Use the following information to answer the next three exercises. A grocery store is interested in how much money,
on average, their customers spend each visit in the produce department. Using their store records, they draw a sample of 1,000 visits and calculate each customer’s average spending on produce. 1. Identify the population, sample, parameter, statistic, variable, and data for this example. 2.
What kind of data is “amount of money spent on produce per visit”? 3. The study finds that the mean amount spent on produce per visit by the customers in the sample is $12.84. This is an example of a: Use the following information to answer the next two exercises. A health club is interested in knowing how many times a typical member uses the club in a week. They decide to ask every tenth customer on a specified day to complete a short survey including information about how many times they have visited the club
in the past week. 4. What kind of a sampling design is this? 5. “Number of visits per week” is what kind of data? 6. Describe a situation
in which you would calculate a parameter, rather than a statistic. 7. The U.S. federal government conducts a survey of high school seniors concerning their plans for future education and employment. One question asks whether they are planning to attend a four-year college or university in the following year. Fifty percent answer yes to this question; that fifty percent is a: 8. Imagine that the U.S. federal government had the means to survey all high school seniors in the U.S. concerning their plans for future education and employment, and found that 50 percent were planning to attend a 4-year college or university in the following year. This 50 percent is an example of a: Use the following information to answer the next three exercises. A survey of a random sample of 100 nurses working at a large hospital asked how many years they had been working in the profession. Their answers are summarized in the following [incomplete] table. 9. Fill in the blanks in the table and round your answers to two decimal places for the Relative Frequency and Cumulative Relative
Frequency cells.
Table B1 10. What proportion of nurses have five or more years of experience? 11. What proportion of nurses have ten or fewer years of experience? 12. Describe how you might draw a random sample of 30 students from a lecture class of 200 students. 13. Describe how you might draw a stratified sample of students from a college, where the strata are the students’ class standing [freshman, sophomore, junior, or senior]. 14. A manager wants to draw a sample, without replacement, of 30 employees from a workforce of 150. Describe how the chance of being selected will change over the course of drawing the sample. 15. The
manager of a department store decides to measure employee satisfaction by selecting four departments at random, and conducting interviews with all the employees in those four departments. What type of survey design is this? 16. A popular American television sports program conducts a poll of viewers to see which team they believe will win the
NFL [National Football League] championship this year. Viewers vote by calling a number displayed on the television screen and telling the operator which team they think will win. Do you think that those who participate in this poll are representative of all football fans in America? 17. Two researchers studying vaccination rates independently draw samples of 50 children, ages 3–18 months, from a large urban area, and determine if they are up to date on
their vaccinations. One researcher finds that 84 percent of the children in her sample are up to date, and the other finds that 86 percent in his sample are up to date. Assuming both followed proper sampling procedures and did their calculations correctly, what is a likely explanation for this discrepancy? 18. A high school increased the length of the school day from 6.5 to 7.5 hours. Students who wished to attend this high school were required to sign
contracts pledging to put forth their best effort on their school work and to obey the school rules; if they did not wish to do so, they could attend another high school in the district. At the end of one year, student performance on statewide tests had increased by ten percentage points over the previous year. Does this improvement prove that a longer school day improves student achievement? 19. You read a newspaper article reporting that eating
almonds leads to increased life satisfaction. The study was conducted by the Almond Growers Association, and was based on a randomized survey asking people about their consumption of various foods, including almonds, and also about their satisfaction with different aspects of their life. Does anything about this poll lead you to question its conclusion? 20. Why is non-response a problem in surveys? 21. Compute the mean of the following numbers, and report your answer using one more decimal place than is present in the original data: 22. A
psychologist is interested in whether the size of tableware [bowls, plates, etc.] influences how much college students eat. He randomly assigns 100 college students to one of two groups: the first is served a meal using normal-sized tableware, while the second is served the same meal, but using tableware that it 20 percent smaller than normal. He records how much food is consumed by each group. Identify the following components of this study. 23. A researcher analyzes the results of the SAT [Scholastic Aptitude Test] over a five-year period and finds that male students on average score higher on the math section, and female students on average score higher on the verbal section. She concludes that these observed differences in test performance are due to genetic factors.
