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mathematics
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Statistics The Art And Science Of Learning From Data 3rd Edition Alan Agresti, Christine A. Franklin - Solutions
a. What is a sample space? Give an example of a sample space for a scenario involving (i) a designed experiment and (ii) an observational study. b. What are disjoint events? Give an example of two events that are disjoint. c. What is a conditional probability? Give an example of two events in your
Short-term aberrations do not affect the long run. To illustrate, suppose that you flip a coin 10 times and you get 10 heads. Find the cumulative proportion of heads, including these first 10 flips, if (a) In the 100 flips after those 10 you get 50 heads (so there are now 60 heads in 110
After the Columbia space shuttle disaster, a former NASA official who faulted the way the agency dealt with safety risk warned (in an AP story, March 7, 2003) that NASA workers believed, “If I’ve flown 20 times, the risk is less than if I’ve flown just once.” a. Explain why it would be
Mrs. Test (see www.mrstest.com) sells diagnostic tests for various conditions. Their Web site gives only imprecise information about the accuracy of the tests. The test for pregnancy is said to be “over 99% accurate.” Describe at least four different probabilities to which this could refer.
a. State in a table the probability distribution for the outcome of rolling a balanced die. (This is called the uniform distribution on the integers 1, 2,., 6.)b. Two balanced dice are rolled. Show that the probability distribution for X = total on the two dice is as shown in the figure. (First
From past experience, a wheat farmer living in Manitoba, Canada finds that his annual profit (in Canadian dollars) is $80,000 if the summer weather is typical, $50,000 if the weather is unusually dry, and $20,000 if there is a severe storm that destroys much of his crop. Weather bureau records
Some companies, such as Demand Tec, have developed software to help retail chains set prices that optimize their profits. An Associated Press story (April 28, 2007) about this software described a case in which a retail chain sold three similar power drills: one for $90, a better one for $120, and
A random number generator is used to generate a real number between 0 and 1, equally likely to fall anywhere in this interval of values. (For instance, 0.3794259832 c is a possible outcome.) a. Sketch a curve of the probability distribution of this random variable, which is the continuous version
A social scientist uses the General Social Survey to study how much time per day people spend watching TV. The variable denoted by TVHOURS at the GSS Web site measures this using the values 0, 1, 2,c, 24. a. Explain how, in theory, TV watching is a continuous random variable. b. An article about
For a normal distribution, use Table A, software, or a calculator to find the probability that an observation is a. At least 1 standard deviation above the mean. b. At least 1 standard deviation below the mean. c. In each case, sketch a curve and show the tail probability.
For the normal distribution shown below, use Table A, software, or a calculator to find the probability that an observation falls in the shaded region.
Verify the empirical rule by using Table A, software, or a calculator to show that for a normal distribution, the probability (rounded to two decimal places) within a. 1 standard deviation of the mean equals 0.68. b. 2 standard deviations of the mean equal 0.95. c. 3 standard deviations of the mean
For a normal distribution, use Table A to verify that the probability (rounded to two decimal places) within a. 1.64 standard deviations of the mean equals 0.90. b. 2.58 standard deviations of the mean equal 0.99. c. Find the probability that falls within 0.67 standard deviations of the mean. d.
For a normal distribution, a. Find the z -score for which a total probability of 0.02 falls more than z standard deviations (in either direction) from the mean, that is, below μ - zσ or above μ + zσ. b. For this z, explain why the probability more than z standard deviations above the mean
For a normal distribution,a. Show that a total probability of 0.01 falls more than z = 2.58 standard deviations from the mean.b. Find the z -score for which the two-tail probability that falls more than that many standard deviations from the mean in either direction equals (a) 0.05, (b) 0.10.
In Monopoly, dice are used to determine which player gets to move first. Suppose there are two players in the game. Each player rolls a die and the player with the higher number gets to move first. If the numbers are the same, the players roll again. a. Using the sample space {(1, 1), (1, 2), (1,
a. For the normal distribution shown below, find the z -score.b. Find the value of z (rounding to two decimal places) for right-tail probabilities of(i) 0.05 and(ii) 0.005.
