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a course in statistics with r
Mind On Statistics 5th Edition Jessica M Utts, Robert F Heckard - Solutions
The probability that Mary will win a game is .01, so the probability that she will not win is .99. If Mary wins, she will be given $100; if she loses, she must pay $5. If X 5 amount of money Mary wins (or loses), what is the expected value of X?
The Brann family wants to be financially prepared to have children. A financial advisor informs them that on the basis of families with similar characteristics, the probability distribution for the random variable X 5 number of children they might have is as follows:No. of Children 0 1 2 3
Suppose that in a gambling game, the probability of winning$4 is .3 and the probability of losing $2 is .7.a. Write a table that gives the probability distribution for X 5 amount won in a single play. Losing $2 can be expressed as “winning” 2$2.b. Calculate the expected value of X 5 amount won
Explain which would be of more interest in each of the following situations—the probability distribution function or the cumulative probability distribution function for X. If you think both would be of interest, explain why.a. A pharmaceutical company wants to show that its new drug is effective
Explain which would be of more interest in each of the following situations, the probability distribution function or the cumulative probability distribution function for X. If you think both would be of interest, explain why.a. A politician wants to know how her constituents feel about a proposed
Refer to Example8.10(pp. 271–272). Find the cumulative distribution function (cdf) for the sum of two fair dice.
A woman decides to have children until she has her first girl or until she has four children, whichever comes first. Let X 5 number of children she has. For simplicity, assume that the probability of a girl is .5 for each birth.a. Write the simple events in the sample space. Use B for boy and G for
Consider three tosses of a fair coin. Write the sample space, and then find the probability distribution function for each of the following random variables:a. X 5 number of tails.b. Y 5 the difference between the number of heads and the number of tails.c. Z 5 the sum of the number of heads and the
Let the random variable X 5 number of phone calls you will get in the next 24 hours. Suppose the possible values for X are 0, 1, 2, or 3, and their probabilities are .1, .1, .3, and .5, respectively. For instance, the probability that you will receive no calls is .1.a. Verify that the “Conditions
A kindergarten class has three left-handed children and seven right-handed children. Two children are selected without replacement for a shoe-tying lesson. Let X 5 the number who are left-handed.a. Write the simple events in the sample space. For instance, one simple event is RL, indicating that
Refer to Example8.10(pp. 271–272), in which the probability distribution is given for the sum of two fair dice. Use the distribution to find the probability that the sum is even.
The following table gives the probability distribution for X 5 number of wins in 3 plays of a game for which the chance of winning each game 5 .2, and plays are independent.X 5 Number of wins 0 1 2 3 Probability .512 .384 .096 .008a. Find P(X # 1), the probability of winning one or fewer games in
The following table gives the probability distribution for X 5 number of classes skipped yesterday by students at a college.No. of Classes skipped, X 0 1 2 3 4 Probability .73 .16 .06 .03 .02a. What is the probability that a randomly selected student skipped either two or three classes yesterday?b.
Suppose the probability distribution for X 5 number of jobs held during the past year for students at a school is as follows:No. of Jobs, X 0 1 2 3 4 Probability .14 .37 .29 .15 .05a. Find P(X # 2), the probability that a randomly selected student held two or fewer jobs during the past year.b. Find
For a fair coin tossed three times, the eight possible simple events are HHH, HHT, HTH, THH, HTT, THT, TTH, and TTT.Let X 5 number of heads. Find the probability distribution for X by writing a table showing each possible value of X along with the probability that the value occurs.
Answer Thought Question8.3 on page 272.
For a large population, probabilities for X 5 the number of meals eaten yesterday are:Meals, X 1 2 3 4 Probability .10 .32 .56 .02a. Write a table that gives the cumulative probability distribution for X.b. What is the value of P(X # 2), the probability that a randomly selected person ate two or
A professor gives a weekly quiz with varying numbers of questions and uses a randomization device to decide how many questions to include. Let the random variable X 5 the number of questions on an upcoming quiz. The probability distribution for X is given in the following table, but one probability
What is the missing value, represented by the question mark(?) in each of the following probability distribution functions?a.k 1 4 5 7 P(X 5 k) 1/6 1/6 1/6 ?b.k 100 200 300 400 500 P(X 5 k) .1 .2 .3 .3 ?
Answer Thought Question8.1 on page 267.Section 8.2 Skillbuilder Exercises
Explain the difference between how probabilities may be represented for discrete and continuous random variables.
Suppose you regularly play a lottery game. Give an example of a discrete random variable in this context.
Refer to Example8.1(p. 266), the scheduling of an outdoor event. Give an example of another continuous random variable (in addition to temperature) and another discrete random variable (in addition to number of planes flying overhead) that would influence the enjoyment of the event. Give the sample
For each characteristic, explain whether the random variable is continuous or discrete.a. Time to read a news article on the Internet.b. Number of losing instant lottery tickets purchased before buying a winning ticket.c. Body weights of 8-year-old children.d. Number of people with brown eyes in a
For each characteristic, explain whether the random variable is continuous or discrete.a. The number of left-handed individuals in a sample of 100 people.b. Time taken to complete an exam for students in a class.c. Vehicle speeds at a highway location.d. The number of accidents reported last year
A book is randomly chosen from a library shelf. For each of the following characteristics of the book, decide whether the characteristic is a continuous or a discrete random variable:a. Weight of the book (e.g.,2.3 pounds).b. Number of chapters in the book (e.g., 10 chapters).c. Width of the book
Decide whether each of the following characteristics of a television news broadcast is a continuous or discrete random variable:a. Number of commercials shown (e.g., five commercials).b. Length of the first commercial shown (e.g., 15 seconds).c. Whether there were any fatal car accidents reported(0
Suppose that two people were randomly selected from that population. Given that one of them had been divorced, what is the probability that the other one had also been divorced?
Suppose that two people were randomly selected from that population, without replacement. What is the probability that they had both been divorced?
Suppose that two people were randomly selected from that population, without replacement. What is the probability that one of them smoked but the other one did not (in either order)?
Given that the person smoked, what is the probability that he or she had been divorced?
Given that the person had been divorced, what is the probability that he or she smoked?
What is the approximate probability that the person had ever been divorced?
What is the approximate probability that the person smoked?
Draw a tree diagram illustrating this situation, where the first set of branches represents smoking status and the second set represents “ever divorced.”
According to the U.S. Census Bureau, “only 35 percent of the foreign-born people in the United States in 1997 were naturalized citizens, compared with 64 percent in 1970”(Sacramento Bee, October 15, 1999, p. A1).a. What is the probability that two randomly selected foreign-born people in the
inches or shorter and the other is62.5 inches or taller?d. What is the probability that two randomly selected women are both 65 inches or taller?
inches or shorter?c. What is the probability that of two randomly selected women, one is
inches or taller?b. What is the probability that two randomly selected women from this university are both
inches. Use this information to answer the following questions:a. What is the probability that a randomly selected woman from this university is
Recall that the Empirical Rule in Chapter 2 stated that for bell-shaped distributions, about 68% of the values fall within one standard deviation of the mean. The heights of women at a large university are approximately bell-shaped, with a mean of 65 inches and standard deviation of
or greater in the next 30 years is 46%.” The report noted that “[t]he final forecast is a sophisticated integration of scientific fact and expert opinion”(source: http://pubs.usgs.gov/fs/2008/3027/fs2008-3027.pdf).Explain whether the figure of 46% is based on personal probability, long-run
The 2007 Working Group on California Earthquake Probabilities reported that “the likelihood of [an earthquake in California] of magnitude
About one-third of all adults in the United States have type O blood. If three randomly selected adults donate blood, find the probability of each of the following events.a. All three are type O .b. None of them is type O .c. Two out of the three are type O .
Recall from Chapter 2 that the median of a dataset is the value with at least half of the observations at or above it and at least half of the observations at or below it. Suppose that four individuals are randomly selected with replacement from a large class and asked how many hours they studied
Refer to Exercises 7.105, 7.106, and 7.107. Given that a randomly chosen student receives an A grade, what is the probability that he or she attended class regularly?
Refer to Exercises7.105 and 7.106. Construct a hypothetical hundred thousand table for this situation.
Refer to Exercise 7.105. Draw a tree diagram, and use it to find the overall percent who receive As.
A professor has noticed that even though attendance is not a component of the grade for his class, students who attend regularly obtain better grades. In fact, 40% of those who attend regularly receive As in the class, while only 10% of those who do not attend regularly receive As. About 70%of
In Chapter 4, we learned that when there is no relationship between two categorical variables in a population, a “statistically significant” relationship will appear in 5% of the samples from that population, over the long run. Suppose that two researchers independently conduct studies to see
New spark plugs have just been installed in a small airplane with a four-cylinder engine. For each spark plug, the probability that it is defective and will fail during its first 20 minutes of flight is 1/10,000, independent of the other spark plugs.a. For any given spark plug, what is the
Refer to Exercise 7.101. A psychologist has noticed that“Teachers” and “Rationalists” get along particularly well with each other, and she thinks that they tend to marry each other. One of her colleagues disagrees and thinks that the types of spouses are independent of each other.a. If the
A psychological test identifies people as being one of eight types. For instance, Type 1 is “Rationalist” and applies to 15% of men and 8% of women. Type 2 is “Teacher” and applies to 12% of men and 14% of women. Each person fits one and only one type.a. What is the probability that a
Answer Thought Question 7.8 on page 252.
