In 716, determine which of the following probability experiments represents a binomial

Problem 4E The phrase “no more than” is represented by the math symbol _______.

Problem 2E For each trial of a binomial experiment, there are two mutually exclusive outcomes: _______ or _______.

Problem 1E A binomial experiment is performed a fixed number of times. Each repetition of the experiment is called a _______.

Problem 3E True or False: In the binomial probability distribution function, nCxrepresents the number of ways of obtaining x successes in n trials.

Problem 5E The expected number of successes in a binomial experiment with n trials and probability of success p is_______.

Problem 6E As a rule of thumb, if _______, the probability distribution of a binomial random variable X is approximately bell shaped.

Problem 7E Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. A random sample of 15 college seniors is obtained, and the individuals selected are asked to state their ages.

Problem 8E Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. A random sample of 30 cars in a used car lot is obtained, and their mileages recorded.

Problem 11E Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. Three cards are selected from a standard 52-card deck without replacement. The number of aces selected is recorded.

Problem 9E Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. An experimental drug is administered to 100 randomly selected individuals, with the number of individuals responding favorably recorded.

Problem 10E Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. A poll of 1200 registered voters is conducted in which the respondents are asked whether they believe Congress should reform Social Security.

Problem 13E Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. A basketball player who makes 80% of her free throws is asked to shoot free throws until she misses. The number of freethrow attempts is recorded.

Problem 14E Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. A baseball player who reaches base safely 30% of the time is allowed to bat until he reaches base safely for the third time. The number of at-bats required is recorded.

Problem 12E Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. Three cards are selected from a standard 52-card deck with replacement. The number of kings selected is recorded.

Problem 17E A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 10, p = 0.4, x = 3

Problem 15E Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. One hundred randomly selected U.S. parents with at least one child under the age of 18 are surveyed and asked if they have ever spanked their child. The number of parents who have spanked their child is recorded.

Problem 16E Determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why. In a town with 400 citizens, 100 randomly selected citizens are asked to identify their religion. The number who identify with a Christian religion is recorded.

Problem 19E A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 40, p = 0.99, x = 38

Problem 18E A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 15, p = 0.85, x = 12

Problem 22E A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 20, p = 0.6, x = 17

Problem 23E A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 9, p = 0.2, x ? 3

Problem 20E A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 50, p = 0.02, x = 3

Problem 21E A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 8, p = 0.35, x = 3

Problem 24E A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 10, p = 0.65, x < 5

Problem 25E A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 7, p = 0.5, x > 3

Problem 26E A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 20, p = 0.7, x ? 12

Problem 27E A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 12, p = 0.35, x ? 4

Problem 28E A binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment. n = 11, p = 0.75, x ? 8

Problem 29E (a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section; (c) compute the mean and standard deviation, using the methods of this section; and (d) draw the probability histogram, comment on its shape, and label the mean on the histogram. n = 6, p = 0.3

Problem 30E (a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section; (c) compute the mean and standard deviation, using the methods of this section; and (d) draw the probability histogram, comment on its shape, and label the mean on the histogram n = 8, p = 0.5

Problem 31E (a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section; (c) compute the mean and standard deviation, using the methods of this section; and (d) draw the probability histogram, comment on its shape, and label the mean on the histogram. n = 9, p = 0.75

Problem 32E (a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section; (c) compute the mean and standard deviation, using the methods of this section; and (d) draw the probability histogram, comment on its shape, and label the mean on the histogram. n = 10, p = 0.2

Problem 33E (a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section; (c) compute the mean and standard deviation, using the methods of this section; and (d) draw the probability histogram, comment on its shape, and label the mean on the histogram. n = 10, p = 0.5

Problem 35E On-Time Flights According to flightstats.com, American Airlines flights from Dallas to Chicago are on time 80% of the time. Suppose 15 flights are randomly selected, and the number of on-time flights is recorded. (a) Explain why this is a binomial experiment. ________________ (b) Find and interpret the probability that exactly 10 flights are on time. ________________ (c) Find and interpret the probability that fewer than 10 flights are on time. ________________ (d) Find and interpret the probability that at least 10 flights are on time. ________________ (e) Find and interpret the probability that between 8 and 10 flights, inclusive, are on time.