Find the z -score such that the interval within z standard deviations of the mean (between μ - zσ and μ + zσ) for a normal distribution contains a. 50% of the probability. b. 90% of the probability. c. Sketch the two cases on a single graph.
The normal distribution for women’s height in North America has μ = 65 inches, σ = 3.5 inches. Most major airlines have height requirements for flight attendants (www.cabincrewjob.com). Although exceptions are made, the minimum height requirement is 62 inches. What proportion of adult females
A World Health Organization study (the MONICA project) of health in various countries reported that in Canada, systolic blood pressure readings have a mean of 121 and a standard deviation of 16. A reading above 140 is considered to be high blood pressure.a. What is the z -score for a blood pressure
According to Current Population Reports, self-employed individuals in the United States work an average of 44.6 hours per week, with a standard deviation of 14.5. If this variable is approximately normally distributed, find the proportion of the self-employed who work more than 40 hours per week.
An energy study in Gainesville, Florida, found that in March 2006, household use of electricity had a mean of 673 and a standard deviation of 556 kilowatt-hours. a. Suppose the distribution of energy use was normal. Using a table, calculator, or software that can give normal probabilities, find the
According to www.mynewplace.com, in January 2011 the average monthly rental rate for one bedroom apartments in Ann Arbor, Michigan, was $796. (Background information from www.mynewplace.com/city/ann_arbor_apartments_for_rent_michigan.) Suppose rental rates across all one-bedroom apartments in Ann
The Mental Development Index (MDI) of the Bayley Scales of Infant Development is a standardized measure used in observing infants over time. It is approximately normal with a mean of 100 and a standard deviation of 16. a. What proportion of children has an MDI of (i) At least 120? (ii) At least
Refer to the previous exercise. a. Find the z -score corresponding to the lower quartile (Q1) of a normal distribution. b. Find and interpret the lower quartile and upper quartile of the MDI. c. Find the inter-quartile range (IQR) of MDI scores. d. Section 2.5 defined an observation to be a
In 2008, the murder rates (per 100,000 residents) for the 50 states and the District of Columbia (D.C.) had a mean of 5.39 and a standard deviation of 4.434 (Statistical Abstract of the United States).8 a. D.C. had a murder rate of 31.4. Find its z -score. If the distribution were roughly normal,
The table shows the probability distribution of the number of bases for a randomly selected time at bat for a San Francisco Giants player in 2010 (excluding times when the player got on base because of a walk or being hit by a pitch). In 74.29% of the at-bats the player was out, 17.04% of the time
A new roller coaster at an amusement park requires individuals to be at least 4' 8" (56 inches) tall to ride. It is estimated that the heights of 10-year-old boys are normally distributed with μ = 54.5 inches and σ = 4.5 inches. a. What proportion of 10-year-old boys is tall enough to ride the
SAT math scores follow a normal distribution with an approximate μ = 500 and σ = 100. Also ACT math scores follow a normal distribution with an approximate μ = 21 and σ = 4.7. You are an admissions officer at a university and have room to admit one more student for the upcoming year. Joe scored
Refer to the normal distributions for women’s height (μ = 65, σ = 3.5) and men’s height (μ = 70, σ = 4.0). A man’s height of 75 inches and a woman’s height of 70 inches are both 5 inches above their means. Which is relatively taller? Explain why.