Answer Thought Question 7.7 on page 247.
A chess club has 5 players with unequal skill. Laurie is the best player. The probability that she will win depends on who her opponent is, with probabilities of winning of.6, .7, .8 and .9, respectively, when playing against members Carrie, David, Paul and Rose. Whether she wins or loses is
Ben drives to school for his class on Monday, Wednesday and Friday. He prefers to park in Parking Lot A because it is next to his classroom building, but he is not always able to find a spot there. If it’s sunny the probability that he finds a spot is .7 but if it’s raining more people drive,
Do you think that all coincidences can be explained by random events? Explain why or why not, using probability as the focus of your explanation. (There is no correct answer;your reasoning is what counts.)Chapter exercises
Suppose that you are seated next to a stranger on an airplane and you start discussing various topics such as where you were born (what state or country), what your favorite movie of all time is, your spouse’s occupation, and so on.For simplicity, assume that the probability that your details
Give an example of a coincidence that has occurred in your life. Using the material from this chapter, try to approximate the probability of exactly that event happening. Discuss whether the answer convinces you that something very odd happened to you.
Tomorrow morning when you first arise, pick a three-digit number (anything from 000 to 999). You can choose randomly or simply decide what number you want to use. As you go through the day, note whether you encounter that number. It could be in a book, on a license plate, in a newspaper or
A friend, quite upset, calls you because she had a dream that a building had been bombed and she was helping to search for survivors. The next day, a terrorist bombed an embassy building in another country. Your friend is convinced that her dream was a warning and that she should have told someone
A friend has three boys and would like to have a girl. She explains to you that the probability that her next baby will be a girl is very high because the law of averages says that half of her children should be girls and she already has three boys. Is she correct? Explain.
Give an example of a situation in which the gambler’s fallacy would not apply because the events are not independent.
Suppose that there are 30 people in your statistics class and you are divided into 15 teams of 2 students each.You happen to mention that your birthday was last week, upon which you discover that your teammate’s mother has the same birthday you have (month and day, not necessarily year). Assume
A rare disease occurs in about 1 out of 1000 people who are similar to you. A test for the disease has sensitivity of 95% and specificity of 90%.a. Create a hypothetical one hundred thousand table illustrating this situation, where the row categories are disease (yes, no) and the column categories
The University of California at Berkeley’s Wellness Encyclopedia(1991) contains the following statement in its discussion of HIV testing: “In a high-risk population, virtually all people who test positive will truly be infected, but among people at low risk the false positives will outnumber
Using material from Section 7.7, explain what is wrong with the following statement: “The probability that you will win the million dollar lottery is about the same as the probability that you will give birth to quintuplets.”General Section Exercises
Which of the following sequences resulting from tossing a fair coin five times is most likely: HHHHH, HTHHT, or HHHTT? Explain your answer.
The Pap smear is a screening test to detect cervical cancer.Estimate the sensitivity and specificity of the test if a study of 200 women with cervical cancer resulted in 160 testing positive, and in another 200 women without cervical cancer, 4 tested positive.
In tossing a fair coin 10 times, if the first nine tosses resulted in all tails, will the chance be greater than .5 that the tenth toss will turn up heads? Explain.
Janice has noticed that on her drive to work, there are several things that can slow her down. First, she hits a red light with probability .3. If she hits the red light, she also has to stop for the commuter train with probability .4, but if she doesn’t hit the red light, she has to stop with
Suppose that in a state lottery game players choose three digits, each from the set 0 to 9, as they do in many state lottery games. But for this game, what counts is the sum of the three chosen digits. The state selects a winning sum from the possibilities 0 (0 0 0) to 27 (9 9 9) by
Refer to Example 7.30(p. 246). Suppose that the probability of a correct guess each time is .40.a. Explain how you would simulate this situation.b. Carry out the simulation, and estimate the probability that a participant will be identified as gifted.
Refer to Example 7.30(p. 246). Explain how you would change the simulation procedure if the assumption was that everyone was randomly guessing, so that the probability of a correct guess was .20 each time.
Refer to the Minitab simulation results given on page 246 for Example 7.30. What is the estimated probability that a participant would guess four or more correctly?
Refer to Example 7.29(p. 245) and use the results given in Table 7.4 to estimate probabilities of the following outcomes:a. Prize 4 is received at least three times.b. Prize 4 is received at least three times given that the full collection of all four prizes was not received.c. Prize 4 is received
The observed risk of an accident per month at a busy intersection without any stoplights was 1%. The potential benefit of adding a stoplight was studied by using a computer simulation modeling the typical traffic flow for a month at that intersection. In 10,000 repetitions (each representing one
Five fair dice were tossed, and the sum of the resulting tosses was recorded. This process was repeated 10,000 times using a computer simulation. The number of times the sum of the five tosses equaled 27 was 45. What is the estimated probability that the sum of the five dice will be 27?
