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PubH 6450 Exam 1 (10/5/06) NAME:_________________________ Lab Session (Day and Time)_________________________ Directions: This is an open-book, open-notes exam; however, sharing of books, notes, homework papers, calculators, or verbal comments is not permitted. You may use a calculator of your choosing, but laptop computers are not permitted. For true/false questions, please clearly indicate your answer. For short answer problems, please show all relevant work necessary to arrive at a solution, as partial credit will be awarded. You will have from 1:30pm to 3:15pm to complete this exam, so do not spend too much time on any one problem. Good Luck!! 1. Below are separate stemplots for the ages of men reporting for a particular colorectal cancer screening procedure at two clinics, one in Minneapolis and the other in St. Paul: Minneapolis St. Paul 2 3 4 5 6 7 2 3 4 5 6 7 5 0 1 4 5 3 9 4 7 8 8 8 7 9 1 2 6 7 8 8 3 5 5 5 3 3 7 a. Find the median ages in both clinics. b. Determine whether each of the following statements is true or false: (i) The Minneapolis plot is bimodal. (ii) The lowest age observed in either clinic is 29. 1 c. Find the five-number summary (min, Q1, median, Q3, max) for the combined data from both clinics. d. Use the 1.5 IQR criterion to find the upper and lower outlier cutoffs for this combined dataset. List any outliers that are identified by these cutoffs. 2. A histogram of University of Minnesota faculty salaries is strongly skewed to the right, with many modest salaries but a few very large ones (corresponding mostly to surgeons and football coaches). Which measure of center, the mean or the median, is larger? 2 3. A company produces packets of soap powder labeled “Giant Size 32 oz.” The actual weight of soap powder in such a box has a normal distribution with a mean of 33 oz. and a standard deviation of 0.7 oz. a. The federal government considers a box of soap to be underweight if it weighs less than 32 oz. What proportion of boxes is underweight? b. What proportion of boxes weighs between 32.65 and 33.7 oz? c. The company loses money on the top (heaviest) 5% of boxes, which it wishes to label as overweight. How heavy does a box have to be in order to receive this label? 3 4. Suppose a bag contains candy of six different colors, for which the proportions of each color are as follows: Color Proportion Brown 0.3 Red 0.3 Yellow ? Green 0.1 Orange 0.1 Blue 0.1 a. Suppose we draw one candy from the bag at random. What is the probability of drawing a yellow candy? c. If you independently select two candies from the bag, with replacement, what is the probability that both are the same color? 5. The weight W of a medium-size tomato selected at random from a bin at the local supermarket is a random variable with mean = 10 oz. and standard deviation = 2 oz. a. Suppose we pick two tomatoes at random from the bin. Let the random variable S = the sum (in ounces) of the weights of the two tomatoes selected. What is the mean and standard deviation of the random variable S? b. Let the random variable Z = the weight of a randomly selected tomato in pounds, where 1 pound = 16 oz. What is the standard deviation of the random variable Z? 4 6. Suppose that out of every 10,000 people age 60 and over, 800 will have colorectal cancer. Of these, 600 will have a positive hemoccult test. Out of the remaining 9200 people without colorectal cancer, 300 will still test positive. How many of those who test positive actually have colorectal cancer? 7. An investigator samples 20 households having two parents and a newborn child, in which both parents have chronic bronchitis. She notices that in 3 of these households, the child also develops chronic bronchitis in the first year of life. The national incidence rate of bronchitis among all children is 5% in the first year of life. Let X be a random variable representing the number of children in the investigator’s sample who will develop bronchitis in the first year of life. a. What is the appropriate distribution for X assuming the children in the investigator’s study are typical of children nationwide? b. Using this distribution, what is the expected number of children in the sample who will develop bronchitis? 5 8. Compute the probability that the mean birth-weight from a sample of 10 infants will fall between 98 and 120 ounces if the mean birth-weight for the population is known to be 112 ounces with a standard deviation of 20.6 ounces. 9. A special study is conducted to test the hypothesis that people with glaucoma have higher blood pressure than average. In the study, the 200 people with glaucoma recruited have a mean systolic blood pressure of 140 mmHg. Assume the population distribution of systolic blood pressure is normally distributed and has a standard deviation of = 25 mmHg. a. Construct a 95% confidence interval for the true mean systolic blood pressure among people with glaucoma. b. What is the margin of error, m, for your confidence interval? What sample size would be needed to deliver a margin of error of just m/3 ? 6