Jane Doe claims to possess extrasensory perception (ESP). She says she can guess more often than not the outcome of a flip of a balanced coin in another room. In an experiment, a coin is flipped three times. If she does not actually have ESP, find the probability distribution of the number of her
In Example 11 on ESP, John Doe had to predict which of five numbers was chosen in each of three trials. Doe did not actually have ESP. Explain why this experiment satisfies the three conditions for the binomial distribution by answering parts a–c. a. For the analogy with coin flipping, what plays
Construct a graph similar to that inFigure 6.1 for each of the following binomial distributions:a. n = 4 and p = 0.50.b. n = 4 and p = 0.30.c. n = 4 and p = 0.10.d. Which if any of the graphs in parts a–c are symmetric? Without actually constructing the graph, would the case n = 10 and p = 0.50
Each newborn baby has a probability of approximately 0.49 of being female and 0.51 of being male. For a family with four children, let X = number of children who are girls. a. Explain why the three conditions are satisfied for X to have the binomial distribution. b. Identify n and p for the
The Internet site www.ItsJustLunch.com advertises itself as a dating service for busy professionals that has set up over two million first dates for lunch or drinks after work. An advertisement for this site stated that a survey of their users found that a woman has chance 1 in 8 of a second date
A quiz in a statistics course has four multiple-choice questions, each with five possible answers. A passing grade is three or more correct answers to the four questions. Allison has not studied for the quiz. She has no idea of the correct answer to any of the questions and decides to guess at
In the National Basketball Association, the top free throw shooters usually have probability of about 0.90 of making any given free throw. a. During a game, one such player (Dolph Schayes) shot 10 free throws. Let X = number of free throws made. What must you assume in order for X to have a
An instructor always assigns final grades such that 20% are A, 40% are B, 30% are C, and 10% are D. The grade point scores are 4 for A, 3 for B, 2 for C, and 1 for D.a. Specify the probability distribution for the grade point score of a randomly selected student of this instructor.b. Find the mean
Refer to the previous exercise. Over the course of a season, this player shoots 400 free throws. a. Find the mean and standard deviation of the probability distribution of the number of free throws he makes. b. By the normal distribution approximation, within what range would you expect the number
A balanced die with six sides is rolled 60 times. a. For the binomial distribution of X = number of 6s, what is n and what is p? b. Find the mean and the standard deviation of the distribution of X. Interpret. c. If you observe x = 0, would you be skeptical that the die is balanced? Explain why,
An exit poll is taken of 3000 voters in a statewide election. Let X denotes the number who voted in favor of a special proposition designed to lower property taxes and raises the sales tax. Suppose that in the population, exactly 50% voted for it. a. Explain why this scenario would seem to satisfy
The juror pool for the upcoming murder trial of a celebrity actor contains the names of 100,000 individuals in the population who may be called for jury duty. The proportion of the available jurors on the population list who are Hispanic is 0.40. A jury of size 12 is selected at random from the
On September 7, 2008, the Pittsburgh Pirates lost their 82nd game of the 2008 season and tied the 1933–1948 Philadelphia Phillies major sport record (baseball, football, basketball, and hockey) for most consecutive losing seasons at 16. One year later on September 7, 2009, they lost their 82nd
For Example 12 on the gender distribution of promotions, the population size was more than one thousand, half of whom were female. The sample size was 10. a. Check whether the guideline was satisfied about the relative sizes of the population and the sample, thus allowing you to use the binomial
For the binomial distribution, the number of trials n is a fixed number. Let X denotes the number of girls in a randomly selected family in Canada that has three children. Let Y denote the number of girls in a randomly selected family in Canada (that is, the number of children could be any number).
For the following random variables, explain why at least one condition needed to use the binomial distribution is unlikely to be satisfied. a. X = number of people in a family of size 4 who go to church on a given Sunday, when any one of them goes 50% of the time in the long run (binomial, n = 4, p
Let X = the number of living grandparents that a randomly selected adult American has. According to recent General Social Surveys, its probability distribution is approximately P(0) = 0.71, P(1) = 0.15, P(2) = 0.09, P(3) = 0.03, P(4) = 0.02. a. Does this refer to a discrete or a continuous random
Let X represents the number of homes a real estate agent sells during a given month. Based on previous sales records, she estimates that P(0) = 0.68, P(1) = 0.19, P(2) = 0.09, P(3) = 0.03, P(4) = 0.01, with negligible probability for higher values of x. a. Explain why it does not make sense to
Consider a Pick-3 lottery such as the one described in Exercise 6.6. Suppose your birthday is May 14, and like many people, you decide to bet $1 on your birthday number (i.e., 514) on your birthday. There are two options to play, straight or boxed. If you choose to play straight, you win $500 if
New Jersey has a six-way combination lottery game in which you pick three digits (each can be any of 0, 1, 2,c, 9) and you win if the digits the lottery picks are the same as yours in any of the six possible orders. For a dollar bet, a winner receives $45.50. Two possible strategies are (a) to pick
You need to choose between two alternative programs for dealing with the outbreak of a deadly disease. In program 1, 200 people are saved. In program 2, there is a 2/3 chance that no one is saved and a 1/3 chance that 600 people are saved. a. Find the expected number of lives saved with each
An insurance company sells a policy to airline passengers for $1. If a flyer dies on a given flight (from a plane crash), the policy gives $100,000 to the chosen beneficiary. Otherwise, there is no return. Records show that a passenger has about a one in a million chance of dying on any given
For a normal distribution, find the probability that an observation is a. Within 1.96 standard deviations of the mean. b. More than 2.33 standard deviations from the mean.