Explain how lurking variables could offer an alternative explanation for the observed differences in test scores. 24. Explain why it would not be possible to use random assignment to study the health effects of smoking. 25. A professor conducts a telephone survey of a city’s population by drawing a sample of numbers from the phone book and having her student assistants call each of the selected numbers once to
administer the survey. What are some sources of bias with this survey? 26. A professor offers extra credit to students who take part in her research studies. What is an ethical problem with this method of recruiting subjects? Use the following information to
answer the next four exercises. The midterm grades on a chemistry exam, graded on a scale of 0 to 100, were: 27. Do you see any outliers in this data? If so, how would you address the situation? 28. Construct a stem plot for this data, using only the
values in the range 0–100. 29. Describe the distribution of exam scores. 30. In a class of 35 students, seven students received scores in the 70–79 range. What is the relative frequency of scores in this range? Use the
following information to answer the next three exercises. You conduct a poll of 30 students to see how many classes they are taking this term. Your results are: 31. You decide to
construct a histogram of this data. What will be the range of your first bar, and what will be the central point? 32. What will be the widths and central points of the other bars? 33. Which bar in this histogram will be the tallest, and what will be its height? 34. You get data from the U.S. Census Bureau on the median household income for your city, and decide to
display it graphically. Which is the better choice for this data, a bar graph or a histogram? 35. You collect data on the color of cars driven by students in your statistics class, and want to display this information graphically. Which is the better choice for this data, a bar graph or a histogram? 36.
Your daughter brings home test scores showing that she scored in the 80th percentile in math and the 76th percentile in reading for her grade. Interpret these scores. 37. You have to wait 90 minutes in the emergency room of a hospital before you can see a doctor. You learn that your wait time was in the 82nd percentile of all wait times. Explain what this means, and whether you think it is good or bad. Use the following information to answer the next three exercises. 1; 1; 2; 3; 4; 4; 5; 5; 6; 7; 7; 8; 9 38. What is the median for this data? 39. What is the first quartile for this data? 40. What is the third quartile for this
data? Use the following information to answer the next four exercises. This box plot represents scores on the final exam for a physics class. Figure B1 41. What is the median for this
data, and how do you know? 42. What are the first and third quartiles for this data, and how do you know? 43. What is the interquartile range for this data? 44. What is the range for this data? 45. In a
marathon, the median finishing time was 3:35:04 [three hours, 35 minutes, and four seconds]. You finished in 3:34:10. Interpret the meaning of the median time, and discuss your time in relation to it. Use the following information to answer the next three exercises. The value, in thousands of dollars, for houses on a block, are: 45; 47; 47.5; 51; 53.5; 125. 46. Calculate the mean for this data. 47. Calculate the median for this data. 48. Which do you think better reflects the average value of the homes on this block? 49. In a left-skewed distribution, which is greater? 50. In a right-skewed distribution, which is greater? 51. In a symmetrical distribution what will be the relationship among the mean, median, and mode? Use the following information to answer the next four exercises. 10; 11; 15; 15; 17; 22 52. Compute the mean and standard deviation for this data; use the sample formula for the standard deviation. 53. What number is two standard deviations above the mean of this data? 54. Express the number 13.7 in terms of the mean and standard deviation of
this data. 55. In a biology class, the scores on the final exam were normally distributed, with a mean of 85, and a standard deviation of five. Susan got a final exam score of 95. Express her exam result as a z-score, and interpret its meaning. Use the following information to
answer the next two exercises. You have a jar full of marbles: 50 are red, 25 are blue, and 15 are yellow. Assume you draw one marble at random for each trial, and replace it before the next trial. 56. Find P[B]. 57. Which is more likely, drawing a red marble or a yellow marble? Justify your answer numerically. Use the following
information to answer the next two exercises. The following are probabilities describing a group of college students. 58. Write the symbols for the probability that a student, selected at random, is both female and a science major. 59. Write the
symbols for the probability that the student is an education major, given that the student is male. 60. Events A and B are independent. 61. C and D are mutually exclusive events. 62. In a high school graduating class of 300, 200 students are going to college, 40 are planning to work full-time, and 80 are taking a gap year. Are these
events mutually exclusive? Use the following information to answer the next two exercises. An archer hits the center of the target [the bullseye] 70 percent of the time. However, she is a streak shooter, and if she hits the center on one shot, her probability of hitting it on the shot immediately following is 0.85. Written in probability notation:
63.
