15.29 Glucose testing. Shelia’s doctor is concerned that she may suffer from
gestational diabetes (high blood glucose levels during pregnancy). There is
variation both in the actual glucose level and in the blood test that measures the
level. In a test to screen for gestational diabetes, a patient is classified as needing
further testing for gestational diabetes if the glucose level is above 130 milligrams
per deciliter (mg/dL) one hour after having a sugary drink. Shelia’s measured
glucose level one hour after the sugary drink varies according to the Normal
distribution with μ = 122 mg/dL and σ = 12 mg/dL.
(a)If a single glucose measurement is made, what is the probability that Shelia is
diagnosed as needing further testing for gestational diabetes?
(b)If measurements are made on four separate days and the mean result is
compared with the criterion 130 mg/dL, what is the probability that Shelia is
diagnosed as needing further testing for gestational diabetes?
15.32 Pollutants in auto exhausts. Light vehicles sold in the United States
must emit an average of no more than 0.05 grams per mile (g/mi) of nitrogen
oxides (NOX) after 50,000 or fewer miles of driving. NOX emissions after 50,000
or fewer miles of driving for one car model vary Normally with mean 0.03 g/mi
and standard deviation 0.01 g/mi.
(a)What is the probability that a single car of this model emits more than 0.05 g/mi
of NOX?
(b)A company has 25 cars of this model in its fleet. What is the probability that the
average NOX level of these cars is above 0.05 g/mi?
15.38 Sampling students, continued. To estimate the mean score μ of those
who took the Medical College Admission Test on your campus, you will obtain the
scores of an SRS of students. From published information, you know that the
scores are approximately Normal with standard deviation about 6.5. You want your
sample mean to estimate μ with an error of no more than 1 point in either
direction.
(a)What standard deviation must have so that 99.7% of all samples give an
within 1 point of μ? (Use the 68-95-99.7 rule.)
(b)How large an SRS do you need in order to reduce the standard deviation of to
the value you found in part (a)?
15.41 Playing the numbers: the house has a business. Unlike Joe (see
Exercise 15.40) the operators of the numbers racket can rely on the law of large
numbers. It is said that the New York City mobster Casper Holstein took as many
as 25,000 bets per day in the Prohibition era. That’s 150,000 bets in a week if he
takes Sunday off. Casper’s mean winnings per bet are $0.40 (he pays out 60 cents
of each dollar bet to people like Joe and keeps the other 40 cents.) His standard
deviation for single bets is about $18.96, the same as Joe’s.
(a)What are the mean and standard deviation of Casper’s average winnings on his
150,000 bets?
(b)According to the central limit theorem, what is the approximate probability that
Casper’s average winnings per bet are between $0.30 and $0.50? After only a
week, Casper can be pretty confident that his winnings will be quite close to $0.40
per bet.