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Introduction To The Practice Of Statistics 6th Edition David S. Moore, George P. McCabe, Bruce A. Craig - Solutions
CHALLENGE What happens when the correlation is 1? We know that variances add if the random variables involved are uncorrelated (ρ = 0), but not otherwise. The opposite extreme is perfect positive correlation (ρ = 1). Show by using the general addition rule for variances that in this case the
Transform the distribution of heights from centimeters to inches. A report of the National Center for Health Statistics says that the heights of 20-year-old men have mean 176.8 centimeters(cm) and standard deviation 7.2 cm. There are 2.54 centimeters in an inch. What are the mean and standard
Will you assume independence? In which of the following games of chance would you be willing to assume independence of X and Y in making a probability model? Explain your answer in each case.(a) In blackjack, you are dealt two cards and examine the total points X on the cards (face cards count 10
Sums of Normal random variables. Continue your work in the previous exercise. Dimensions of mechanical parts are often roughly Normal.According to the 68–95–99.7 rule, 95% of rods have lengths within ±d1 of 12 cm and 95% of bearings have lengths within ±d2 of 2 cm.(a) What are the values of
A mechnanical assembly. A mechanical assembly(Figure 4.15) consists of a rod with a bearing on each end. The three parts are manufactured independently, and all vary a bit from part to part.The length of the rod has mean 12 centimeters(cm) and standard deviation 0.004 millimeters(mm). The length of
CHALLENGE Toss a 4-sided die twice. Role-playing games like Dungeons & Dragons use many different types of dice. Suppose that a four-sided die has faces marked 1, 2, 3, 4. The intelligence of a character is determined by rolling this die twice and adding 1 to the sum of the spots. The faces are
Means and variances of sums. The rules for means and variances allow you to find the mean and variance of a sum of random variables without first finding the distribution of the sum, which is usually much harder to do.(a) A single toss of a balanced coin has either 0 or 1 head, each with
The effect of correlation. Find the mean and standard deviation of the total number of errors(nonword errors plus word errors) in an essay if the error counts have the distributions given in Exercise 4.74 and(a) the counts of nonword and word errors are independent.(b) students who make many
Standard deviations of numbers of rooms in housing units. Which of the two distributions of room counts appears more spread out in the probability histograms you made in Exercise 4.53(page 268)? Why? Find the standard deviation for both distributions. The standard deviation provides a numerical
Find the mean of the sum. Figure 4.12 (page 269)displays the density curve of the sum Y = X1 + X2 of two independent random numbers, each uniformly distributed between 0 and 1.(a) The mean of a continuous random variable is the balance point of its density curve. Use this fact to find the mean of Y
Means of the numbers of rooms in housing units.How do rented housing units differ from units occupied by their owners? Exercise 4.53 (page 268)gives the distributions of the number of rooms for owner-occupied units and renter-occupied units in San Jose, California. Find the mean number of rooms for
Mean of the distributions of errors. Typographical and spelling errors can be either “nonword errors”or “word errors.” A nonword error is not a real word, as when “the” is typed as “teh.” A word error is a real word, but not the right word, as when “lose” is typed as
Mean of the grade distribution. Example 4.22 gives the distribution of grades (A = 4, B = 3, and so on) in English 210 at North Carolina State University as Value of X 0 1 2 3 4 Probability 0.05 0.04 0.20 0.40 0.31 Find the average (that is, the mean) grade in this course.
Mean of the distribution for the number of aces. In Exercise 4.50 you examined the probability distribution for the number of aces when you are dealt two cards in the game of Texas hold ’em.Let X represent the number of aces in a randomly selected deal of two cards in this game. Here is the
Find the variance and the standard deviation. The random variable X has the following probability distribution:Value of X 0 2 Probability 0.5 0.5 Find the variance σ2 X and the standard deviation σX for this random variable.
Find μW. The random variable U has mean μU = 20 and the random variable V has mean μV = 20. If W = 0.5U + 0.5V, find μW.
Find μY. The random variable X has mean μX = 10. If Y = 15 + 8X, what is μY?
Use the Law of Large Numbers applet. The Law of Large Numbers applet animates a graph like Figure 4.14. Use it to better understand the law of large numbers by making a similar graph.
Find the mean of the probability distribution. You toss a fair coin.If the outcome is heads, you win $1.00; if the outcome is tails, you win nothing. Let X be the amount that you win in a single toss of a coin.Find the probability distribution of this random variable and its mean.