Calculate the probability that she will hit the center of the target on two consecutive shots. 64. Are P[A] and P[B] independent in this example? Use the following information
to answer the next three exercises. The following contingency table displays the number of students who report studying at least 15 hours per week, and how many made the honor roll in the past semester.Practice Test 1
1.1: Definitions of Statistics, Probability, and Key Terms
1.2: Data, Sampling, and Variation in Data and Sampling
# of yearsFrequencyRelative FrequencyCumulative Relative Frequency < 5
25
5–10
30
> 10
empty
1.3: Frequency, Frequency Tables, and Levels of Measurement
14, 5, 18, 23, 61.4: Experimental Design and Ethics
2.1: Stem-and Leaf Graphs [Stemplots], Line Graphs, and Bar Graphs
62, 64, 65, 65, 68, 70, 72, 72, 74, 75, 75, 75, 76,78, 78, 81, 83, 83, 84, 85, 87, 88, 92, 95, 98, 98, 100, 100, 7402.2: Histograms, Frequency Polygons, and Time Series Graphs
1; 1; 1; 1
2; 2; 2; 2; 2
3; 3; 3; 3; 3; 3; 3; 3
4; 4; 4; 4; 4; 4; 4; 4; 4
5; 5; 5; 52.3: Measures of the Location of the Data
2.4: Box Plots
2.5: Measures of the Center of the Data
2.6: Skewness and the Mean, Median, and Mode
2.7: Measures of the Spread of the Data
3.1: Terminology
Let P[R] = the probability of drawing a red marble.
Let P[B] = the probability of drawing a blue
marble.
Let P[Y] = the probability of drawing a yellow marble.
Let P[M] = the probability that the student is male
Let P[F] = the probability that the student is female
Let
P[E] = the probability the student is majoring in education
Let P[S] = the probability the student is majoring in science3.2: Independent and Mutually Exclusive Events
If P[A] = 0.3 and
P[B] = 0.5, find P[A AND B].
If P[C] = 0.18 and
P[D] = 0.03, find P[C OR D].3.3: Two Basic Rules of Probability
P[A] = P[B] = P[hitting the center on one shot] = 0.70
P[B|A] = P[hitting the center on a second shot, given that she hit it on the first] = 0.853.4: Contingency Tables
| Study at least 15 hours/week | 200 | ||
| Study less than 15 hours/week | 125 | 193 | |
| Total | 1,000 |
Table B2
65. Complete the table.
66. Find P[honor roll|study at least 15 hours per week].
67. What is the probability a student studies less than 15 hours per week?
68. Are the events “study at least 15 hours per week” and “makes the honor roll” independent? Justify your answer numerically.
3.5: Tree and Venn Diagrams
69. At a high school, some students play on the tennis team, some play on the soccer team, but neither plays both tennis and soccer. Draw a Venn diagram illustrating this.
70. At a high school, some students play tennis, some play soccer, and some play both. Draw a Venn diagram illustrating this.
Practice Test 1 Solutions
1.1: Definitions of Statistics, Probability, and Key Terms
1.
- population: all the shopping visits by all the store’s customers
- sample: the 1,000 visits drawn for the study
- parameter: the average expenditure on produce per visit by all the store’s customers
- statistic: the average expenditure on produce per visit by the sample of 1,000
- variable: the expenditure on produce for each visit
- data: the dollar amounts spent on produce; for instance, $15.40, $11.53, etc
2. c
3. d
1.2: Data, Sampling, and Variation in Data and Sampling
4. d
5. c
6. Answers will vary.
Sample Answer: Any solution in which you use data from the entire population is acceptable. For instance, a professor might calculate the average exam score for her class: because the
scores of all members of the class were used in the calculation, the average is a parameter.
7. b
8. a
9.
| < 5 | 25 | 0.25 | 0.25 |
| 5–10 | 30 | 0.30 | 0.55 |
| > 10 | 45 | 0.45 | 1.00 |
Table B3
10. 0.75
11. 0.55
12. Answers will vary.
Sample Answer: One possibility is to obtain the class roster and assign each student a number from 1 to 200. Then use a random number generator or table of random number to generate
30 numbers between 1 and 200, and select the students matching the random numbers. It would also be acceptable to write each student’s name on a card, shuffle them in a box, and draw 30 names at random.