Suppose that only 2% of individuals undergoing a particular medical test have the disease that the test is intended to identify. If an individual has the disease, the probability that the test indicates the presence of the disease is .98. If an individual does not have the disease, the probability
Suppose that in a population of adults, the probability that a woman is taller than 68 inches is .15 and the probability that a man is taller than 68 inches is .75. The population is 50%women and 50% men. Suppose that a randomly selected individual from this population is taller than 68 inches.
In a test for ESP, a picture is randomly selected from four possible choices and hidden away. Participants are asked to describe the hidden picture, which is unknown to anyone in the environment. They are later shown the four possible choices and asked to identify the one they thought was the
Refer to the part of Example 7.21(p. 238) in which two students are drawn without replacement from a class of 30 students in which three are left-handed. Draw a tree diagram to illustrate that the probability of selecting two left-handed students is 1/145.
An airline has noticed that 40% of its customers who buy tickets don’t take advantage of advance-purchase fares and the remaining 60% do. The no-show rate for those who don’t have advance-purchase fares is 30%, while for those who do have them, it is only 5%.a. Create a tree diagram for this
Refer to Exercise 7.68. Suppose that a calculus student is randomly selected to accompany the math teachers to a conference. What is the probability that the student is a junior?
Suppose that a magnet high school includes grades 11 and 12, with half of the students in each grade. Half of the senior class and 30% of the junior class are taking calculus. Create a hypothetical table of 100 students for this situation. Use grade (junior, senior) as the row variable.
A standard poker deck of cards contains 52 cards of which four are aces. Suppose that two cards are drawn se quentially, so that one random circumstance is the result of the first card drawn and a second random circumstance is the result of a second card drawn. Find the probability that the first
Suppose that 20% of a population of registered voters are women who identify themselves as Democrats. The percent of women in the population of registered voters is 50%. A randomly selected voter in the population is a woman. What is the conditional probability this individual will identify herself
In an Italian breakfast café, a waitress has observed that 80% of her customers order coffee and 25% of her customers order both biscotti and coffee. Define:A a randomly selected customer orders coffee.B a randomly selected customer orders biscotti.a. Express the waitress’s observations as
In a computer store, 30% (.3) of the computers in stock are laptops and 70% (.7) are desktops. Five percent (.05) of the laptops are on sale, while 10% (.1) of the desktops are on sale. Use a tree diagram to determine the percentage of the computers in the store that are on sale.
Refer to Exercise 7.62.a. Create a tree diagram for this situation.b. Use the tree diagram in part (a) to determine what percentage of the class are seniors.
In a large general education class, 60% (.6) are science majors and 40% (.4) are liberal arts majors. Twenty percent(.2) of the science majors are seniors, while 30% (.3) of the liberal arts majors are seniors.a. If there are 100 students in the class, how many of them are science majors?b. If
A public library carries 50 magazines, each of which focuses on either news or sports. Thirty of the magazines focus on news and the remaining 20 focus on sports. Among the 30 news magazines, 20 include international news and 10 include national, state, or local news only. Among the 20 sports
A waitperson in a restaurant defines a large tip as one that exceeds 20% of the bill. She has learned that the probability that she will receive a large tip from a table of customers is.25 and is independent from one table of customers to the next. On a night when she serves five tables, what is
Suppose the probability of winning a particular casino game is .2 and is independent from one game to the next. If someone plays the game 4 times, what is the probability of winning at least once?
A robbery has been committed in an isolated town.Witnesses all agree that the criminal was driving a red pickup truck and had blond hair. Evidence at the scene indicates that the criminal also smoked cigarettes. Police determine that 1/50 of the vehicles in town are red pickup trucks, 30% of the
A raffle is held in a club in which 10 of the 40 members who bought raffle tickets are good friends with the club president.The club president draws two winners.a. If the two winners are drawn with replacement, what is the probability that a friend of the president wins each time?b. If the two
Harold and Maude plan to take a cruise together, but they live in separate cities. The cruise departs from Miami, and they each book a flight to arrive in Miami an hour before they need to be on the ship. Their travel planner explains that Harold’s flight has an 80% chance of making it on time
In Example 7.16, we found that the probability that a woman with two children either has two girls or two boys is .5002. What is the probability that she has one child of each sex?
Refer to Exercise 7.53. In this exercise, another method is used for finding the probability that at least one of two unrelated strangers shares your birth month.a. What is the probability that the first stranger shares your birth month?b. What is the probability that the second stranger shares
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