MT 272 Homework 5 HW: 5.2.4, 5.2.5, 5.2.6, 5.2.7, 5.2.10, 5.2.11, 5.2.12, 5.2.13 Due Mon Feb 22 5.2.4 The serum cholesterol levels of a population of 12-to 14-year-olds follow a normal distribution with mean 155 mg/dl and standard deviation 27 mg/dl (a) What percentage of the 12- to 14-year-olds have serum cholesterol values between 145 and 165 mg/dl? (b) Suppose we were to choose at random from the population a large number of groups of nine 12- to14-year-olds each. In what percentage of the groups would the group mean cholesterol value be between 145 and 165? (c) If represents the mean cholesterol value of a random sample of nine 12- to 14-year-ols form the population, what is Pr 145 165]? [ 5.2.5 Refer to exercise 5.2.4. Suppose we take a random sample of sixteen 12- to 14-year-olds from the population. (a) What is the probability that the mean cholesterol value for the group will be between 145 and 165? (b) What is the probability that the mean cholesterol value for the group will be between 140 and 170? 5.2.6 An important indicator of lung function is forced expiratory volume (FEV), which is the volume of air that a person can expire in one second. Dr. Hernandez plans to measure FEV in a random sample of n young women from a certain population, and to use the sample mean as an estimate of the population mean. Let E be the event that Hernandez's sample mean will be within 100 ml of the population mean. Assume that the population distribution is normal with mean 3,000 ml and standard deviation 400 ml. (a) Find Pr ] if = 15 [ (b) Find Pr ] if = 60 [ (c) How does Pr ] depend on the sample size? That is, as n increases, does [ increase, decrease,or stay the same? 5.2.7 Refer to Exercise 5.2.6. Assume that the population distribution of FEV is normal with 1 standard deviation 400 ml. (a) Find Pr ] if = 15 and the population mean is 2,800 ml. [ (b) Find Pr ] if = 15 if and the population mean is 2,600 ml. [ (c) How does Pr ] depend on the population mean? [ 5.2.10 In a certain population of fish, the lengths of the individual fish follow approximately a normal distribution with mean 54.0 mm and standard deviation 4.5 mm. In this situation 65.68% of the fish are between 51 and 60 mm long. Suppose a random sample of four fish is chosen from the population. Find the probability that (a) all four fish are between 51 and 60 mm long. (b) the mean length of the four fish is between 51 and60 mm. 5.2.11 In Exercise 5.2.10, the answer to part (b) was larger than the answer to part (a). Argue that this must necessarily be true, no matter what the population mean and standard deviation might be. [Hint: Can it happen that the event in part (a) occurs but the event in part (b) does not?] 5.2.13 A certain assay for serum alanine aminotransferase (ALT) is rather imprecise. The results of repeated assays of a single specimen follow a normal distribution with mean equal to the ALT concentration for that specimen and standard deviation equal to 4 U/l Suppose a hospital lab measures many specimens every day, and specimens with reported ALT values of 40 or more are flagged as \"unusually high.\" If a patient's true ALT concentration is 35 U/l, find the probability that his specimen will be flagged as \"unusually high\" (a) if the reported value is the result of a single assay. (b) if the reported value is the mean of three independent assays of the same specimen. 2