Normal approximation for a sample proportion.A sample survey contacted an SRS of 663 registered voters in Oregon shortly after an election and asked respondents whether they had voted. Voter records show that 56% of registered voters had actually voted. We will see in the next chapter that in this
How many close friends? How many close friends do you have? Suppose that the number of close friends adults claim to have varies from person to person with mean μ = 9 and standard deviationσ = 2.5. An opinion poll asks this question of an SRS of 1100 adults. We will see in the next chapter that
The sum of two uniform random numbers.Generate two random numbers between 0 and 1 and take Y to be their sum. Then Y is a continuous random variable that can take any value between 0 and 2. The density curve of Y is the triangle shown in Figure 4.12.(a) Verify by geometry that the area under this
Uniform numbers between 0 and 2. Many random number generators allow users to specify the range of the random numbers to be produced. Suppose that you specify that the range is to be all numbers between 0 and 2. Call the random number generated Y. Then the density curve of the random variable Y has
Find the probabilities. Let the random variable X be a random number with the uniform density curve in Figure 4.9. Find the following probabilities:(a) P(X ≥ 0.35)(b) P(X = 0.35)(c) P(0.35 < X < 1.35)(d) P(0.15 ≤ X ≤ 0.25 or 0.8 ≤ X ≤ 0.9)(e) The probability that X is not in the interval
Uniform random numbers. Let X be a random number between 0 and 1 produced by the idealized uniform random number generator described in Example 4.25 and Figure 4.9. Find the following probabilities:(a) P(X < 0.4)(b) P(X ≤ 0.4)(c) What important fact about continuous random variables does
Select the members of a student advisory board.Weary of the low turnout in student elections, a college administration decides to choose an SRS of three students to form an advisory board that represents student opinion. Suppose that 40% of all students oppose the use of student fees to fund
Foreign-born residents of California. The Census Bureau reports that 27% of California residents are foreign-born. Suppose that you choose three Californians at random, so that each has probability 0.27 of being foreign-born and the three are independent of each other. Let the random variable W be
CHALLENGE Dungeons & Dragons. Role-playing games like Dungeons & Dragons use many different types of dice, usually having either 4, 6, 8, 10, 12, or 20 sides. Roll a balanced 8-sided die and a balanced 6-sided die and add the spots on the up-faces. Call the sum X. What is the probability
CHALLENGE Nonstandard dice. Nonstandard dice can produce interesting distributions of outcomes. You have two balanced, six-sided dice.One is a standard die, with faces having 1, 2, 3, 4, 5, and 6 spots. The other die has three faces with 0 spots and three faces with 6 spots. Find the probability
Tossing two dice. Some games of chance rely on tossing two dice. Each die has six faces, marked with 1, 2, . . . , 6 spots called pips. The dice used in casinos are carefully balanced so that each face is equally likely to come up. When two dice are tossed, each of the 36 possible pairs of faces is
Find the probabilities. Let the random variable X be the number of rooms in a randomly chosen owner-occupied housing unit in San Jose, California.Exercise 4.53 gives the distribution of X.(a) Express “the unit has 6 or more rooms” in terms of X. What is the probability of this event?(b) Express
Households and families in government data.In government data, a household consists of all occupants of a dwelling unit, while a family consists of two or more persons who live together and are related by blood or marriage. So all families form households, but some households are not families.Here
Owner-occupied and rented housing units. How do rented housing units differ from units occupied by their owners? Here are the distributions of the number of rooms for owner-occupied units and renter-occupied units in San Jose, California:15 Rooms 1 2 3 4 5 Owned 0.003 0.002 0.023 0.104 0.210 Rented
Length of human pregnancies. The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. Call the length of a randomly chosen pregnancy Y.(a) Make a sketch of the density curve for this
Spell-checking software. Spell-checking software catches “nonword errors,” which result in a string of letters that is not a word, as when “the” is typed as “teh.” When undergraduates are asked to write a 250-word essay (without spell-checking), the number X of nonword errors has the
Texas hold ’em. The game of Texas hold ’em starts with each player receiving two cards. Here is the probability distribution for the number of aces in two-card hands:Number of aces 0 1 2 Probability 0.559 0.382 0.059(a) Verify that this assignment of probabilities satisfies the requirement that
Discrete or continuous. In each of the following situations decide if the random variable is discrete or continuous and give a reason for your answer.(a) YourWeb page has five different links and a user can click on one of the links or can leave the page.You record the length of time that a user
Find the probability. For the uniform distribution described in Example 4.25, find the probability that X is between 0.1 and 0.4.
