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Questions and Answers of
Statistics
1. The results of a trial used to determine the capabilities of a new diagnostic test are shown in Table 1. Use the empirical probabilities obtained from the table to calculate the positive
1. Refer to Exercise 3. What is the probability that a randomly chosen hiree will be assigned to a management position at the end of the training period? 2. Refer to Exercise 4. What is the
1. Suppose that there is a white urn containing two white balls and one red ball and there is a red urn containing one white ball and three red balls. An experiment consists of selecting at random a
An automobile insurance company has determined the accident rate (probability of having at least one accident during a year) for various age groups. (See Table 2.) Suppose that a policyholder calls
A specially made pair of dice has only one-and two-spots on the faces. One of the dice has three faces with a one-spot and three faces with a two-spot. The other die has two faces with a one-spot and
The New York Times of January 24, 1997, discussed the recommendation of a special panel concerning mammograms for women in their 40s. About 2% of women aged 40 to 49 years old develop breast cancer
A drug-testing laboratory produces false negative results 2% of the time and false positive results 5% of the time. Suppose that the laboratory has been hired by a company in which 10% of the
An over-the-counter pregnancy test claims to be 99% accurate. Actually, what the insert says is that if the test is performed properly, it is 99% sure to detect a pregnancy. (a) What is the
A test for a condition has a high probability of false positives, 20%. Its rate of false negatives is 10%. The condition is estimated to exist in 65% of all patients sent for screening. If the test
It is estimated that 10% of Olympic athletes use steroids. The test currently being used to detect steroids is said to be 93% effective in correctly detecting steroids in users. It yields false
The results of a trial used to determine the capabilities of a new diagnostic test are shown in Table 9. Use the empirical probabilities obtained from the table to calculate the probability that a
According to exit polling from the 2016 Virginia Democratic primary, 43% of primary voters were men and 57% were women. Fifty-seven percent of Democratic men voting in the primary supported Hillary
According to exit polling from the 2014 U.S. midterm elections, 36% of voters had a household income less than $50,000, while 64% had a household income of at least $50,000. Forty-three percent of
Thirteen cards are dealt from a deck of 52 cards. (a) What is the probability that the ace of spades is one of the 13 cards? (b) Suppose that one of the 13 cards is chosen at random and found not to
1. A scoreboard has six different types of LEDs. For each type of LED, Table 3 gives the proportion of the total number of LEDs of that type and the failure rate (probability of failing within one
1. There are three sections of English 101. In Section I, there are 25 students, of whom 5 are mathematics majors. In Section II, there are 20 students, of whom 6 are mathematics majors. In Section
Ten percent of the pens made by Apex are defective. Only 5% of the pens made by its competitor, B-ink, are defective. Since Apex pens are cheaper than B-ink pens, an office orders 70% of its stock
Use the method of natural frequencies to calculate the probabilities.1. Rework Exercise 3, starting with a class size of 50 people.Refer to Exercise 3,The enrollment in a certain course is 10%
1. A metropolitan police department maintains statistics of larcenies reported in the various precincts of the city. It records the proportion of the city population in each precinct and the precinct
1. Table 6 gives the distribution of voter registration and voter turnouts for a certain city. A randomly chosen person is questioned at the polls. What is the probability that the person is an
1. Table 7 shows the percentages of various portions of the U.S. population in 2012, based on age and gender. Suppose that a person is chosen at random from the entire population.Table 7(a) What is
A multinational company has five divisions: A, B, C, D, and E. The percentage of employees from each division who speak at least two languages fluently is shown in Table 8.Table 8:(a) Find the
1. Simulate 36 rolls of a fair die. Give the relative frequency and the corresponding theoretical probability of each of the outcomes, and compare them. 2. Simulate 96 rolls of a pair of dice where
1. Simulate 10 free-throws for Kobe Bryant, whose career free throw average is 84%. How many of the shots were successful? 2. A baseball player is a .331 hitter. Simulate 10 at-bats for this player,
1. A student who has not studied for a 10-question multiple-choice test, with 4 choices among the answers (a, b, c, d) for each question, decides to simulate such a test and answer the questions
1. What is the sample space of an experiment? 2. Using the language of sets and assuming that A and B are events in a sample space S, write the following events in set notation: (A or B); (A and B);
