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essential statistics
Elementary Statistics 3rd International Edition William Navidi, Barry Monk - Solutions
A sample of size n = 55 is drawn from a population with proportion p = 0.34. Let ̂p be the sample proportion.a. Find ????̂p and ????̂p.b. Find P( ̂p > 0.21).c. Find P( ̂p < 0.40).
The running time for videos submitted to YouTube in a given week is normally distributed with ???? = 390 seconds and standard deviation ???? = 148 seconds.a. If a single video is randomly selected, what is the probability that the running time of the video exceeds 6 minutes(360 seconds)?b. Suppose
Compute ????̄ x and ????̄ x for samples of size n = 64.
A population has mean ???? = 193 and standard deviation ???? =
State the Central Limit Theorem.
Suppose that in a bowling league, the scores among all bowlers are normally distributed with mean ???? = 182 points and standard deviation ???? = 14 points. A trophy is given to each player whose score is at or above the 97th percentile. What is the minimum score needed for a bowler to receive a
A normal population has mean ???? = 242 and standard deviation ???? = 31.a. What proportion of the population is greater than 233?b. What is the probability that a randomly chosen value will be less than 249?
Suppose that salaries of recent graduates from a certain college are normally distributed with mean ???? = $42,650 and standard deviation ???? = $3800. What two salaries bound the middle 50%?
Find z0.15.
Find the z-scores that bound the middle 80% of the area under the normal curve.
Find the z-score that hasa. An area of 0.33 to its leftb. An area of 0.79 to its right
Find the area under the standard normal curvea. To the left of z = 1.77b. To the right of z = 0.41c. Between z = −2.12 and z = 1.37
Following is a probability density curve for a population.a. What proportion of the population is between 2 and 4?b. If a value is chosen at random from this population, what is the probability that it will be greater than 2? 0.4 0.3 0.2- 0.1 0+ 2 Area = 0.59 Area = 0.09 00- 10
Provide an example of a Poisson random variable and explain how each condition for the Poisson distribution is fulfilled.
Twenty percent of the men in a certain community are more than 6 feet tall. An anthropologist samples five men from a large family in the community and counts the number X who are more than 6 feet tall. Explain why the binomial distribution is not appropriate in this situation. Is P(X = 0) likely
Provide an example of a binomial random variable and explain how each condition for the binomial distribution is fulfilled.
When a population mean is unknown, people will often approximate it with the mean of a large sample. Explain why this is justified.
If a business decision has an expected gain, is it possible to lose money? Explain.
Provide an example of a continuous random variable and explain why it is continuous.
Provide an example of a discrete random variable and explain why it is discrete.
Congenital disease: The connexin-26 mutation is a genetic mutation that results in deafness. The probability that a person carries a gene with this mutation is 0.0151. Use the Poisson approximation to find the probability that exactly 10 people in a sample of 500 carry this mutation.
Standard deviation: Refer to Exercise 12.Find the standard deviation of the number of particles that are withdrawn.
Mean: Refer to Exercise 12.Find the mean number of particles that are withdrawn.
Suspensions: The concentration of particles in a suspension is 2 per milliliter. A volume of 3 milliliters is withdrawn.Find the following probabilities.a. P(Exactly five particles are withdrawn)b. P(Fewer than two particles are withdrawn)c. P(More than one particle is withdrawn)
Survey sample: In a college with 5000 students, 100 are randomly chosen to complete a survey in which they rate the quality of the cafeteria food. Let X be the number of freshmen who are chosen. Does X have a binomial distribution?Why or why not?
Rain, rain, go away: Let X be the number of days during the next month that it rains. Does X have a binomial distribution?Why or why not?