Problem 36E Smokers According to the American Lung Association, 90% of adult smokers started smoking before turning 21 years old. Ten smokers 21 years old or older are randomly selected, and the number of smokers who started smoking before 21 is recorded. (a) Explain why this is a binomial experiment. ________________ (b) Find and interpret the probability that exactly 8 of them started smoking before 21 years of age. ________________ (c) Find and interpret the probability that fewer than 8 of them started smoking before 21 years of age. ________________ (d) Find and interpret the probability that at least 8 of them started smoking before 21 years of age. ________________ (e) Find and interpret the probability that between 7 and 9 of them, inclusive, started smoking before 21 years of age.

Problem 37E Morality In a recent poll, the Gallup Organization found that 45% of adult Americans believe that the overall state of moral values in the United States is poor. Suppose a survey of a random sample of 25 adult Americans is conducted in which they are asked to disclose their feelings on the overall state of moral values in the United States. (a) Find and interpret the probability that exactly 15 of those surveyed feel the state of morals is poor. ________________ (b) Find and interpret the probability that no more than 10 of those surveyed feel the state of morals is poor. ________________ (c) Find and interpret the probability that more than 16 of those surveyed feel the state of morals is poor. ________________ (d) Find and interpret the probability that 13 or 14 believe the state of morals is poor. ________________ (e) Would it be unusual to find 20 or more adult Americans who believe the overall state of moral values is poor in the United States? Why?

Problem 34E (a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section; (c) compute the mean and standard deviation, using the methods of this section; and (d) draw the probability histogram, comment on its shape, and label the mean on the histogram. n = 9, p = 0.8

Problem 38E Allergy Sufferers Clarinex-D is a medication whose purpose is to reduce the symptoms associated with a variety of allergies. In clinical trials of Clarinex-D, 5% of the patients in the study experienced insomnia as a side effect. A random sample of 20 Clarinex-D users is obtained, and the number of patients who experienced insomnia is recorded. (a) Find the probability that exactly 3 experienced insomnia as a side effect. ________________ (b) Find the probability that 3 or fewer experienced insomnia as a side effect. ________________ (c) Find the probability that between 1 and 4 patients inclusive, experienced insomnia as a side effect. ________________ (d) Would it be unusual to find 4 or more patients who experienced insomnia as a side effect? Why?

Problem 40E Sneeze Revisited According to a study done by Nick Wilson of Otago University Wellington, the probability a randomly selected individual will cover his or her mouth with a tissue, handkerchief, or elbow (the method recommended by public health officials) is 0.047. Suppose you sit on a bench in a mall and observe people’s habits as they sneeze. (a) What is the probability that among 15 randomly observed individuals exactly 2 cover their mouth with a tissue, handkerchief, or elbow? ________________ (b) What is the probability that among 15 randomly observed individuals fewer than 3 cover their mouth with a tissue, handkerchief, or elbow? ________________ (c) Would you be surprised if, after observing 15 randomly observed individuals, more than 4 covered the mouth with a tissue, handkerchief, or elbow?

Problem 41E Jury Selection Twelve jurors are randomly selected from a population of 3 million residents. Of these 3 million residents, it is known that 45% are Hispanic. Of the 12 jurors selected, 2 are Hispanic. (a) What proportion of the jury described is Hispanic? ________________ (b) If 12 jurors are randomly selected from a population that is 45% Hispanic, what is the probability that 2 or fewer jurors will be Hispanic? ________________ (c) If you were the lawyer of a Hispanic defendant, what might you argue?

Problem 39E Sneeze According to a study done by Nick Wilson of Otago University Wellington, the probability a randomly selected individual will not cover his or her mouth when sneezing is 0.267. Suppose you sit on a bench in a mall and observe people’s habits as they sneeze. (a) What is the probability that among 10 randomly observed individuals exactly 4 do not cover their mouth when sneezing? ________________ (b) What is the probability that among 10 randomly observed individuals fewer than 3 do not cover their mouth? ________________ (c) Would you be surprised if, after observing 10 individuals, fewer than half covered their mouth when sneezing? Why?