1. What is the definition of Pr (EF )? 2. What is Bayes' Theorem? 3. What is a tree diagram?
1. State the addition principle.2. Suppose that the probability of an event is k/n. What are the odds that the event will occur?3. Suppose that the odds that an event occurs are a to b. What is the
1. Which gender do the individual professional schools appear to be favoring? 2. Without using probability, justify the apparent contradiction between the answers for part 7 and part 10.
1. Calculate the probability of Alice getting the $100 reward if she chooses the sequence CBC. 2. Calculate the probability of Alice getting the $100 reward if she chooses the sequence BCB. 3. Which
1. How would you explain to someone who didn't know probability why the sequence that you chose is best? 2. What is the probability that a male applicant was accepted to a professional program? A
1. Which gender does the university appear to be favoring? 2. What is the probability that a male applicant was accepted to law school? A female applicant 3. What is the probability that a male
In Exercises 1-4, display the data in a bar chart that shows frequencies on the y-axis.1. 2016 school Enrollments2. 2015 U.S. Defense Employees 3. Areas of the Great Lakes 4. Bachelor's Degrees
The number of tie-breaking votes cast by each of the 21 vice presidents of the United States who served during the twentieth century are shown. Draw a histogram for this data. 0, 0, 4, 10, 0, 2, 3,
The ages at inauguration of the 44 presidents from George Washington to Barack Obama are shown. Draw a histogram for the ages. 57, 61, 57, 57, 58, 57, 61, 54, 68, 51, 49, 64, 50, 48, 65, 52, 56, 46,
In following Exercises, find the five-number summary and the inter quartile range for the given set of numbers, and then draw the box plot. 1. 10, 11, 13, 14, 16, 17, 19, 20, 21, 23, 24 2. 3, 6, 8,
1. The data shown gives the top 10 batting averages at the end of the 2015 season for the American League and the National League. Compare the data, using a box plot, and discuss which of the leagues
1. Display the data from the table in following Exercise in a bar chart showing percentages on the y-axis. 2. Display the data from the table in following Exercise in a bar chart showing percentages
1. In 2015, the interest on the public debt accounted for about 10% of the federal budget. If the federal budget is displayed in a pie chart, what should be the size of the central angle of the
Table 11 gives the frequency distribution for the final grades in a course. (Here, A = 4, B = 3, C = 2, D = 1, F = 0.) Determine the relative frequency distribution associated with this data, and
In a certain carnival game, a player pays $1 and then tosses a fair coin until either a heads occurs or he has tossed the coin four times. He receives 50 cents for each toss. Determine the
1. Figure 9 is the histogram for a probability distribution. What is the probability that the outcome is between 5 and 7, inclusive?2. Figure 10 on the next page is the histogram for a probability
Let the random variables X and Y have the probability distributions listed in Table 15. Determine the probability distributions of the random variables in Exercises.1. X2 2. Y2 3. X - 1 4. Y - 15 5.
