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Elements of Statistics (Math 106) – Exam 2 Fall 2007 – Brad Hartlaub Name:________________________ Directions: Please answer all of the questions below and show your work. The point values for each question are indicated in parentheses. You may use your formula sheets, Minitab, and the course web page during the exam. Good luck! 1. In the game “Plinko” on The Price is Right, a contestant drops a chip, which randomly falls into a slot that is marked with a dollar designation. There are nine slots, two of which are marked $0, two are marked $100, two are marked $500, two are marked $1,000, and 1 is marked $5,000. a. If a chip is dropped, what’s the probability that it will land in a $1,000 slot? (5) b. What’s the probability that a chip will land in a slot valued at less than $1,000? (5) c. What is the expected value of earnings (a contestant does not have to pay anything to play games on this television show) for a contestant? (5) d. What is the variance of earnings for a contestant? (5) 2. A nationwide study of academic dishonesty among junior and senior college students was published in Psychology Today. The survey found that 70 percent of men and women confessed to cheating during high school. If a sample of 20 college students is surveyed, what is the probability that a. b. c. d. e. more than 12 confess to high school cheating? (3) at most 9 confess to high school cheating? (3) at least 10 confess to high school cheating? (3) exactly 12 confess to high school cheathing? (3) between 10 and 15 (inclusive) confess to high school cheating? (5) 3. The New England Journal of Medicine reported that the population mean and standard deviation for the cost per case for patients with medical back problems are $406 and $98, respectively. What’s the probability that the mean cost per case for a random sample of 45 patients with medical back problems is between $375 and $450. (15) 4. An industrial engineer wishes to estimate the mean life of a calculator battery (in hours) to within 2 hours of the true value. Past experience has shown that the standard deviation is 15.2 hours. How large a sample should she select to be within 2 hours with 90% confidence? (10) 5. The American Journal of Psychiatry published the results of a study to determine if there was a possible type of brain dysfunction associated with infantile autism. Each child in the study was given a behavioral test and scored on a scale from 0 to 116, where 0 = absence of symptoms and 116 = maximum severity. The scores for a random sample of 21 children in the study are listed below and in the file p:\data\math\stats\behavior.mtw. 27, 35, 65, 67, 47, 46, 63, 44, 34, 51, 17, 40, 41, 60, 24, 48, 29, 73, 60, 41, 27 a. What is the value of the point estimate you would use to estimate the mean score of all children with symptoms of infantile autism? (3) b. Calculate the estimated standard error. (3) c. Construct and interpret a 90% confidence interval for the mean score of all such children. (10) 6. In a 1997 class project, Jodi Sauer reported that hemoglobin (Hgb), a quaternary blood protein that transports oxygen in the blood to different tissues, varies in level based on sex and age. Medical journals report that normal adult Hgb levels in men are between 14 and 18 g/ml while women are reported to have levels between 12 and 16 g/ml. To examine the validity of these results she obtained a sample of the Hgb levels of 151 males and 174 females from Los Palos Medical Center. The data are in the file p:\data\math\stats\hemoglob.mtw. a. Do the data support the researchers claim that the mean adult Hgb level is higher for the population of males than females? (15) b. What assumptions are necessary for the inference procedure you used in part (a) to be valid? Are these assumptions satisfied? (10) 7. A study was carried out involving 12 patients with aldosterone-producing adenomas before and after adrenalectomy. All patients had high blood pressure, took antihypertensive drugs, and showed a tendency toward hypernatraemia. Profiles of these patients with primary aldosteronism were taken at admission and 2 months after the operation. Some of the variables reported in the study include systolic blood pressure (SBP in mm Hg), diastolic blood pressure (DBP in mm Hg), serum Na concentration (SNa in mmol/l), serum concentration (SK in mmol/l), plasma aldosterone concentration (PAC in pmol/l). These measurements can be found in the file p:\data\math\stats\adenoma.mtw. a. Are the SBP levels significantly lower after the operation? Explain. (15) b. Are the SK levels significantly higher after the operation? Explain. (15) c. Construct and interpret a 95% confidence interval for the median change in PAC. (10)