Problem 43E On-Time Flights According to flightstats.com, American Airlines flights from Dallas to Chicago are on time 80% of the time. Suppose 100 flights are randomly selected. (a) Compute the mean and standard deviation of the random variable X, the number of on-time flights in 100 trials of the probability experiment. ________________ (b) Interpret the mean. ________________ (c) Would it be unusual to observe 75 on-time flights in a random sample of 100 flights from Dallas to Chicago? Why?

Problem 42E Sullivan Survey: Car Color According to paint manufacturer DuPont, 6% of all cars worldwide are red. In the Sullivan Statistics Survey, of 175 respondents, 17, or 9.7%, indicated the color of their car is red. Determine if the results of the Sullivan Survey contradict those of DuPont by computing P(X ? 17), where X is a binomial random variable with n = 175 and p = 0.06. Explain what the probability represents.

Problem 47E Spanking In March 1995, The Harris Poll reported that 80% of parents spank their children. Suppose a recent poll of 1030 adult Americans with children finds that 781 indicated that they spank their children. If we assume parents’ attitude toward spanking has not changed since 1995, how many of 1030 parents surveyed would we expect to spank? Do the results of the survey suggest that parents’ attitude toward spanking may have changed since 1995? Why?

Problem 46E Allergy Sufferers Clarinex-D is a medication whose purpose is to reduce the symptoms associated with a variety of allergies. In clinical trials of Clarinex-D, 5% of the patients in the study experienced insomnia as a side effect. (a) If 240 users of Clarinex-D are randomly selected, how many would we expect to experience insomnia as a side effect? ________________ (b) Would it be unusual to observe 20 patients experiencing insomnia as a side effect in 240 trials of the probability experiment? Why?

Problem 48E Government Solutions? In May, 2000, the Gallup Organization reported that 11% of adult Americans had a great deal of trust and confidence in the federal government handling domestic issues. Suppose a survey of a random sample of 1100 adult Americans finds that 84 have a great deal of trust and confidence in the federal government handling domestic issues. Would these results be considered unusual? Why?

Problem 45E Morality In a recent poll, the Gallup Organization found that 45% of adult Americans believe that the overall state of moral values in the United States is poor. (a) Compute the mean and standard deviation of the random variable X, the number of adults who believe that the overall state of moral values in the United States is poor based on a random sample of 500 adult Americans. ________________ (b) Interpret the mean. ________________ (c) Would it be unusual to identify 240 adult Americans who believe that the overall state of moral values in the United States is poor based on a random sample of 500 adult Americans? Why?

Problem 44E Smokers According to the American Lung Association, 90% of adult smokers started smoking before turning 21 years old. (a) Compute the mean and standard deviation of the random variable X, the number of smokers who started before turning 21 years old in 200 trials of the probability experiment. ________________ (b) Interpret the mean. ________________ (c) Would it be unusual to observe 185 smokers who started smoking before turning 21 years old in a random sample of 200 adult smokers? Why?

Problem 49E Internet Searches In August 2007, Google™ accounted for 53.6% of all U.S. Internet searches. Assuming this percentage is still accurate today, would it be unusual to observe 600 searches using Google™ in a random sample of 1000 U.S. Internet searches? Why? Source: Nielsen Net Ratings

Problem 52E Simulation According to the U.S. National Center for Health Statistics, there is a 98% probability that a 20-year-old male will survive to age 30. (a) Using statistical software, simulate taking 100 random samples of size 30 from this population. ________________ (b) Using the results of the simulation, compute the probability that exactly 29 of the 30 males survive to age 30. ________________ (c) Compute the probability that exactly 29 of the 30 males survive to age 30, using the binomial probability distribution. Compare the results with part (b). ________________ (d) Using the results of the simulation, compute the probability that at most 27 of the 30 males survive to age 30. ________________ (e) Compute the probability that at most 27 of the 30 males survive to age 30 using the binomial probability distribution. Compare the results with part (d). ________________ (f) Compute the mean number of male survivors in the 100 simulations of the probability experiment. Is it close to the expected value? ________________ (g) Compute the standard deviation of the number of male survivors in the 100 simulations of the probability experiment. Compare the result to the theoretical standard deviation of the probability distribution. ________________ (h) Did the simulation yield any unusual results?