Find the z -score such that the interval within z standard deviations of the mean contains probability (a) 0.95 (b) 0.99 for a normal distribution. Sketch the two cases on a single graph.
a. Find the z -score for the number that is less than only 1% of the values of a normal distribution. Sketch a graph to show where this value is. b. Find the z -scores corresponding to the (i) 90th and (ii) 99th percentiles of a normal distribution.
If z is the positive number such that the interval within z standard deviations of the mean contains 50% of a normal distribution, then a. Explain why this value of z is about 0.67. b. Explain why for any normal distribution the first and third quartiles equal μ - 0.67σ and μ + 0.67σ. c. The
Female heights in North America follow a normal distribution with μ = 65 inches and σ = 3.5 inches. Find the proportion of females who are a. Under five feet. b. Over six feet. c. Between 60 and 70 inches. d. Repeat parts a–c for North American males, the heights of whom are normally
The state of Ohio has several statewide lottery options. One is the Pick 3 game in which you pick one of the 1000 three-digit numbers between 000 and 999. The lottery selects a three-digit number at random. With a bet of $1, you win $500 if your number is selected and nothing ($0) otherwise. (Many
The wingspans of recently cloned monarch butterflies follow a normal distribution with mean 9 inches and standard deviation 0.75 inches. What proportion of the butterflies has a wingspan a. Less than 8 inches? b. Wider than 10 inches? c. Between 8 and 10 inches? d. Ten percent of the butterflies
A study of water use in Gainesville, Florida, indicated that in 2006 residential water consumption had a mean of 78 and a standard deviation of 119, in thousands of gallons. a. If the distribution of water consumption were approximately normal, then what proportion of the residences used less than
Mac’s fast-food restaurant finds that its daily profits have a normal distribution with mean $140 and standard deviation $80. a. Find the probability that the restaurant loses money on a given day (that is, daily profit less than 0). b. Find the probability that the restaurant makes money for the
A Dutch researcher reads that male height in the Netherlands has a normal distribution with μ = 72.0 inches and σ = 4.0 inches. She prefers to convert this to the metric scale (1 inch = 2.54 centimeters). The mean and standard deviation then have the same conversion factor. a. In centimeters,
According to the rules of tennis, a tennis ball is supposed to weigh between 56.7 grams (2 ounces) and 58.5 grams (2 1/16 ounces). A machine for manufacturing tennis balls produces balls with a mean of 57.6 grams and a standard deviation of 0.3 grams, when it is operating correctly. Suppose that
In a study carried out at the Infant Cognition Center at Yale University, researchers showed 16 infants two videos: one featured a character that could be perceived as helpful, and the other featured a character that could be perceived as hindering. After the infants viewed the videos, the
A roulette wheel consists of 38 numbers, 0 through 36 and 00. Of these, 18 numbers are red, 18 are black, and 2 are green (0 and 00). You are given $10 and told that you must pick one of two wagers, for an outcome based on a spin of the wheel: (1) Bet $10 on number 23. If the spin results in 23,
A weather forecaster states, “The chance of rain is 50% on Saturday and 50% again on Sunday. So there’s a 100% chance of rain sometime over the weekend.” If whether or not it rains on Saturday is independent of whether or not it rains on Sunday, find the actual probability of rain at least
Based on past experience, Julio believes he has a 60% chance of success when he calls a woman and asks for a date. a. State assumptions needed for the binomial distribution to apply to the number of times he is successful on his next five requests. b. If he asks the same woman each of the five
In one Canadian lottery option, you bet on one of the million six-digit numbers between 000000 and 999999. For a $1 bet, you win $100,000 if you are correct. In playing n times, let X be the number of times you win. a. Find the mean of the distribution of X, in terms of n . b. How large an n is
In a given year, the probability that an adult American female dies in a motor vehicle accident equals 0.0001. a. In a city having 1 million adult American females, state assumptions for a binomial distribution to apply to X = the number of them who die in the next year from motor vehicle
Refer to the previous exercise. The probability of a motor vehicle death for adult American males is 0.0002. Repeat part b and c for a city having 1 million of them, and compare results to those for females.a. If the binomial applies, find the mean and standard deviation of X.b. Based on the normal