Let the random variables X and Y have the probability distributions listed in Table 15. Determine the probability distributions of the random variables in Exercises.1. 2X2 2. (X+1)2 3. (1/5Y + 1)2
The number of cars waiting to be served at a gas station was counted at the beginning of every minute during the morning rush hour. The frequency distribution is given in Table 12. Determine the
Two classes take the same examination, and the grades are recorded in Table 16. We assigned the integers 0 through 4 to the grades F through A, respectively. Table 16 gives the frequency distribution
Use Table 16 to answer the following questions: (a) What percentage of the students in the 9 a.m. class have grades of C or less? (b) What percentage of the 10 a.m. class have grades of C or
Below Exercises refer to the tables and relative frequency distributions associated with following Exercises. 1. Use the data in Table 13. (a) For what percentage of the 60 minutes from 5 a.m. to 6
Table 17 gives the probability distribution of the random variable U.(a) Determine the probability that U = 4. (b) Find Pr (U ¥ 2). (c) Find the probability that U is at most 3. (d) Find
Table 18 gives the probability distribution of the random variable X.(a) Draw the histogram for X. (b) Find Pr (X = 2 or 3) (c) Find the probability that X is at least 2. (d) Find the probability
Table 19 gives the probability distribution of the random variable Y.(a) Draw the histogram for Y. (b) Find Pr (Y = 2 or 3) (c) Find the probability that Y2 is at most 9. (d) Find the probability
A local news website counted the number of visits to their weather page each minute on a rainy morning from 5 a.m. to 6 a.m. The frequency distribution is given in Table 13. Determine the relative
Refer back to Tables 18 and 19. Which of the random variables X and Y has the higher average value? Why would you think so?
A production manager counted the number of items produced each hour during a 40-hour workweek. The frequency distribution is given in Table 14. Determine the relative frequency distribution
A fair coin is tossed three times, and the number of heads is observed. Determine the probability distribution for this experiment, and draw its histogram.
An archer can hit the bull's-eye of the target with probability 1>3. She shoots until she hits the bull's-eye or until four shots have been taken. The number of shots is observed. Determine the
An urn contains three red balls and four white balls. A sample of three balls is selected at random and the number of red balls observed. Determine the probability distribution for this experiment,
A die is rolled, and the number on the top face is observed. Determine the probability distribution for this experiment, and draw its histogram.
In a certain carnival game, the player selects two balls at random from an urn containing two red balls and four white balls. The player receives $5 if he draws two red balls and$1 if he draws one
single die is rolled four times. In following Exercises, find the probabilities that the number of 6s that appear is as stated. 1. Four 2. Exactly two 3. Two or three
single die is rolled four times. In following Exercises, find the probabilities that the number of 6s that appear is as stated. 1. One or two 2. At most two 3. At least two
Fourteen percent of U.S. residents are in their twenties. Consider a group of eight U.S. residents selected at random. In following Exercises, find the probabilities that the number of people in the
Fourteen percent of U.S. residents are in their twenties. Consider a group of eight U.S. residents selected at random. In following Exercises, find the probabilities that the number of people in the
According to the study The American Freshman: National Norms 2015, 76.1% of college freshmen said that "to get training for a specific career" was a very important reason for their going to college.
According to the study The American Freshman: National Norms 2015, 76.1% of college freshmen said that "to get training for a specific career" was a very important reason for their going to college
A family chosen at random has four children. (a) What is the probability that there are two boys and two girls? (b) What is the probability that there are three children of one gender and one of the
A manager at a call center notices that 60% of new hires quit within their first year. In a group of 20 new hires, find the probability that (a) exactly 15 quit within their first year. (b) fewer
Below Exercises refer to Fig. 4, the histogram for Example 4.The histogram in Fig. 4 shows that the values of Pr (X = 3) and Pr (X = 5) are nearly the same. Calculate the two probabilities to
Below Exercises refer to Fig. 4, the histogram for Example 4.Use the histogram in Fig. 4 to determine the larger of Interpret your result in words.
Below Exercises refer to Fig. 5, the histogram for Example 2.Use the histogram in Fig. 5 to determine the larger of
Below Exercises refer to Fig. 5, the histogram for Example 2.The histogram in Fig. 5 shows that the values of Pr(X = 5) and Pr(X = 15) are nearly the same. Calculate the two probabilities to
Suppose that the probability of a person's having an adverse reaction to a certain drug is 2%. If the drug is administered to 56 people, what is the probability that three or more people will have an
Nine customers at a supermarket are asked independently if they use brand X laundry soap. In general, 30% of the population use this brand. What is the probability that, among the nine, more than two
Suppose that 60% of the voters in a state intend to vote for a certain candidate. What is the probability that a survey polling five people reveals that two or fewer intend to vote for that candidate?