Problem 51E Racial Profiling in New York City The following excerpt is from the Racial Profiling Data Collection Resource Center (www.racialprofilinganalysis.neu.edu/). In 2006, the New York City Police Department stopped a halfmillion pedestrians for suspected criminal involvement. Raw statistics for these encounters suggest large racial disparities—89 percent of the stops involved nonwhites. Do these statistics point to racial bias in police officers’ decisions to stop particular pedestrians? Do they indicate that officers are particularly intrusive when stopping nonwhites? Write a report that answers the questions posed using the fact that 44% of New York City residents were classified as white in 2006. In your report, cite some shortcomings in using the proportion of white residents in the city to formulate likelihoods.

Problem 50E Asthma Control Singulair is a medication whose purpose is to control asthma attacks. In clinical trials of Singulair, 18.4% of the patients in the study experienced headaches as a side effect. Would it be unusual to observe 86 patients who experience headaches in a random sample of 400 patients who use this medication? Why?

Problem 53E Athletics Participation According to the High School Athletics Participation Survey, approximately 55% of students enrolled in high schools participate in athletic programs. You are performing a study of high school students and would like at least 11 students in the study to be participating in athletics. Source: National Federation of State High School Associations (a) How many high school students do you expect to have to randomly select? ________________ (b) How many high school students do you have to select to have a 99% probability that the sample contains at least 12 who participate in athletics?

Geometric Probability Distribution A probability distribution for the random variable X, the number of trials until a success is observed, is called the geometric probability distribution. It has the same criteria as the binomial distribution (see page 334), except that the number of trials is not fixed. Its probability distribution function (pdf) is \(P(x)=p(1-p)^{x-1}, x=1,2,3, \ldots\) where p is the probability of success. 1. What is the probability that Shaquille O’Neal misses his first two free throws and makes the third? Over his career, he made 52.4% of his free throws. That is, find P(3). 2. Construct a probability distribution for the random variable X, the number of free-throw attempts of Shaquille O’Neal until he makes a free throw. Construct the distribution for x = 1, 2, 3, ... , 10. The probabilities are small for \(x>10\). 3. Compute the mean of the distribution, using the formula presented in Section 6.1. 4. Compare the mean obtained in part (c) with the value \(\frac{1}{p}\) Conclude that the mean of a geometric probability distribution is \(\mu x=\frac{1}{p}\). How many free throws do we expect Shaq to take before we observe a made free throw? Equation Transcription: Text Transcription: P(x)=p(1-p)^{x-1}, x=1,2,3, \ldots x>10 \frac{1}{p} \mu x=\frac{1}{p}

Problem 54E Educational Attainment According to the 2008 American Community Survey, 27% of residents of the United States 25 years old or older had earned at least a bachelor’s degree. You are performing a study and would like at least 10 people in the study to have earned at least a bachelor’s degree. (a) How many residents of the United States 25 years old or older do you expect to randomly select? ________________ b) How many residents of the United States 25 years old or older do you have to randomly select to have a probability 0.99 that the sample contains at least 10 who have earned at least a bachelor’s degree?

Problem 57E State the criteria for a binomial probability experiment.

Problem 60E Explain how the value of p, the probability of success, affects the shape of the binomial probability histogram.

Negative Binomial Probability Distribution The negative binomial probability distribution can be used to compute the probability of the random variable X, the number of trials necessary to observe r successes of a binomial experiment. The probability distribution function is given by \(P(x)=\left(_{x-1} C_{r-i}\right) p^{r}(1-p)^{x-r}\) \(X=r, r+1, r+2, \ldots\) Consider a roulette wheel. Remember, a roulette wheel has 2 green slots, 18 red slots, and 18 black slots. 1. What is the probability that it will take x = 1 trial before observing r = 1 green? 2. What is the probability that it will take x = 20 trials before observing r = 2 greens? 3. What is the probability that it will take x = 30 trials before observing r = 3 greens? 4. The expected number of trials before observing r successes is \(\frac{r}{p}\). What is the expected number of trials before observing 3 greens? Equation Transcription: Text Transcription: P(x)=(_{x-1} C_{r-i}) p^{r}(1-p)^{x-r} X=r, r+1, r+2, \ldots \frac{r}{p}

Problem 59E Explain how the value of n, the number of trials in a binomial experiment, affects the shape of the binomial probability

Problem 58E Explain what “success” means in a binomial probability experiment.

