z z 0 ) = 0.7 A) -0.525 B) -0.985 C) -0.815 D) 0.475 10. A study of college students stated that 25% of all college students have at least one tattoo. In a random sample of 80 college students, let X be the number of the students that have at least one tattoo. Can the normal approximation be used to estimate the probability of X in this problem? A) Yes B) No 11. The probability of success, p, in a binomial experiment is a parameter, while the mean and standard deviation, µ and σ, are statistics. A) True B) False 12. The central limit theorem states that the sampling distribution of the sample mean is approximately normal under certain conditions. Which of the following is a necessary condition for the central limit theorem to be used. A) The population size must be large (e.g. at least 30). B) The population from which we are sampling must be normally distributed. C) The sample size must be large (e.g. at least 30). D) The population from which we are sampling must not be normally distributed. 13. Suppose students ages follow a skewed right distribution with a mean of 21 years old and a standard deviation of 2 years. if we randomly sample 450 students, which of the following statements about the sampling distribution of the sample mean age is incorrect? A) The standard deviation of the sampling distribution is equal to 2 years. B) The shape of the sampling distribution is approximately normal. C) The mean of the sampling distribution is approxmately 21 years old. D) All of the above statements are not correct. 3
Section 2 Instruction: You are to answer at most 2 questions from this section. You must answer Question 2 and either Question 1 or 3. Clearly outline your answers. Show your work and answer each question carefully. Problem 1 (20 points) Consider the population described by the probability distribution below x 2 5 7 p(x) 0.2 0.5 0.3 The random variable X is observed twice. The observations are indpendent. The different samples of size 2 and their probabilities are shown below. Sample Probability 2,2 0.04 2,5 0.10 2,7 0.06 Sample Probability 5,2 0.10 5,5 0.25 5,7 0.15 Sample Probability 7,2 0.06 7,5 0.15 7,7 0.09 a) Compute the sample mean for each sample of size 2. (4 points) b) Compute P ( X = 3.5). (2 points) c) Find the sampling distribution of the sample mean x. (5 points) d) Compute the expected value of X. (4 points) e) Is x an unbiased estimator of the population mean µ? Why? (5 points) 4
Problem 2 (20 points) Spray drift is a constant concern for pesticide applicators and agricultural producers. In a study on the effects of herbicide formulation on spray atomization it was found that a normal distribution with mean 1050 µm and standard deviation 150 µm is a reasonable model for droplet size sprayed through a 760 ml/min nozzle. a) What is the probability that the size of a single droplet is at least 1000 µm? (5 points) b) How would you characterize the smallest 2% of all droplets? (5 points) c) If the sizes of five independently selected droplets are measured, what is the probability that at least one exceeds 1500 µm? (10 points) 5
Problem 3 (20 points) Consider randomly selecting a student at Memorial University and let A denote the event that the selected individual has a Visa credit card and B be the analogous event for a Mastercard. Suppose that P (A) = 0.5, P (B) = 0.4, and P (A B) = 0.25 a) Compute the probability that the selected individual has at least one of the two types of cards. (5 points) b) What is the probability that the selected individual has neither type of card? (5 points) c) Find P (A c B). (5 points) d) Find P (B c A). (5 points) 6