A salesperson uses random digit dialing to call people and try to interest them in applying for a charge card for a large department store chain. From past experience, she is successful on 2% of her calls. In a typical working day, she makes 200 calls. Let X be the number of calls on which she is
In basketball, when the probability of making a free throw is 0.50 and successive shots are independent, the probability distribution of the longest streak of shots made has μ = 4 for 25 shots, μ = 5 for 50 shots, μ = 6 for 100 shots, and μ = 7 for 200 shots. a. How does the mean change for
For the Boston to Chicago route, an airline flies a Boeing 737–800 with 170 seats. Based on past experience, the airline finds that people who purchase a ticket for this flight have 0.80 probability of showing up for the flight. They routinely sell 190 tickets for the flight, claiming it is
Consider a game of poker being played with a standard 52-card deck (four suits, each of which has 13 different denominations of cards). At a certain point in the game, six cards have been exposed. Of the six, four are diamonds. Your opponent makes a bet of $20, and you must decide whether to call
The sex distribution of new babies is close to 50% each, with the percentage of males usually being just slightly higher. In China in recent years, the percentage of female births seems to have dropped, a combination of policy limiting family size, the desire of most families to have at least one
Lake Wobegon Junior College admits students only if they score above 1200 on the sum of their critical reading, mathematics, and writing scores. Applicants from ethnic group A have a mean of 1500 and a standard deviation of 300 on this test, and applicants from ethnic group B have a mean of 1350
The National Center for Educational Statistics reported that in 2009 the ACT college placement and admission examination had a mean of 21.1 and standard deviation of 5.1. (Source: Data from nces.ed.gov/programs/digest/d09/tables/dt09_147.asp.) a. Which probability distribution would you expect to
Justify the px(1 - p)n-x part of the binomial formula for the probability P (x) of a particular sequence with x successes, using what you learned in Section 5.2 about probabilities for intersections of independent events.
Most discrete random variables can take on a finite number of values. Let X = the number of rolls of two dice necessary until doubles (the same number on each die) first appears. The possible values for this discrete random variable (called the geometric) are 1, 2, 3, 4, 5, 6, 7, and so on, still
Let X denotes the response of a randomly selected person to the question, €œWhat is the ideal number of children for a family to have?€ The probability distribution of X in the United StatesThe probabilities do not sum to exactly 1 due to rounding error.a. Show that the means
Simulate an exit poll of 100 voters, using the Sampling Distributions applet on the text CD, assuming that the population proportion is 0.53. Refer to Activity 1 for guidance on using the applet.a. What sample proportion did you get? Why do you not expect to get exactly 0.53?b. Simulate this exit
The figure illustrates two sampling distributions for sample proportions when the population proportion p = 0.50. a. Find the standard deviation for the sampling distribution of the sample proportion with (i) n = 100 and (ii) n = 1000. b. Explain why the sample proportion would be very likely (as
You’d like to estimate the proportion of the 14,201 (www.syr.edu/about/facts.html) undergraduate students at Syracuse University who are full-time students. You poll a random sample of 100 students, of whom 94 are full-time. Unknown to you, the proportion of all undergraduate students who are
At a university, 60% of the 7,400 students are female. The student newspaper reports results of a survey of a random sample of 50 students about various topics involving alcohol abuse, such as participation in binge drinking. They report that their sample contained 26 females. a. Explain how you
a. With random sampling, does the shape of the data distribution tend to resemble more closely the sampling distribution or the population distribution? Explain. b. Explain carefully the difference between a data distribution of sample observations and the sampling distribution of the sample
According to a recent Current Population Reports, the population distribution of number of years of education for self-employed individuals in the United States has a mean of 13.6 and a standard deviation of 3.0. a. Identify the random variable X whose population distribution is described here. b.