Problem 61E Explain how the value of p, the probability of success, affects the shape of the binomial probability histogram. When can the Empirical Rule be used to identify unusual results in a binomial experiment? Why can the Empirical Rule be used to identify results in a binomial experiment?

A binomial experiment is performed a fixed number of times. Each repetition of the experiment is called a _______.

For each trial of a binomial experiment, there are two mutually exclusive outcomes: _______ or _______.

True or False: In the binomial probability distribution function, nCx represents the number of ways of obtaining x successes in n trials.

The phrase no more than is represented by the math symbol _______.

The expected number of successes in a binomial experiment with n trials and probability of success p is_______.

As a rule of thumb, if _______, the probability distribution of a binomial random variable X is approximately bell shaped.

In Problems 716, determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why A random sample of 15 college seniors is obtained, and the individuals selected are asked to state their ages.

In Problems 716, determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state whyA random sample of 30 cars in a used car lot is obtained, and their mileages recorded.

In Problems 716, determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state whyAn experimental drug is administered to 100 randomly selected individuals, with the number of individuals responding favorably recorded.

In Problems 716, determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state whyA poll of 1200 registered voters is conducted in which the respondents are asked whether they believe Congress should reform Social Security.

In Problems 716, determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state whyThree cards are selected from a standard 52-card deck without replacement. The number of aces selected is recorded.

In Problems 716, determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why Three cards are selected from a standard 52-card deck with replacement. The number of kings selected is recorded

In Problems 716, determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state whyA basketball player who makes 80% of her free throws is asked to shoot free throws until she misses. The number of freethrow attempts is recorded.

In Problems 716, determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state whyA baseball player who reaches base safely 30% of the time is allowed to bat until he reaches base safely for the third time. The number of at-bats required is recorded.

In Problems 716, determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state whyOne hundred randomly selected U.S. parents with at least one child under the age of 18 are surveyed and asked if they have ever spanked their child. The number of parents who have spanked their child is recorded.

In Problems 716, determine which of the following probability experiments represents a binomial experiment. If the probability experiment is not a binomial experiment, state why In a town with 400 citizens, 100 randomly selected citizens are asked to identify their religion. The number who identify with a Christian religion is recorded.

In Problems 1728, a binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.n = 10, p = 0.4, x = 3

In Problems 1728, a binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.n = 15, p = 0.85, x = 12

In Problems 1728, a binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.n = 40, p = 0.99, x = 38

In Problems 1728, a binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.n = 50, p = 0.02, x = 3

In Problems 1728, a binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.n = 8, p = 0.35, x = 3

In Problems 1728, a binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.n = 20, p = 0.6, x = 17

In Problems 1728, a binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.n = 9, p = 0.2, x 3

In Problems 1728, a binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.n = 10, p = 0.65, x 6 5

In Problems 1728, a binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.n = 7, p = 0.5, x 7 3

In Problems 1728, a binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.n = 20, p = 0.7, x 12

In Problems 1728, a binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.n = 12, p = 0.35, x 4

In Problems 1728, a binomial probability experiment is conducted with the given parameters. Compute the probability of x successes in the n independent trials of the experiment.n = 11, p = 0.75, x 8

In Problems 2934, (a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section 6.1; (c) compute the mean and standard deviation, using the methods of this section; and (d) draw the probability histogram, comment on its shape, and label the mean on the histogram.. n = 6, p = 0.3

In Problems 2934, (a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section 6.1; (c) compute the mean and standard deviation, using the methods of this section; and (d) draw the probability histogram, comment on its shape, and label the mean on the histogram.n = 8, p = 0.5

In Problems 2934, (a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section 6.1; (c) compute the mean and standard deviation, using the methods of this section; and (d) draw the probability histogram, comment on its shape, and label the mean on the histogram.n = 9, p = 0.75

In Problems 2934, (a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section 6.1; (c) compute the mean and standard deviation, using the methods of this section; and (d) draw the probability histogram, comment on its shape, and label the mean on the histogram.n = 10, p = 0.2

In Problems 2934, (a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section 6.1; (c) compute the mean and standard deviation, using the methods of this section; and (d) draw the probability histogram, comment on its shape, and label the mean on the histogram.n = 10, p = 0.5

In Problems 2934, (a) construct a binomial probability distribution with the given parameters; (b) compute the mean and standard deviation of the random variable using the methods of Section 6.1; (c) compute the mean and standard deviation, using the methods of this section; and (d) draw the probability histogram, comment on its shape, and label the mean on the histogram.n = 9, p = 0.8

On-Time Flights According to flightstats.com, American Airlines flights from Dallas to Chicago are on time 80% of the time. Suppose 15 flights are randomly selected, and the number of on-time flights is recorded. (a) Explain why this is a binomial experiment. (b) Find and interpret the probability that exactly 10 flights are on time. (c) Find and interpret the probability that fewer than 10 flights are on time. (d) Find and interpret the probability that at least 10 flights are on time. (e) Find and interpret the probability that between 8 and 10 flights, inclusive, are on time.4

Smokers According to the American Lung Association, 90% of adult smokers started smoking before turning 21 years old. Ten smokers 21 years old or older are randomly selected, and the number of smokers who started smoking before 21 is recorded. (a) Explain why this is a binomial experiment. (b) Find and interpret the probability that exactly 8 of them started smoking before 21 years of age. (c) Find and interpret the probability that fewer than 8 of them started smoking before 21 years of age. (d) Find and interpret the probability that at least 8 of them started smoking before 21 years of age. (e) Find and interpret the probability that between 7 and 9 of them, inclusive, started smoking before 21 years of age.

Morality In a recent poll, the Gallup Organization found that 45% of adult Americans believe that the overall state of moral values in the United States is poor. Suppose a survey of a random sample of 25 adult Americans is conducted in which they are asked to disclose their feelings on the overall state of moral values in the United States. (a) Find and interpret the probability that exactly 15 of those surveyed feel the state of morals is poor. (b) Find and interpret the probability that no more than 10 of those surveyed feel the state of morals is poor.(c) Find and interpret the probability that more than 16 of those surveyed feel the state of morals is poor. (d) Find and interpret the probability that 13 or 14 believe the state of morals is poor. (e) Would it be unusual to find 20 or more adult Americans who believe the overall state of moral values is poor in the United States? Why?

Allergy Sufferers Clarinex-D is a medication whose purpose is to reduce the symptoms associated with a variety of allergies. In clinical trials of Clarinex-D, 5% of the patients in the study experienced insomnia as a side effect. A random sample of 20 Clarinex-D users is obtained, and the number of patients who experienced insomnia is recorded. (a) Find the probability that exactly 3 experienced insomnia as a side effect. (b) Find the probability that 3 or fewer experienced insomnia as a side effect. (c) Find the probability that between 1 and 4 patients inclusive, experienced insomnia as a side effect. (d) Would it be unusual to find 4 or more patients who experienced insomnia as a side effect? Why?

Sneeze According to a study done by Nick Wilson of Otago University Wellington, the probability a randomly selected individual will not cover his or her mouth when sneezing is 0.267. Suppose you sit on a bench in a mall and observe peoples habits as they sneeze. (a) What is the probability that among 10 randomly observed individuals exactly 4 do not cover their mouth when sneezing? (b) What is the probability that among 10 randomly observed individuals fewer than 3 do not cover their mouth? (c) Would you be surprised if, after observing 10 individuals, fewer than half covered their mouth when sneezing? Why?

Sneeze Revisited According to a study done by Nick Wilson of Otago University Wellington, the probability a randomly selected individual will cover his or her mouth with a tissue, handkerchief, or elbow (the method recommended by public health officials) is 0.047. Suppose you sit on a bench in a mall and observe peoples habits as they sneeze. (a) What is the probability that among 15 randomly observed individuals exactly 2 cover their mouth with a tissue, handkerchief, or elbow? (b) What is the probability that among 15 randomly observed individuals fewer than 3 cover their mouth with a tissue, handkerchief, or elbow? (c) Would you be surprised if, after observing 15 randomly observed individuals, more than 4 covered the mouth with a tissue, handkerchief, or elbow?