Reading tests: According to the National Center for Education Statistics, 66% of fourth graders could read at a basic level in 2007. Suppose that eight fourth graders are randomly selected.a. Find the probability that exactly five of them can read at a basic level.b. Find the probability that more
Looking for a job: According to the General Social Survey conducted at the University of Chicago, 59% of employed adults believe that if they lost their job, it would be easy to find another one with a similar salary. Suppose that 10 employed adults are randomly selected.a. Find the probability
If the dice come up 11, your profit is $15. If the dice don’t come up 11, you lose $1. The probability that the dice come up 11 is 1∕18. What is the expected value of your profit? Is it an expected gain or an expected loss?
Craps: In the game of craps, you may bet $1 that the next roll of the dice will be an
Genetic disease: Sickle-cell anemia is a disease that results when a person has two copies of a certain recessive gene.People with one copy of the gene are called carriers. Carriers do not have the disease, but can pass the gene on to their children. A child born to parents who are both carriers
Lottery tickets: Several million lottery tickets are sold, and 60% of the tickets are held by women. Five winning tickets will be drawn at random.a. What is the probability that three or fewer of the winners will be women?b. What is the probability that three of the winners will be of one gender
AP tests again: During a recent academic year, approximately 1.7 million students took one or more AP tests. Following is the frequency distribution of the number of AP tests taken by students who took one or more AP tests.Let X represent the number of exams taken by a student who took one or
AP tests: Advanced Placement (AP) tests are graded on a scale of 1 (low) through 5 (high). The College Board reported that the distribution of scores on the AP Statistics Exam in 2009 was as follows:A score of 3 or higher is generally required for college credit. What is the probability that a
Mean, variance, and standard deviation: A random variable X has the following probability distribution.a. Find the mean of X.b. Find the variance of X.c. Find the standard deviation of X. x 6 7 89 10 11 P(x) 0.21 0.12 0.29 0.11 0.01 0.26
Which are distributions? Which of the following tables represent probability distributions? a. x P(x) b. x P(x) C. x P(x) d. x P(x) 3 0.35 5 0.27 0 0.02 2 0.10 4 0.20 6 0.45 1 0.34 3. 0.07 5 0.18 7 -0.06 2 1.02 4 0.75 6 0.09 8 0.44 3 0.01 5 0.08 7 0.18 4 0.43 5 0.14
Huntington’s disease is a genetically transmitted disease that causes degeneration of nerve cells in the brain. The probability that a person carries a gene for Huntington’s disease is 0.00005 (i.e., 1∕20,000). In a sample of 100,000 people, what is the probability that exactly four people
What are the variance and standard deviation of the number of messages received in an 8-hour period?
Refer to Problem
The number of text messages received on a certain person’s cell phone follows a Poisson distribution with the rate of 10 messages per hour. What is the mean number of messages received in an 8-hour period?
The number of large packages delivered by a courier service follows a Poisson distribution with a rate of 5 per day. Find the probability thata. Exactly 4 large packages are delivered on a given day.b. Fewer than 6 large packages are delivered over a 2-day period.c. At least one large package is
A meteorologist states that the probability of rain tomorrow is 0.4 and the probability of rain on the next day is 0.6.Assuming these probabilities are accurate, and that the rain events are independent, find the probability distribution for X, the number of days out of the next two that it rains.
Find the mean and standard deviation for the number of defective batteries in the sample of size 50.
Refer to Problem
At a cell phone battery plant, 5% of cell phone batteries produced are defective. A quality control engineer randomly collects a sample of 50 batteries from a large shipment from this plant and inspects them for defects. Find the probability thata. None of the batteries are defective.b. At least
Find the standard deviation of the number of pets.
Refer to Problem
Find the mean number of pets.
Refer to Problem
Find the probability that a randomly selected family has:a. 1 or 2 petsb. More than 2 petsc. No more than 3 petsd. At least 1 pet
Refer to Problem
The following table presents a probability distribution for the number of pets each family has in a certain neighborhood.Construct a probability histogram. Number of pets Probability 01 2 3 4 0.4 0.2 0.2 0.1 0.1
Find the missing value that makes the following a valid probability distribution. x 2 3 5 8 10 P(x) 0.23 0.12 0.09 ? 0.37
Refer to Problem 2.a. Find the variance of the random variable X.b. Find the standard deviation of the random variable X.
