Py ⬍ 200 b. Py ⬎ 100 Lyman Ott Michael Longnecker

first person on the list and 1,000 to the last person. You need to next obtain a random sample of 50 numbers from the numbers 1 to 1,000. The names on the sampling frame corresponding to these 50 numbers will be the 50 persons selected for the poll. A Minitab program is shown here for purposes of illustration. Note that you would need to run this program 230 separate times to obtain a new random sample for each of the 230 precincts. Follow these steps: Click on Calc. Click on Random Data. Click on Integer. Type 5 in the Generate rows of data box. Type c1– c10 in the Store in Columns: box. Type 1 in the Minimum value: box. Type 1000 in the Maximum value: box. Click on OK. Click on File. Click on Print Worksheet.

a. Using either a random number table or a computer program, generate a second ran-

dom sample of 50 numbers from the numbers 1 to 1,000.

b. Give several reasons why you need to generate a different set of random numbers for

each of the precincts. Why not use the same set of 50 numbers for all 230 precincts? 4.12 Sampling Distributions 4.77 A random sample of 16 measurements is drawn from a population with a mean of 60 and a standard deviation of 5. Describe the sampling distribution of , the sample mean. Within what interval would you expect to lie approximately 95 of the time?

4.78 Refer to Exercise 4.77. Describe the sampling distribution for the sample sum . Is it un-

likely improbable that would be more than 70 units away from 960? Explain. Psy. 4.79 Psychomotor retardation scores for a large group of manic-depressive patients were approximately normal, with a mean of 930 and a standard deviation of 130. a. What fraction of the patients scored between 800 and 1,100? b. Less than 800? c. Greater than 1,200? Soc. 4.80 Federal resources have been tentatively approved for the construction of an outpatient clinic. In order to design a facility that will handle patient load requirements and stay within a lim- ited budget, the designers studied patient demand. From studying a similar facility in the area, they found that the distribution of the number of patients requiring hospitalization during a week could be approximated by a normal distribution with a mean of 125 and a standard deviation of 32.

a. Use the Empirical Rule to describe the distribution of y, the number of patients

requesting service in a week.

b. If the facility was built with a 160-patient capacity, what fraction of the weeks might

the clinic be unable to handle the demand? a y i a y i y y C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 1 340 701 684 393 313 312 834 596 321 739 2 783 877 724 498 315 282 175 611 725 571 3 862 625 971 30 766 256 40 158 444 546 4 974 402 768 593 980 536 483 244 51 201 5 232 742 1 861 335 129 409 724 340 218 4.81 Refer to Exercise 4.80. What size facility should be built so the probability of the patient load’s exceeding the clinic capacity is .10? .30? Soc. 4.82 Based on the 1990 census, the number of hours per day adults spend watching television is approximately normally distributed with a mean of 5 hours and a standard deviation of 1.3 hours.

a. What proportion of the population spends more than 7 hours per day watching

television?

b. In a 1998 study of television viewing, a random sample of 500 adults reported that the

average number of hours spent viewing television was greater than 5.5 hours per day. Do the results of this survey appear to be consistent with the 1990 census? Hint: If the census results are still correct, what is the probability that the average viewing time would exceed 5.5 hours? Env. 4.83 The level of a particular pollutant, nitrogen oxide, in the exhaust of a hypothetical model of car, the Polluter, when driven in city traffic has approximately a normal distribution with a mean level of 2.1 grams per mile gm and a standard deviation of 0.3 gm.

a. If the EPA mandates that a nitrogen oxide level of 2.7 gm cannot be exceeded, what

proportion of Polluters would be in violation of the mandate?

b. At most, 25 of Polluters exceed what nitrogen oxide level value that is, find the

75th percentile?

c. The company producing the Polluter must reduce the nitrogen oxide level so that

at most 5 of its cars exceed the EPA level of 2.7 gm. If the standard deviation remains 0.3 gm, to what value must the mean level be reduced so that at most 5 of Polluters would exceed 2.7 gm? 4.84 Refer to Exercise 4.83. A company has a fleet of 150 Polluters used by its sales staff. Describe the distribution of the total amount, in gm, of nitrogen oxide produced in the exhaust of this fleet. What are the mean and standard deviation of the total amount, in gm, of nitrogen oxide in the exhaust for the fleet? Hint: The total amount of nitrogen oxide can be represented as , where W i is the amount of nitrogen oxide in the exhaust of the ith car. Thus, the Central Limit Theorem for sums is applicable. Soc. 4.85 The baggage limit for an airplane is set at 100 pounds per passenger. Thus, for an airplane with 200 passenger seats there would be a limit of 20,000 pounds. The weight of the baggage of an individual passenger is a random variable with a mean of 95 pounds and a standard deviation of 35 pounds. If all 200 seats are sold for a particular flight, what is the probability that the total weight of the passengers’ baggage will exceed the 20,000-pound limit? Med. 4.86 A patient visits her doctor with concerns about her blood pressure. If the systolic blood pressure exceeds 150, the patient is considered to have high blood pressure and medication may be prescribed. The problem is that there is a considerable variation in a patient’s systolic blood pressure readings during a given day.

a. If a patient’s systolic readings during a given day have a normal distribution with

a mean of 160 mm mercury and a standard deviation of 20 mm, what is the probability that a single measurement will fail to detect that the patient has high blood pressure?

b. If five measurements are taken at various times during the day, what is the probability

that the average blood pressure reading will be less than 150 and hence fail to indicate that the patient has a high blood pressure problem?

c. How many measurements would be required so that the probability is at most 1 of

failing to detect that the patient has high blood pressure? 4.13 Normal Approximation to the Binomial Bus. 4.87 Critical key-entry errors in the data processing operation of a large district bank occur approximately .1 of the time. If a random sample of 10,000 entries is examined, determine the following:

a. The expected number of errors b. The probability of observing fewer than four errors

c. The probability of observing more than two errors

a 150 i⫽1 W i