STATISTICS 20 Shobhana M. Stoyanov FALL 2019 MIDTERM Name: SID: Section number and time: • No calculators, and no notes except for the provided sheet. • EXPLAIN all your reasoning and SHOW your work UNLESS otherwise instructed. • The test begins on the next page. You may use this page for scratch work. • WRITE YOUR NAME ON EACH PAGE Name: Part I In this part of the exam, the problems are multiple choice and worth one point each. You do not need to show your work. 1. According to a report by Dr. John Olney in the Journal of Neuropathology and Experimental Neurology, data from the US government show that brain cancer rates jumped 10% shortly after NutraSweet was approved by the Food and Drug Administration for widespread use in 1983. This study was (circle one): (a) an observational study (b) a simple random sample (c) a randomized control study (d) none of the above 2. Which of the following conditions is sufficient to ensure that the SD of a list of numbers is zero (circle all that apply): (a) All the numbers on the list are equal (b) The median of the list is zero (c) The average of the list is zero (d) None of the above 3. In a large statistics course, the scores for the final followed the normal curve closely. The average was 70 points, and three-fourths of the class scored between 60 and 80 points. The SD of the scores was . Circle one of the following options and explain: (2 points) (i) larger than 10 points (ii) smaller than 10 points (iii) can’t say with the information given 4. Which of the following statements is always true? (i) If A ⊂ B and B ⊂ A,then either A is empty or B is empty (ii) If A ∪B = A, then A ∩B = B Choose one of the following options: (a) Only (i) is always true. (b) Only (ii) is always true. (c) Both the statements are always true. (d) Both the statements are always false. 5. Consider tossing a fair coin 5 times, and then rolling a fair six-sided die 5 times. Assume that all the tosses and rolls are independent. The number of tails in all the tosses plus the number of times that the die lands showing three or four spots ( or ) (a) has a Binomial distribution with n = 5 and p = 1/2 (b) has a Binomial distribution with n = 5 and p = 1/6 (c) has a Binomial distribution with n = 10 and p = 1/2 (d) has a Binomial distribution with n = 10 and p = 1/6 (e) does not have a Binomial distribution. Name: Part II In this part you must show ALL your reasoning. If you write the answer without indicating how you arrived at this answer, you will not get any points. You may leave your answers as fractions etc, for example ( 1 2 )2 . 1. Below we have a table giving the distribution of weight (in kilograms) for adult women in the US (from sampling the NHANES survey of 2009-2010). The class intervals include the right endpoint, but not the left. Using this table, estimate the interquartile range for the weight of adult US women. (3 points) Weight (kg) Percent in interval 30-60 20 60-70 25 70-80 20 80-100 20 100-150 10 150-200 5 2. Consider a list with average 5 and standard deviation 2. If I take each number on the list, and subtract 3 from it, and then multiply the result by -3, what is the average and standard deviation of the new list? (3 points) Name: 3. Stat 20 usually has a large lecture and multiple sections. Consider two of the sections from last fall. Suppose the average of the midterm in the first section was 80 points and the average of the midterm in the second section was 65 points. In both classes, the standard deviation was about 10 points. Suppose the first section had 20 students, and the second section had 10 students. What was the average of the two sections combined? (3 points) 4. In each of the following, identify if we can use the binomial distribution or the geometric distribution to investigate the probabilities, and write down the parameters (n, p for the binomial, and p for the geometric). If neither distribution can be used, then write “neither”. You do NOT need to compute probabilities at this point. (2 points each) (a) A group of 5 students, 3 juniors and 2 seniors, take turns taking notes, picking a student at random without replacement each time. What is the probability that the third student picked is a junior, but not the first two. (b) A class of 20 statistics majors and 10 data science majors picks 5 students to present a topic. What is the chance that at least one of them is a statistics major? (c) A fair six-sided die is rolled 10 times. What is the probability that we never see a five ( )? Name: 5. Compute the probabilities for the previous problem. (3 points each) (a) What is the probability that the third student picked is a junior, but not the first two? (b) What is the chance that at least one of the 5 students picked is a statistics majors? (c) What is the probability that we never see a five in 10 rolls of a fair die? 6. Say we have a standard deck of 52 cards and deal 7 cards. What is the chance that we will get two triples in our hand of 7? (A triple is 3 cards of the same rank, such as 2♦, 2♥, 2♣.) (3 points) Name: 7. A password must consist of 6 characters that must be letters from the English alphabet or the digits 0, 1, 2, 3, . . ., 9 (no special characters). The letters may be lower case and upper case, and case matters, so abC is not the same as abc. The password may not begin with the number 0, but other than that there are no restrictions (characters may be repeated). How many such passwords are there? (2 points) 8. Let X be a discrete random variable such that X takes the values -1 with probability 1 5 , 0 with probability 3 10 and the value 2 with probability 1 2 . Find the expected value of X. (3 points) 9. Consider the following box of tickets: -2 -1 -1 0 4 (a) What is the standard deviation of the tickets in the box? (2 points) (b) If we draw 100 times from this box, what will be the standard error of the sum of these 100 draws? (2 points) Name: 10. Suppose that A, B and C are three events with probabilities 0.5, 0.7 and 0.2 respectively. (a) What is the largest that P (A ∩B) can be? (1 point) (b) What is the smallest that P (A ∩B) can be? (2 points) (c) What is the smallest that P (A ∩B ∩ C) can be? (1 point) (d) Now, suppose we assume that A and B are independent. What is P (A∩BC)? Find the probability that exactly one of the events A or B occurs. (3 points) Part III 1. In R, write the code to create a vector that consists of 100 2’s, and 200 0’s and 50 1’s. (1 points) 2. How would you use R to compute the probability that a fair coin will land heads fewer than 3 times in 10 tosses (2 points) 3. I would like to simulate the following: roll a fair die 3 times and sum the result. Is the following code correct? If not, please correct it. sum(sample(1:6, size=3)) (2 points) Name: