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Economics 3640-001 Instructor: Sanghoon Lee Lecture 15: Ch.4 Random Variables and Probability Distributions 4.5 The Normal Distribution Eg. 4.17 Assume that the length of time, x, between charges of a cellular phone is normally distributed with a mean of 10 hours and a standard deviation of 1.5 hours. Find the probability that the cell phone will last between 8 and 12 hours between charges. Eg. 4.18 Suppose an automobile manufacturer introduces a new model that has an advertised mean in-city mileage of 27 miles per gallon. Although such advertisements seldom report any measure of variability, suppose you write the manufacturer for the details of the tests and you find that the standard deviation is 3 miles per gallon. This information leads you to formulate a probability model for the random variable x, the in-city mileage for this car model. You believe that the probability distribution of x can be approximated by a normal distribution with a mean of 27 and a standard deviation of 3. a. If you were to buy this model of automobile, what is the probability that you would purchase one that averages less than 20 miles per gallon for in-city driving? In other words, find P(x < 20). Eg. 4.21 Suppose the scores, x, on a college entrance examination are normally distributed with μ = 550 and σ = 100. A certain university will consider for admission only those applicants whose scores exceed the 90%. Find the minimum score an applicant must achieve in order to receive consideration for admission to the university. Ex. 4.70 Psychology students completed the Dental Anxiety Scale questionnaire. Scores on the scale range from 0 (no anxiety) to 20 (extreme anxiety). The mean score was 11 and the standard deviation was 3.5. Assume that the distribution of all scores on the dental Anxiety Scale is normal. a. Suppose you score a 16. Find the z value for this score. b. Find the probability that someone scores between a 10 and 15. c. Find the probability that someone scores above a 17. Ex. 4.72 Refer to the Chance (Winter 2001) study of students who paid a private tutor to help them improve their SAT scores, Exercise 2.95 (p.80). Assume this distribution is normal. SAT-Math SAT-Verbal Mean changes in score 19 7 σ of score changes 65 49 a. What is the probability that a student increases his/her score on SAT-Math test by at least 50 points? b. What on SAT-Verbal test by at least 50 points? Ex. 4.76 The physical fitness of a patient is often measured by the patient's maximum oxygen uptake (ml/kg). The mean maximum oxygen uptake for cardiac patients who regularly participate in sports or exercise programs was found to be 24.1 with a standard deviation of 6.30. Assume this distribution is normal. a. What is the probability that a cardiac patient who regularly participates in sports has a maximum oxygen uptake of at least 20 ml/kg? b. What is the probability that a cardiac patient who regularly exercises has a maximum oxygen uptake of 10.5 ml/kg or lower? 1 Ex. 4.78 A group of psychologists examined the effects of alcohol on the reactions of people to a threat. After obtaining a specified blood alcohol level, experimental subjects were placed in a room and threatened with electric shocks. Using sophisticated equipments to monitor the subjects' eye movements, the startle response (measured in milliseconds) was recorded for each subject. The μ and σ of the startle response were 37.9 and 12.4, respectively. Assume that the startle response x is approximately normally distributed. a. Find the probability that x is between 40 and 50 milliseconds. b. Find the probability that x is less than 30 millisecond. c. Give an interval for x, centered around 37.9 milliseconds, so that the probability that x falls in the interval is .95. d. Ten percent of the experimental subjects have startle responses above what value? Ex. 4.84 A machine used to regulate the amount of dye dispensed for mixing shades of paint can be set so that it discharges an average of μ milliliters (mL) of dye per can of paint. The amount of dye discharged is known to have a normal distribution with σ = .4mL. If more than 6 mL of dye are discharged when making a certain shade of blue paint, the shade is unacceptable. Determine the setting for μ so that only 1% of the cans of paint will be unacceptable. Ex. 4.160 In baseball, a "no-hitter" is a regulation 9-inning game in which the pitcher yields no hits to the opposing batters. Chance (Summer 1994) reported on a study of no-hitters in Major League Baseball (MLB). The initial analysis focused on the total number of hits yielded per game per team for all 9-inning MLB games played between 1989 and 1993. The distribution of hits/9-innings is approximately normal with mean 8.72 and standard deviation 1.10. a. What percentage of 9-inning MLB games results in fewer than 6 hits? b. Demonstrate, statistically, why a no-hitter is considered an extremely rare occurrence in MLB. 2