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Unformatted text preview: 386 5.9 5.11 5.12 Chapter 5: Sampling Distributions I Each entry in a table of random digits like Table B has probability 0.1 of
being a 0, and digits are independent of each other. (a) What is the probability that a group of ﬁve digits from the table will
contain at least one 0? (b) What is the mean number of Us in lines 40 digits long? The What [3 Probability? applet available at mm . whfr eeman . com/ ips
simulates tosses of a coin. You can choose the number of tosses n and the probability p of a head. That is, you can use the applet to simulate binomial _ random variables. The count of nonconforming switches in Example 5.12 (page 382) has
the binomial distribution with n 2 10 and p I 0.1. Set these values for
the number of tosses and probability of heads in the applet. The example calculates that the probability of getting a sample with exactly 1 bad switch I is 0.3874. Of course, when we inspect only a few lots, the proportion of
samples with exactly 1 bad switch will differ from this probability. Click
"Toss" and "Reset" repeatedly to simulate inspecting 20 lots. Record the
number of bad switches (the count of heads) in each of the 20 samples.
What proportion of the 20 lots had exactly 1 bad switch? Remember that
probability tells us only what happens in the long run. In 1998, Mark McGwire of the St. Louis Cardinals hit 70 home runs, a new '
major league record. Was this feat as surprising as most of us thought? In the three seasons before 1998, McGwire hit a home run in 11.6% of his times at bat. He went to bat 509 times in 1998. If he continues his past performance, McGwire’s home run count in 509 times at bat has
approximately the binomial distribution with n r 509 and p r 0.116. (a) What is the mean number of home runs McGwire will hit in 509 tinis
at bat? (b) What is the probability that he hits 70 or more home runs? (c) In 2001, Barry Bonds of the San Francisco Giants hit 73 home runs, I
breaking McGwire's record. This was surprising. In the three previous
seasons, Bonds hit a home run in 8.65% of his times at bat. He batted
476 times in 2001. Considering his home run count as a binomial
random variable with n z 476 andp r— 00865, what is the probabili '_
of 73 or more home runs?  You operate a restaurant. You read that a sample survey by the National Restaurant Association shows that 40% of adults are committed to eating '
nutritious food when eating away from home. To help plan your menu, decide to conduct a sample survey in your own area. You will use random
digit dialing to contact an SR8 of 200 households by telephone. ' (a) If the national result holds in your area, it is reasonable to use the
B (200, 0.4) distribution to describe the count X of respondents who
seek nutritious food when eating out. Explain why. (b) What is the mean number of nutritionconscious people in your samp .1
p = 0.4 is true? What is the probability that Xlies between 75 and 85.? . ._il:. _' ?(>< >310) :2 ?(£> (c) Hm :
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km) ﬁrm 'JWL ""‘J z._ astributio . Section 5.1 Exercises y [)1 of E _ (c) You ﬁnd 100 of your 200 respondents concerned about nutrition. Is
_' this reason to believe that the percent in your area is higher than the
ble will . national 40%? To answer this question, ﬁnd the probability that X is 100 or larger if p r 0.4 is true. If this probability is very small, that is
reason to think that p is actually greater than 0.4. 5.13 The Harvard College Alcohol Study ﬁnds that 67% of college students n/ ligh support efforts to “crack down on underage drinking." The study took
I a; t (la; ' a sample of almost 15,000 students, so the population proportion who
6 mom . support a crackdown is very close to p : 0.67.1 The administration of
382) has _ your college surveys an SRS of 100 students and ﬁnds that 62 support a
MES for crackdown on underage drinking.
example (a) What is the sample proportion who support a crackdown on underage
bad switch . ' drinking?
tion of (b) If in fact the proportion of all students on your campus who support a
y. Click " crackdown is the same as the national 67%, what is the probability that
ord the .' _ the proportion in an SRS of 100 students is as small or smaller than the
mples. _ I result of the administration’s sample?