Jury Selection Twelve jurors are randomly selected from a population of 3 million residents. Of these 3 million residents, it is known that 45% are Hispanic. Of the 12 jurors selected, 2 are Hispanic. (a) What proportion of the jury described is Hispanic? (b) If 12 jurors are randomly selected from a population that is 45% Hispanic, what is the probability that 2 or fewer jurors will be Hispanic? (c) If you were the lawyer of a Hispanic defendant, what might you argue?

. Sullivan Survey: Car Color According to paint manufacturer DuPont, 6% of all cars worldwide are red. In the Sullivan Statistics Survey, of 175 respondents, 17, or 9.7%, indicated the color of their car is red. Determine if the results of the Sullivan Survey contradict those of DuPont by computing P(X 17), where X is a binomial random variable with n = 175 and p = 0.06. Explain what the probability represents.

. On-Time Flights According to flightstats.com, American Airlines flights from Dallas to Chicago are on time 80% of the time. Suppose 100 flights are randomly selected. (a) Compute the mean and standard deviation of the random variable X, the number of on-time flights in 100 trials of the probability experiment. (b) Interpret the mean. (c) Would it be unusual to observe 75 on-time flights in a random sample of 100 flights from Dallas to Chicago? Why?

Smokers According to the American Lung Association, 90% of adult smokers started smoking before turning 21 years old. (a) Compute the mean and standard deviation of the random variable X, the number of smokers who started before turning 21 years old in 200 trials of the probability experiment. (b) Interpret the mean. (c) Would it be unusual to observe 185 smokers who started smoking before turning 21 years old in a random sample of 200 adult smokers? Why?

Morality In a recent poll, the Gallup Organization found that 45% of adult Americans believe that the overall state of moral values in the United States is poor. (a) Compute the mean and standard deviation of the random variable X, the number of adults who believe that the overall state of moral values in the United States is poor based on a random sample of 500 adult Americans. (b) Interpret the mean. (c) Would it be unusual to identify 240 adult Americans who believe that the overall state of moral values in the United States is poor based on a random sample of 500 adult Americans? Why?

Allergy Sufferers Clarinex-D is a medication whose purpose is to reduce the symptoms associated with a variety of allergies. In clinical trials of Clarinex-D, 5% of the patients in the study experienced insomnia as a side effect. (a) If 240 users of Clarinex-D are randomly selected, how many would we expect to experience insomnia as a side effect? (b) Would it be unusual to observe 20 patients experiencing insomnia as a side effect in 240 trials of the probability experiment? Why?

Spanking In March 1995, The Harris Poll reported that 80% of parents spank their children. Suppose a recent poll of 1030 adult Americans with children finds that 781 indicated that they spank their children. If we assume parents attitude toward spanking has not changed since 1995, how many of 1030 parents surveyed would we expect to spank? Do the results of the survey suggest that parents attitude toward spanking may have changed since 1995? Why?

Government Solutions? In May, 2000, the Gallup Organization reported that 11% of adult Americans had a great deal of trust and confidence in the federal government handling domestic issues. Suppose a survey of a random sample of 1100 adult Americans finds that 84 have a great deal of trust and confidence in the federal government handling domestic issues. Would these results be considered unusual? Why?

Internet Searches In August 2007, Google accounted for 53.6% of all U.S. Internet searches. Assuming this percentage is still accurate today, would it be unusual to observe 600 searches using Google in a random sample of 1000 U.S. Internet searches? Why? Source: Nielsen NetRatings

Asthma Control Singulair is a medication whose purpose is to control asthma attacks. In clinical trials of Singulair, 18.4% of the patients in the study experienced headaches as a side effect. Would it be unusual to observe 86 patients who experience headaches in a random sample of 400 patients who use this medication? Why?

Racial Profiling in New York City The following excerpt is from the Racial Profiling Data Collection Resource Center (www.racialprofilinganalysis.neu.edu/). In 2006, the New York City Police Department stopped a halfmillion pedestrians for suspected criminal involvement. Raw statistics for these encounters suggest large racial disparities 89 percent of the stops involved nonwhites. Do these statistics point to racial bias in police officers decisions to stop particular pedestrians? Do they indicate that officers are particularly intrusive when stopping nonwhites? Write a report that answers the questions posed using the fact that 44% of New York City residents were classified as white in 2006. In your report, cite some shortcomings in using the proportion of white residents in the city to formulate likelihoods.