Two tosses of a fair coin. Find the probability distribution for the number of heads that appear in two tosses of a fair coin.
Will the course satisfy the requirement? Refer to Example 4.22.Suppose that a grade of D or F in English 210 will not count as satisfying a requirement for a major in linguistics. What is the probability that a randomly selected student will not satisfy this requirement?
Three children. Jasmine has alleles A and O. Joshua has alleles B and O.(a) What is the probability that a child of these parents has blood type O?(b) If Jasmine and Joshua have three children, what is the probability that all three have blood type O?What is the probability that the first child has
Two children. Jennifer has alleles A and O. Jos´e has alleles A and B. They have two children. What is the probability that both children have blood type A?What is the probability that both children have the same blood type?
Parents with alleles B and O. Nancy and David both have alleles B and O.(a) What blood types can their children have?(b) What is the probability that their next child has each of these blood types?
Blood types of children. Hannah and Jacob both have alleles A and B.(a) What blood types can their children have?(b) What is the probability that their next child has each of these blood types?
CHALLENGE Independence of complements. Show that if events A and B obey the multiplication rule, P(A and B) = P(A)P(B), then A and the complement Bc of B also obey the multiplication rule, P(A and Bc) = P(A)P(Bc). That is, if events A and B are independent, then A and Bc are also independent.(Hint:
CHALLENGE Axioms of probability. Show that any assignment of probabilities to events that obeys Rules 2 and 3 on page 246 automatically obeys the complement rule (Rule 4). This implies that a mathematical treatment of probability can start from just Rules 1, 2, and 3. These rules are sometimes
Random walks and stock prices. The “random walk” theory of securities prices holds that price movements in disjoint time periods are independent of each other. Suppose that we record only whether the price is up or down each year, and that the probability that our portfolio rises in price in
Colored dice. Here’s more evidence that our intuition about chance behavior is not very accurate.A six-sided die has four green and two red faces, all equally probable. Psychologists asked students to say which of these color sequences is most likely to come up at the beginning of a long set of
Is this calculation correct? Government data show that 6% of the American population are at least 75 years of age and that about 51% are women.Explain why it is wrong to conclude that because(0.06)(0.51) = 0.0306 about 3% of the population are women aged 75 or over.
Random digit dialing. Most sample surveys use random digit dialing equipment to call residential telephone numbers at random. The telephonepolling firm Zogby International reports that the probability that a call reaches a live person is 0.2.9 Calls are independent.(a) A polling firm places 5
Disappearing Internet sites. Internet sites often vanish or move, so that references to them can’t be followed. In fact, 13% of Internet sites referenced in papers in major scientific journals are lost within two years after publication.8 If a paper contains seven Internet references, what is the
Universal blood donors. People with type Onegative blood are universal donors. That is, any patient can receive a transfusion of O-negative blood. Only 7% of the American population have O-negative blood. If 10 people appear at random to give blood, what is the probability that at least 1 of them
PINs. The personal identification numbers (PINs)for automatic teller machines usually consist of four digits. You notice that most of your PINs have at least one 0, and you wonder if the issuers use lots of 0s to make the numbers easy to remember.Suppose that PINs are assigned at random, so that
Winning the lottery. A state lottery’s Pick 3 game asks players to choose a three-digit number, 000 to 999. The state chooses the winning three-digit number at random, so that each number has probability 1/1000. You win if the winning number contains the digits in your number, in any order.(a)
Roulette. A roulette wheel has 38 slots, numbered 0, 00, and 1 to 36. The slots 0 and 00 are colored green, 18 of the others are red, and 18 are black. The dealer spins the wheel and at the same time rolls a small ball along the wheel in the opposite direction.The wheel is carefully balanced so
Are the events independent? Exercise 4.28 assigns probabilities for the ethnic background of a randomly chosen resident of the United States.Let A be the event that the person chosen is Hispanic, and let B be the event that he or she is white. Are events A and B independent? How do you know?