A jury has 12 jurors. A vote of at least 10 of 12 for "guilty" is necessary for a defendant to be convicted of a crime. Assume that each juror acts independently of the others and that the
Refer to following Example. What is the probability that a box of washers contains at most two defective washers?
Every offspring inherits a gene for hair color from each parent. We denote the dominant gene by A and the recessive gene by a. If a person has AA or Aa, then the person exhibits the dominant
A coin is tossed 10 times. In following Exercises, find the probabilities that the number of heads is as stated. 1. Exactly three 2. None 3. Seven or eight
A small country owns 10,000 gold ingots that are stored in 100 guarded bins, each holding 100 ingots. One of the guards has stolen an ingot from each bin and replaced it with a bar of pyrite (fool's
A .300 hitter comes to bat four times in a game. What is the probability of his getting no hits? Three hits?
A basketball player makes 82% of her free throws. What is the most likely number of free throws for her to make in her next 10 tries? Note: First make a guess in order to test your intuition. Then
The probability is .64 that a dartist scores a bull's-eye on a single toss of a dart. What is the most probable number of bull's-eyes for them to score in their next 10 tosses? Note: First make a
A single match at the men's U.S. Open consists of a sequence of at most five sets that terminates when one person wins his third set. Suppose that the stronger person (the favorite) has probability p
The World Series consists of a sequence of at most seven games that terminates when one team wins its fourth game. Suppose that the stronger team (the favorite) has probability p (where p > 1/2)
1. How many times must a person roll a die so that the probability of obtaining at least one 6 is greater than 95%? 2. How many times must a person roll a die so that the probability of obtaining at
Refer to following Exercise. How large a group of 20-year-old women is required so that the probability of at least two of them becoming centenarians is over 90%?
1. Find the probability of obtaining between 40 and 60 heads when tossing a coin 100 times. 2. Find the probability of obtaining 6s between 20 and 40 times when rolling a die 200 times.
1. Sixty percent of the voters in a city intend to vote for Proposition A in the next election. If a pollster samples 100 people selected at random, what is the probability that 60 or more of them
Refer to following Exercise. (a) Use the formula in part (d) to calculate the probability that the underdog wins the World Series when p = .6. (b) If the World Series consisted of more than seven
1. Calculate the probability of at most 25 successes for a binomial random variable of 50 trials with a 45% probability of success. 2. Calculate the probability of at least 40 successes for a
1. Calculate the probability of between 45 and 55 successes for a binomial random variable of 90 trials with a 40% probability of success. 2. Calculate the probability of between 10 and 20 successes
A coin is tossed 10 times. In following Exercises, find the probabilities that the number of heads is as stated. 1. Two or three 2. At least one 3. At most seven
In following Exercises, find the expected value for the random variable having the specified histogram.1.2. 3. 4.
1. In American roulette, a bettor may place a $1 bet on any one of the 38 numbers on the roulette wheel. They win $35 (plus the return of their bet) if the ball lands on their number; otherwise, they
Using life insurance tables, a retired man determines that the probability of living 5 more years is .9. He decides to take out a life insurance policy that will pay $10,000 in the event that he dies
Using life insurance tables, a retired couple determines that the probability of living 5 more years is .9 for the man and .95 for the woman. They decide to take out a life insurance policy that will
1. A pair of dice is rolled, and the larger of the two numbers showing is recorded. Find the expected value of this experiment. 2. A pair of dice is rolled, and the smaller of the two numbers showing
1. According to the National Center for Education Statistics, 20.5% of all bachelor's degrees are awarded to students majoring in business. If 15 students receiving bachelor's degrees are selected at
A basketball player makes 40% of their three-point shots and 60% of their free throws. If they are fouled while taking a three-point shot, they are given three free throws. Which is greater: the
What is the probability of success for a binomial random variable with 20 trials whose expected value is 3?
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