Find the mean of the random variable X with the following probability distribution. -2 1 4 5 P(x) 0.3 0.2 0.1 0.4
Explain why the following is not a probability distribution. x 6 7 89 10 P(x) 0.32 0.11 0.19 0.28 0.03
Describe circumstances under which you would use a combination.
Describe circumstances under which you would use a permutation.
In practice, one must decide whether to treat two events as independent based on an understanding of the process that creates them. For example, in a manufacturing process that produces electronic circuit boards for calculators, assume that the probability that a board is defective is 0.01. You
Sometimes events are in the form ‘‘at least’’ a given number. For example, if a coin is tossed five times, an event could be getting at least two heads. What would be the complement of the event of getting at least two heads?
Explain why the General Addition Rule P (A or B) = P (A) + P (B) − P (A and B) may be used even when A and B are mutually exclusive events.
If the odds of an event occurring are 5:8, what is the probability that the event will occur?
Since there are two winning outcomes out of six equally likely outcomes, the odds of winning are 2:4.Suppose that a pair of dice is rolled and the player wins if it comes up ‘‘doubles,’’ that is, if the same number of dots appears on each die. What are the odds of winning?
When it comes to betting, the chance of winning or losing may be expressed as odds. If there are n equally likely outcomes and m of them result in a win, then the odds of winning are m:(n − m), read ‘‘m to n − m.’’ For example, suppose that a player rolls a die and wins if the number of
Explain how you could use the law of large numbers to show that a coin is unfair by tossing it many times.
Bart chooses three books at random.a. What is the probability that the books on his shelf are statistics textbook, dictionary, and comic book, in that order?b. What is the probability that the statistics textbook, dictionary, and comic book are the three books chosen, in any order?
Bookshelf: Refer to Exercise
Bookshelf: Bart has six books: a novel, a biography, a dictionary, a self-help book, a statistics textbook, and a comic book.a. Bart’s bookshelf has room for only three of the books. In how many ways can Bart choose and order three books?b. In how many ways may the books be chosen and ordered if
Assume the student chooses three courses at random. What is the probability that she chooses English, mathematics, and biology?
Required courses: Refer to Exercise
Required courses: A college student must take courses in English, history, mathematics, biology, and physical education.She decides to choose three of these courses to take in her freshman year. In how many ways can this choice be made?
Rainy weekend: Sally is planning to go away for the weekend this coming Saturday and Sunday. At the place she will be going, the probability of rain on any given day is 0.10. Sally says that the probability that it rains on both days is 0.01. She reasons as follows:a. What assumption is being made
Heart attack: The following table presents the number of hospitalizations for myocardial infarction (heart attack) for men and women in various age groups.a. What is the probability that a randomly chosen patient is a woman?b. What is the probability that a randomly chosen patient is aged
Female business majors: At a certain university, the probability that a randomly chosen student is female is 0.55, the probability that the student is a business major is 0.20, and the probability that the student is female and a business major is 0.15.a. What is the probability that the student is
Music to my ears: Jeri is listening to the songs on a new CD in random order. She will listen to two different songs, and will buy the CD if she likes both of them. Assume there are 10 songs on the CD, and that she would like five of them.a. What is the probability that she likes the first song?b.