Rh blood types.Human blood is typed as O, A, B, or AB and also as Rh-positive or Rh-negative. ABO type and Rh-factor type are independent because they are governed by different genes. In the American population, 84% of people are Rh-positive. Use the information about ABO type in Exercise 4.21 to
Race in the census. The 2000 census allowed each person to choose from a long list of races.That is, in the eyes of the Census Bureau, you belong to whatever race you say you belong to.“Hispanic/Latino” is a separate category; Hispanics may be of any race. If we choose a resident of the United
CHALLENGE Loaded dice. There are many ways to produce crooked dice. To load a die so that 6 comes up too often and 1 (which is opposite 6)comes up too seldom, add a bit of lead to the filling of the spot on the 1 face. Because the spot is solid plastic, this works even with transparent dice. If a
Spam topics. A majority of email messages are now“spam.” Choose a spam email message at random.Here is the distribution of topics:7 Topic Adult Financial Health Leisure Products Scams Probability 0.145 0.162 0.073 0.078 0.210 0.142(a) What is the probability that a spam email does not concern
Education levels of young adults. Choose a young adult (age 25 to 34 years) at random. The probability is 0.12 that the person chosen did not complete high school, 0.31 that the person has a high school diploma but no further education, and 0.29 that the person has at least a bachelor’s
French and English in Canada. Canada has two official languages, English and French. Choose a Canadian at random and ask, “What is your mother tongue?” Here is the distribution of responses, combining many separate languages from the broad Asian/Pacific region:6 Language English French
Are the probabilities legitimate? In each of the following situations, state whether or not the given assignment of probabilities to individual outcomes is legitimate, that is, satisfies the rules of probability.If not, give specific reasons for your answer.record gender and enrollment status:
Blood types in China. The distribution of blood types in China differs from the U.S. distribution given in the previous exercise:Blood type A B AB O China probability 0.27 0.26 0.12 0.35 Choose an American and a Chinese at random, independently of each other. What is the probability that both have
Distribution of blood types. All human blood can be “ABO-typed” as one of O, A, B, or AB, but the distribution of the types varies a bit among groups of people. Here is the distribution of blood types for a randomly chosen person in the United States:Blood type A B AB O U.S. probability 0.40
Record the length of time spent on the page.Refer to the previous exercise. You also decide to measure the length of time a visitor spends on your page. Give the sample space for this measure.
Evaluating Web page designs. You are a Web page designer and you set up a page with five different links. A user of the page can click on one of the links or he or she can leave that page. Describe the sample space for the outcome of a visitor to your Web page.
The probability of a second ace. A deck of 52 cards contains 4 aces, so the probability that a card drawn from this deck is an ace is 4/52. If we know that the first card drawn is an ace, what is the probability that the second card drawn is also an ace? Using the idea of independence, explain why
Two tails in two tosses. What is the probability of obtaining two tails on two tosses of a fair coin?
Possible outcomes for rolling a die. A die has six sides with 1 to 6“spots” on the sides. Give the probability distribution for the six possible outcomes that can result when a perfect die is rolled.
Use the addition rule. Use the addition rule with the probabilities for the events A and B from Example 4.13 to find the probability that a first digit is either 1 or 6 or greater.
Benford’s law. Using the probabilities for Benford’s law, find the probability that a first digit is anything other than 1.
Not on Wednesday. Find the probability that a phone-related accident occurred on a day other than a Wednesday.
Phone-related accidents on Monday or Friday. Find the probability that a phone-related accident occurred on a Monday or a Friday.
Sample space for heights. You record the height in inches of a randomly selected student. What is the sample space?
When do you study? A student is asked on which day of the week he or she spends the most time studying. What is the sample space?
APPLET A question about dice. Here is a question that a French gambler asked the mathematicians Fermat and Pascal at the very beginning of probability theory: what is the probability of getting at least one six in rolling four dice? The Law of Large Numbers applet allows you to roll several dice
APPLE T Use the Probability applet. The idea of probability is that the proportion of heads in many tosses of a balanced coin eventually gets close to 0.5. But does the actual count of heads get close to one-half the number of tosses? Let’s find out. Set the “Probability of heads” in the
APPLE T Simulate free throws. The basketball player Shaquille O’Neal makes about half of his free throws over an entire season. Use Table B or the Probability applet to simulate 100 free throws shot independently by a player who has probability 0.5 of making each shot.(a) What percent of the 100
Side effects of eyedrops. You go to the doctor and she prescribes a medicine for an eye infection that you have. Suppose that the probability of a serious side effect from the medicine is 0.00001. Explain in simple terms what this number means.