Simulation According to the U.S. National Center for Health Statistics, there is a 98% probability that a 20-year-old male will survive to age 30. (a) Using statistical software, simulate taking 100 random samples of size 30 from this population. (b) Using the results of the simulation, compute the probability that exactly 29 of the 30 males survive to age 30. (c) Compute the probability that exactly 29 of the 30 males survive to age 30, using the binomial probability distribution. Compare the results with part (b). (d) Using the results of the simulation, compute the probability that at most 27 of the 30 males survive to age 30. (e) Compute the probability that at most 27 of the 30 males survive to age 30 using the binomial probability distribution. Compare the results with part (d). (f ) Compute the mean number of male survivors in the 100 simulations of the probability experiment. Is it close to the expected value? (g) Compute the standard deviation of the number of male survivors in the 100 simulations of the probability experiment. Compare the result to the theoretical standard deviation of the probability distribution. (h) Did the simulation yield any unusual results?

Athletics Participation According to the High School Athletics Participation Survey, approximately 55% of students enrolled in high schools participate in athletic programs. You are performing a study of high school students and would like at least 11 students in the study to be participating in athletics. Source: National Federation of State High School Associations (a) How many high school students do you expect to have to randomly select? (b) How many high school students do you have to select to have a 99% probability that the sample contains at least 12 who participate in athletics?

. Educational Attainment According to the 2008 American Community Survey, 27% of residents of the United States 25 years old or older had earned at least a bachelors degree. You are performing a study and would like at least 10 people in the study to have earned at least a bachelors degree. (a) How many residents of the United States 25 years old or older do you expect to randomly select? (b) How many residents of the United States 25 years old or older do you have to randomly select to have a probability 0.99 that the sample contains at least 10 who have earned at least a bachelors degree?

Geometric Probability Distribution A probability distribution for the random variable X, the number of trials until a success is observed, is called the geometric probability distribution. It has the same criteria as the binomial distribution (see page 334), except that the number of trials is not fixed. Its probability distribution function (pdf) is P(x) = p(1 - p) x-1 , x = 1, 2, 3, p where p is the probability of success. (a) What is the probability that Shaquille ONeal misses his first two free throws and makes the third? Over his career, he made 52.4% of his free throws. That is, find P(3). (b) Construct a probability distribution for the random variable X, the number of free-throw attempts of Shaquille ONeal until he makes a free throw. Construct the distribution for x = 1, 2, 3, p , 10. The probabilities are small for x 7 10. (c) Compute the mean of the distribution, using the formula presented in Section 6.1. (d) Compare the mean obtained in part (c) with the value 1 p . Conclude that the mean of a geometric probability distribution is mX = 1 p . How many free throws do we expect Shaq to take before we observe a made free throw?

Negative Binomial Probability Distribution The negative binomial probability distribution can be used to compute the probability of the random variable X, the number of trials necessary to observe r successes of a binomial experiment. The probability distribution function is given by P(x) = (x-1Cr-1 )pr (1 - p) x-r x = r, r + 1, r + 2, p Consider a roulette wheel. Remember, a roulette wheel has 2 green slots, 18 red slots, and 18 black slots. (a) What is the probability that it will take x = 1 trial before observing r = 1 green? (b) What is the probability that it will take x = 20 trials before observing r = 2 greens? (c) What is the probability that it will take x = 30 trials before observing r = 3 greens? (d) The expected number of trials before observing r successes is r p . What is the expected number of trials before observing 3 greens?

State the criteria for a binomial probability experiment

Explain what success means in a binomial probability experiment.

Explain how the value of n, the number of trials in a binomial experiment, affects the shape of the binomial probability histogram.

Explain how the value of p, the probability of success, affects the shape of the binomial probability histogram.

When can the Empirical Rule be used to identify unusual results in a binomial experiment? Why can the Empirical Rule be used to identify results in a binomial experiment?