Defective parts: A process manufactures microcircuits that are used in computers. Twelve percent of the circuits are defective. Assume that three circuits are installed in a computer. Denote a defective circuit by ‘‘D’’ and a good circuit by ‘‘G.’’a. List all eight items in the
Stop that car: A drag racer has two parachutes, a main and a backup, that are designed to bring the vehicle to a stop at the end of a run. Suppose that the main chute deploys with probability 0.99, and that if the main fails to deploy, the backup deploys with probability 0.98.a. What is the
Start a business: Suppose that start-up companies in the area of biotechnology have probability 0.2 of becoming profitable, and that those in the area of information technology have probability 0.15 of becoming profitable. A venture capitalist invests in one firm of each type. Assume the companies
Blood types: Human blood may contain either or both of two antigens, A and B. Blood that contains only the A antigen is called type A, blood that contains only the B antigen is called type B, blood that contains both antigens is called type AB, and blood that contains neither antigen is called type
Statistics, anyone? Let S be the event that a randomly selected college student has taken a statistics course, and let C be the event that the same student has taken a chemistry course. Suppose P (S) = 0.4, P (C) = 0.3, and P (S and C) = 0.2.a. Find the probability that a student has taken
How are your grades? There were 30 students in last semester’s statistics class. Of these, 6 received a grade of A, and 12 received a grade of B. What is the probability that a randomly chosen student received a grade of A or B?
Colored dice: A six-sided die has one face painted red, two faces painted white, and three faces painted blue. Each face is equally likely to turn up when the die is rolled.a. Construct a sample space for the experiment of rolling this die.b. Find the probability that a blue face turns up.
In a standard game of pool, there are 15 balls labeled 1 through 15.a. In how many ways can the 15 balls be ordered?b. In how many ways can 3 of the 15 balls be chosen and ordered?
A caterer offers 24 different types of dessert. In how many ways can 5 of them be chosen for a banquet if the order doesn’t matter?
The Roman alphabet (the one used to write English) consists of five vowels (a,e, i, o, u), along with 21 consonants (we are considering y to be a consonant). Gregory needs to make up a computer password containing seven characters. He wants the first six characters to alternate—consonant, vowel,
Suppose that the slot machine is played 5 times in a row. What is the probability of winning at least once?
Refer to Problem
Individual plays on a slot machine are independent. The probability of winning on any play is 0.38. What is the probability of winning 3 plays in a row?
A student is chosen at random. Which of the following pairs of events are independent?i. A: The student was born on a Monday. B: The student’s mother was born on a Monday.ii. A: The student is above average in height. B: The student’s mother is above average in height.
A jar contains 4 red marbles, 3 blue marbles, and 5 green marbles. Two marbles are drawn from the jar one at a time without replacement. What is the probability that the second marble is red, given that the first was blue?
The owner of a bookstore has determined that 80% of people who enter the store will buy a book. Of those who buy a book, 60% will pay with a credit card. Find the probability that a randomly selected person entering the store will buy a book and pay for it using a credit card.
A certain neighborhood has 100 households. Forty-eight households have a dog as a pet. Of these, 32 also have a cat.Given that a household has a dog, what is the probability that it also has a cat?
In a group of 100 teenagers, 61 received their driver’s license on their first attempt on the driver’s certification exam and 18 received their driver’s license on their second attempt. What is the probability that a randomly selected teenager received their driver’s license on their first
Which of the following pairs of events are mutually exclusive?i. A: A randomly chosen student is 18 years old. B: The same student is 20 years old.ii. A: A randomly chosen student owns a red car. B: The same student owns a blue car.
The following table presents the results of a survey in which 400 college students were asked whether they listen to music while studying.a. Find the probability that a randomly selected student does not listen to music while studying.b. Find the probability that a randomly selected student listens
State each of the following rules:a. General Addition Ruleb. Addition Rule for Mutually Exclusive Eventsc. Rule of Complementsd. General Multiplication Rulee. Multiplication Rule for Independent Events
A pollster will draw a simple random sample of voters from a large city to ask whether they support the construction of a new light rail line. Assume that there are one million voters in the city, and that 560,000 of them support this proposition.One voter is sampled at random.a. Identify the
Fill in the blank: The probability that a fair coin lands heads is 0.5. Therefore, we can be sure that if we toss a coin repeatedly, the proportion of times it lands heads will .i. approach 0.5 ii. be equal to 0.5 iii. be greater than 0.5 iv. be less than 0.5
Consider the case where there are only two ordered pairs in the data set. What must be true about the residuals?
Explain how it is possible for a point to be an outlier without being an influential point.
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