Let X denotes the outcome of rolling a die. a. Construct a graph of the (i) probability distribution of X and (ii) sampling distribution of the sample mean for n = 2. (You can think of (i) as the population distribution you would get if you could roll the dice an infinite number of times. The first
A roulette wheel in Las Vegas has 38 slots. If you bet a dollar on a particular number, you’ll win $35 if the ball ends up in that slot and $0 otherwise. Roulette wheels are calibrated so that each outcome is equally likely. a. Let X denote your winnings when you play once. State the probability
In one lottery option in Canada, you bet on a six-digit number between 000000 and 999999. For a $1 bet, you win $100,000 if you are correct. The mean and standard deviation of the probability distribution for the lottery winnings are μ = 0.10 (that is, 10 cents) and σ = 100.00. Joe figures that
For the population of farm workers in New Zealand, suppose that weekly income has a distribution that is skewed to the right with a mean of μ = +500 (N.Z.) and a standard deviation of σ = +160. A researcher, unaware of these values, plans to randomly sample 100 farm workers and use the sample
The previous exercise reported that for the population, μ = +500 and σ = +160, and that the sample mean income for a random sample of 100 farm workers would have a standard deviation of 16.0. Sketch the sampling distribution of the sample mean and find the probability that the sample mean falls
A company that is selling condos in Florida plans to send out an advertisement for the condos to 4000 potential customers, in which they promise a free weekend at a resort on the Florida coast in exchange for agreeing to attend a four-hour sales presentation. The company would like to know how many
Jan’s All You Can Eat Restaurant charges $8.95 per customer to eat at the restaurant. Restaurant management finds that its expense per customer, based on how much the customer eats and the expense of labor, has a distribution that is skewed to the right with a mean of $8.20 and a standard
A study investigating the relationship between age and annual medical expenses randomly samples 100 individuals in Davis, California. It is hoped that the sample will have a similar mean age as the entire population. a. If the standard deviation of the ages of all individuals in Davis is σ = 15,
Vincenzo Baranello was diagnosed with high blood pressure. He was able to keep his blood pressure in control for several months by taking blood pressure medicine (amlodipine besylate). Baranello’s blood pressure is monitored by taking three readings a day, in early morning, at mid-day, and in the
According to the 2010 U.S. Census Bureau Current Population Survey (www.census.gov/population/www/socdemo/hh-fam/cps2010.html), the average number of people in family households which contain both family and non-family members is 4.43 with a standard deviation of 2.02. This is based on census
A large corporation employs 27,251 individuals. The average income in 2008 for all employees was $74,550 with a standard deviation of $19,872. You are interested in comparing the incomes of today’s employees with those of 2008. A random sample of 100 employees of the corporation yields =
Refer to Examples 1 and 2 about the California exit poll, for which the sample size was 3889 and 42.4% of the sample voted for Whitman. We found that after the election, 40.9% voted for Whitman in the population. a. Identify n and p for the binomial distribution that is the sampling distribution of
Use the Sampling Distributions applet. Set the population to Binary, p = 0.6. The graph of the population distribution for a categorical variable with p , the population proportion, equal to 0.60 should appear. Let’s simulate a sample of size n = 100 from this population. Set n = 100, then click
Consider the sampling distribution you were simulating in parts a and b of the previous exercise, assuming p = 0.10 with samples of size 4000 each. Using the appropriate formulas from this section, find the mean and the standard deviation of the sampling distribution of the sample proportion: a.
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