1b“ that _' I (c) A writer in the student paper says that support for a crackdown is
‘ lower on your campus than nationally. Write a short letter to the editor
um] a new I. explaining why the survey does not support this conclusion.
Ought? .' 5.14 “How would you describe your own physical health at this time? Would you
6%.Of I say your physical health ice—excellent. good, only fair, or poor?" The Gallup
is hls Poll asked this question of 1005 randomly selected adults, of whom 29%
"t has said “excellent.”2 Suppose that in fact the proportion of the adult population
0116' . who say their health is excellent is p 2 0.29.
l 509 “mm , (a) What is the probability that the sample proportion p of an SR8 of size
n : 1000 who say their health is excellent lies between 26% and 32%?
(That is, within i3% of the truth about the population.)
me runs, ' (b) Repeat the probability calculation of (a) for SRSs of sizes n = 250
:e previous _ and it : 4000. What general conclusion can you draw from your
He batted .' calculations?
[giggibihw I' 5.15 “How would you describe your own personal weight situation right now—very overweight, somewhat overweight, about right, somewhat
‘. underweight, or very underweight?” When the Gallup Poll asked an SRS
National of 1005 adults this question, 51% answered "about right."3 Suppose that l to eating _' ' in fact 51% of the entire adult population think their weight is about 7 menu. you {j  right. 58 random I (a) Many opinion polls announce a r‘margin of error” of about i3%. What
 is the probability that an SR8 of size 1005 has a sample proportion p use the . ' that is within :3% (i003) of the population proportion p 2 0.5}? “NS Who !' I (b) Answrtr the same question if the population proportion is p = 0.06. (This is the proportion who told Gallup that they were "very
our sample it overweight") How does the probability change as p moves from near
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ﬁlmy. MW W7“ 1M ﬁz’m’ﬁﬂ SFEﬂ» 388 Chapter 5: Sampling Distributions 5.}6 A student organization is planning to ask a sample of 50 students if they
have noticed alcohol abuse brochures on campus. The sample percentage
who say “Yes” will be reported. The organization's statistical advisor says
that the standard deviation of this percentage will be about 7%. (a) \Vhat would the standard deviation be if the sample contained 100
students rather than 50? (b) How large a sample is required to reduce the standard deviation of the
percentage who say “Yes” from 7% to 3.5%? Explain to someone who knows no statistics the advantage of taking a larger sample in a survey
of opinion. 5.17 Children inherit their blood type from their parents, with probabilities that
reﬂect the parents’ genetic makeup. Children of Juan and Maria each have
probability 1/4 of having blood type A and inherit independently of each other. Juan and Maria plan to have 4 children; let X be the number who
have blood type A. (a) What are it and p in the binomial distribution ofX? (b) Find the probability of each possible value of X, and draw a probability
histogram for this distribution. (c) Find the mean number of children with type A blood, and mark the
location of the mean on your probability histogram. 5.18 A believer in the “random walk” theory of the behavior of stock prices
thinks that an index of stock prices has probability 0.65 of increasing in any
year. Moreover, the change in the index in any given year is not inﬂuenced
by whether it rose or fell in earlier years. Let X be the number of years
among the next 6 years in which the index rises. {a} What are n and p in the binomial distribution ofX? (‘0) Give the possible values thatX can take and the probability of each
value. Draw a probability histogram for the distribution of X. {c} Find the mean of the number X of years in which the stock price index
rises and mark the mean on your probability histogram. (d) Find the standard deviation of X. What is the probability that X takes a
value within one standard deviation of its mean? ‘ 5.19 In a test for ESP (extrasensor‘y perception), the experimenter looks at cards .I
that are hidden From the subject. Each card contains either a star, a circle, a:
wave, or a square. As the experimenter looks at each of 20 cards in turn, the
subject names the shape on the card. (a) If a subject simply guesses the shape on each card, what is the I‘ '
probability of a successth guess on a single card? Because the cards _' are independent, the count of successes in 20 cards has a binomial
distribution. (13) What is the probability that a subject correctly guesses at least 10 of 20 shapes? p, 6%. mu Lk 5’0"?) . “i919 Zn 1 #1 $ :11"; 3d. h)
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’2/ 1’) 4V) 404, 5.35 Chapter 5: Sampling Distributions {b} Accounting 20] is a large course. We can take the grades of an SRS of