The color of candy. It is reasonable to think that packages of M&M’s Milk Chocolate Candies are filled at the factory with candies chosen at random from the very large number produced. So a package of M&M’s contains a number of repetitions of a random phenomenon: choosing a candy at random and
Winning at craps. The game of craps starts with a“come-out” roll where the shooter rolls a pair of dice.If the total is 7 or 11, the shooter wins immediately(there are ways that the shooter can win on later rolls if other numbers are rolled on the come-out roll). Roll a pair of dice 25 times
Wait 5 seconds between each observation. Refer to the previous exercise. Explain why you would not want to wait only 5 seconds between each time you turn the radio station on.
Is music playing on the radio? Turn on your favorite music radio station 10 times at least 10 minutes apart. Each time record whether or not music is playing. Calculate the number of times music is playing divided by 10. This number is an estimate of the probability that music is playing when you
Use Table B. We can use the random digits in Table B in the back of the text to simulate tossing a fair coin. Start at line 109 and read the numbers from left to right. If the number is 0, 1, 2, 3, or 4, you will say that the coin toss resulted in a head; if the number is a 5, 6, 7, 8, or 9, the
A cheating experiment. A psychologist conducts the following experiment: she measures the attitude of subjects toward cheating, then has them play a game rigged so that winning without cheating is impossible. The computer that organizes the game also records—unknown to the subjects—whether or
A theft experiment. Students sign up to be subjects in a psychology experiment. When they arrive, they are told that interviews are running late and are taken to a waiting room.The experimenters then stage a theft of a valuable object left in the waiting room. Some subjects are alone with the
Should we ask for the consent of the parents?The Centers for Disease Control and Prevention, in a survey of teenagers, asked the subjects if they were sexually active. Those who said “Yes” were then asked, How old were you when you had sexual intercourse for the first time?Should consent of
Use of data produced by the government.Data produced by the government are often available free or at low cost to private users. For example, satellite weather data produced by the U.S. National Weather Service are available free to TV stations for their weather reports and to anyone on the Web.
Your institutional review board. Your college or university has an institutional review board that screens all studies that use human subjects. Get a copy of the document that describes this board(you can probably find it online).(a) According to this document, what are the duties of the board?(b)
CHALL ENGE Two ways to ask sensitive questions.Sample survey questions are usually read from a computer screen. In a Computer Aided Personal Interview (CAPI), the interviewer reads the questions and enters the responses.In a Computer Aided Self Interview (CASI), the interviewer stands aside and the
CHALL ENGE Informed consent. The requirement that human subjects give their informed consent to participate in an experiment can greatly reduce the number of available subjects. For example, a study of new teaching methods asks the consent of parents for their children to be randomly assigned to be
Would the results be different for men and women? The drug that is the subject of the experiment in Exercise 3.124 may behave differently in men and women. How would you modify your experimental design to take this into account?
Discolored french fries. Few people want to eat discolored french fries. Potatoes are kept refrigerated before being cut for french fries to prevent spoiling and preserve flavor. But immediate processing of cold potatoes causes discoloring due to complex chemical reactions.The potatoes must
Compare two doses of a drug. A drug manufacturer is studying how a new drug behaves in patients. Investigators compare two doses: 5 milligrams (mg) and 10 mg. The drug can be administered by injection, by a skin patch, or by intravenous drip. Concentration in the blood after 30 minutes (the
CHALLENGE Design a survey. You want to investigate the attitudes of students at your school about the faculty’s commitment to teaching.The student government will pay the costs of contacting about 500 students.(a) Specify the exact population for your study;for example, will you include part-time
CHALLENGE Design an experiment. The previous two exercises illustrate the use of statistically designed experiments to answer questions that arise in everyday life. Select a question of interest to you that an experiment might answer and carefully discuss the design of an appropriate experiment.
Bicycle gears. How does the time it takes a bicycle rider to travel 100 meters depend on which gear is used and how steep the course is? It may be, for example, that higher gears are faster on the level but lower gears are faster on steep inclines.Discuss the design of a two-factor experiment to
Compare the burgers. Do consumers prefer the taste of a cheeseburger from McDonald’s or from Wendy’s in a blind test in which neither burger is identified? Describe briefly the design of a matched pairs experiment to investigate this question. How will you use randomization?
Choose the type of study. Give an example of a question about college students, their behavior, or their opinions that would best be answered by(a) a sample survey.(b) an observational study that is not a sample survey.(c) an experiment.
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