50 students to be independent of each other. It}? is the average of these
50 grades, what are the mean and standard deviation of 2?? (c) What is the probability P(X 2 3) that a randomly chosen Accounting
201 student gets a B or better? What is the approximate probability
PO? 2 3) that the grade point average for 50 randomly chosen
Accounting 201 students is B or better? An automatic grinding machine in an auto parts plant prepares axles with
a target diameter ,u : 40.125 millimeters (mm). The machine has some
variability, so the standard deviation of the diameters is Cr : 0.002 mm. A
sample of 4 axles is inspected each hour for process control purposes, and
records are kept of the sample mean diameter. What will be the mean and
standard deviation of the numbers recorded? Sheila’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. A patient
is classiﬁed as having gestational diabetes if the glucose level is above 140
milligrams per deciliter (mg/dl) one hour after a sugary drink is ingested.
Sheila’s measured glucose level one hour after ingesting the sugary drink
varies according to the normal distribution with p. I 125 nig/dl and a = l0
mg/dl. (a) If a single glucose measurement is made, what is the probability that
Sheila is diagnosed as having gestational diabetes? (b) If measurements are made instead on 4 separate days and the mean
result is compared with the criterion 140 mg/dl, what is the probability
that Sheila is diagnosed as having gestational diabetes? A roulette wheel has 38 slots, oEwhich 18 are black, 18 are red, and 2 are
green When the wheel is spun, the ball is equally likely to come to rest in
any of the slots. Gamblers can place a number of different bets in roulette. One of the simplest wagers chooses red or black. A bet of $1 on red will pay '
off an additional dollar if the ball lands in a red slot. Otherwise, the player
loses his dollar. When gamblers bet on red or black, the two green slots
belong to the house. (a) A gambler’s winnings on a $1 bet are either $1 or —$1. Give the n
probabilities of these outcomes. Find the mean and standard deviation ;_
of the gamblers winnings. (2)} Explain briefly what the law of large numbers tells us about what will
happen if the gambler makes a large number of bets on red, (c) The central limit theorem tells us the approximate distribution of the :
gambler’s mean winnings in 50 bets. What is this distribution? Use
the 6895—997 rule to give the range in which the mean winnings
will fall 95% of the time. Multiply by 50 to get the middle 95% of the
distribution of the gambler’s winnings on nights when he places 50 I
bets. 5 with
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James 50 5.37 5.38 5.39 Section 5.2 Exercises 405 (d) What is the probability that the gambler will lose money if he makes 50
bets? (This is the probability that the mean is less than 0.) (e) The casino takes the other side of these bets. if 100,000 bets on red are
placed in a week at the casino, what is the distribution ofthe mean winnings
of gamblers on these bets? What range covers the middle 95% of mean
winnings in 100,000 bets? Multiply by 100,000 to get the range of gamblers
losses. (Gamblers losses are the casinos winnings. Part (c) shows that a
gambler gets excitement. Now we see that the casino has a business.) In Exercise 5.35, Sheila’s measured glucose level one hour after ingesting
the sugary drink varies according to the normal distribution with u : 125 mg/dl and 0' = 10 mg/dl. What is the level L such that there is
probability only 0.05 that the mean glucose level of 4 test results falls above
L for Sheila’s glucose level distribution? A company that owns and services a ﬂeet of cars for its sales force has
found that the service lifetime of disc brake pads varies from car to car
according to a normal distribution with mean [.L : 55, 000 miles and
standard deviation (7 : 4500 miles. The company installs a new brand of
brake pads on 8 cars. (a) If the new brand has the same lifetime distribution as the previous
type, what is the distribution of the sample mean lifetime for the 8
cars? (b) The average life of the pads on these 8 cars turns out to be 2? I 51, 800
miles. What is the probability that the sample mean lifetime is 51,800
miles or less if the lifetime distribution is unchanged? The company
takes this probability as evidence that the average lifetime of the new
brand of pads is less than 55,000 miles. The number of flaws per square yard in a type of carpet material varies
with mean 1.6 flaws per square yard and standard deviation 1.2 flaws per
square yard. This population distribution cannot be normal, because a
count takes only wholenumber values. An inspector studies 200 square
yards of the material, records the number of flaws found in each square
yard, and calculates 9?, the mean number of flaws per square yard inspected.
Use the central limit theorem to ﬁnd the approximate probability that the
mean number of flaws exceeds 2 per square yard. The number of accidents per week at a hazardous intersection varies with
mean 2.2 and standard deviation 1.4. This distribution takes only whole
number values, so it is certainly not normal. (a) Let x“ be the mean number of accidents per week at the intersection
during a year (52 weeks}. What is the approximate distribution off
according to the central limit theorem? {b} What is the approximate probability that 2? is less than 2? (c) What is the approximate probability that there are fewer than 100
accidents at the intersection in a year? (Hint: Restate this event in
terms of f.) 406 Chapter 5: Sampling Distributions I 5.41 The idea of insurance is that we all face risks that are unlikely but carry .
high cost. Think of a ﬁre destroying your home. So we form a group to
share the risk: we all pay a small amount, and the insurance policy pays
a large amount to those few of us whose homes burn down. An insurance
company looks at the records for millions of homeowners and sees that the
mean loss from ﬁre in a year is ,u : $250 per house and that the standard
deviation of the loss is or : $1000. (The distribution of losses is extremely
rightskewed: most people have $0 loss, but a few have large losses.) The
company plans to sell ﬁre insurance for $250 plus enough to cover its costs
and proﬁt. (a) Explain clearly why it would be unwise to sell only 12 policies. Then
explain why selling many thousands of such policies is a safe business (b) If the company sells 10,000 policies, what is the approximate
probability that the average loss in a year will be greater than $275? 5.42 The distribution of annual returns on common stocks is roughly symmetric,
but extreme observations are more frequent than in a normal distribution,
Because the distribution is not strongly nonnormal, the mean return over
even a moderate number of years is close to normal. In the long run,
annual real returns on common Stocks have varied with mean about 9%
and standard deviation about 28%. Andrew plans to retire in 45 years
and is considering investing in stocks. What is the probability (assuming
that the past pattern of variation continues) that the mean annual return
on common stocks over the next 45 years will exceed 15%? What is the :
probability that the mean return will be less than 5%? 5.43 Children in kindergarten are sometimes given the Ravin Progressive
Matrices Test (RPMT) to assess their readiness for learning. Experience
at Southwark Elementary School suggests that the RPMT scores for
its kindergarten pupils have mean 13.6 and standard deviation 3.1. The distribution is close to normal. Mr. Lavin has 22 children in his
kindergarten class this year. He suspects that their RPMT scores will be
unusually low because the test was interrupted by a ﬁre drill, To check
this suspicion, he wants to ﬁnd the level L such that there is probability
only 0.05 that the mean score of 22 children falls below L when the usual
Southwark distribution remains true. What is the value of L? 5.44 The design of an electronic circuit calls for a 100~ohrn resistor and a
250ohm resistor connected in series so that their resistances add. The
components used are not perfectly uniform, so that the actual resistances
vary independently according to normal distributions. The resistance of
lOOohm resistors has mean 100 ohms and standard deviation 2.5 ohmsy i while that of 250—ohm resistors has mean 250 ohms and standard deviation 
2.8 ohms. (a) What is the distribution of the total resistance of the two components
in series? (b) What is the probability that the total resistance lies between 345 and
355 ohms? 23/5 [email protected] unloaﬂu
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