13. One possibility would be to obtain a roster of students enrolled in the college, including the class standing for each student. Then you would draw a proportionate random sample from within each class [for instance, if 30 percent of the students in the college are freshman, then 30 percent of your sample would be drawn from the freshman class].
14. For the first person picked, the chance of any individual being selected is one in 150. For the second person, it is one in 149, for the third it is one in 148, and so on. For the 30th person selected, the chance of selection is one in 121.
15. a
16. No. There are at least two chances for bias. First, the viewers of this particular program may not be representative of American football fans as a whole. Second, the sample will be self-selected, because people have to make a phone call in order to take part, and those people are probably not representative of the American football fan population as a whole.
17. These results [84 percent in one sample, 86 percent in the other] are probably due to sampling variability. Each researcher drew a different sample of children, and you would not expect them to get exactly the same result, although you would expect the results to be similar, as they are in this case.
18. No. The improvement could also be due to self-selection: only motivated students were willing to sign the contract, and they would have done well even in a school with 6.5 hour days. Because both changes were implemented at the same time, it is not possible to separate out their influence.
19. At least two aspects of this poll are troublesome. The first is that it was conducted by a group who would benefit by the result—almond sales are likely to increase if people believe that eating almonds will make them happier. The second is that this poll found that almond consumption and life satisfaction are correlated, but does not establish that eating almonds causes satisfaction. It is equally possible, for instance, that people with higher incomes are more likely to eat almonds, and are also more satisfied with their lives.
20. You want the sample of people who take part in a survey to be representative of the population from which they are drawn. People who refuse to take part in a survey often have different views than those who do participate, and so even a random sample may produce biased results if a large percentage of those selected refuse to participate in a survey.
1.3: Frequency, Frequency Tables, and Levels of Measurement
21. 13.2
1.4: Experimental Design and Ethics
22.
- population: all college students
- sample: the 100 college students in the study
- experimental units: each individual college student who participated
- explanatory variable: the size of the tableware
- treatment: tableware that is 20 percent smaller than normal
- response variable: the amount of food eaten
23. There are many lurking variables that could influence the observed differences in test scores. Perhaps the boys, on average, have taken more math courses than the girls, and the girls have taken more English classes than the boys. Perhaps the boys have been encouraged by their families and teachers to prepare for a career in math and science, and thus have put more effort into studying math, while the girls have been encouraged to prepare for fields like communication and psychology that are more focused on language use. A study design would have to control for these and other potential lurking variables [anything that could explain the observed difference in test scores, other than the genetic explanation] in order to draw a scientifically sound conclusion about genetic differences.
24. To use random assignment, you would have to be able to assign people to either smoke or not smoke. Because smoking has many harmful effects, this would not be an ethical experiment. Instead, we study people who have chosen to smoke, and compare them to others who have chosen not to smoke, and try to control for the other ways those two groups may differ [lurking variables].
25. Sources of bias include the fact that not everyone has a telephone, that cell phone numbers are often not listed in published directories, and that an individual might not be at home at the time of the phone call; all these factors make it likely that the respondents to the survey will not be representative of the population as a whole.
26. Research subjects should not be coerced into participation, and offering extra credit in exchange for participation could be construed as coercion. In addition, this method will result in a volunteer sample, which cannot be assumed to be representative of the population as a whole.
2.1: Stem-and Leaf Graphs [Stemplots], Line Graphs, and Bar Graphs
27. The value 740 is an outlier, because the exams were graded on a scale of 0 to 100, and 740 is far outside that range. It may be a data entry error, with the actual score being 74, so the professor should check that exam again to see what the actual score was.
28.
| 6 | 2 4 5 5 8 |
| 7 | 0 2 2 4 5 5 5 6 8 8 |
| 8 | 1 3 3 4 5 7 8 |
| 9 | 2 5 8 8 |
| 10 | 0 0 |
Table B4
29. Most scores on this exam were in the range of 70–89, with a few scoring in the 60–69 range, and a few in the 90–100 range.
2.2: Histograms, Frequency Polygons, and Time Series Graphs
30. RF=735=0.2RF=735=0.2
31. The range will be 0.5–1.5, and the central point will be 1.
32. Range 1.5–2.5, central point 2; range 2.5–3.5, central point 3; range 3.5–4.5, central point 4; range 4.5–5.5., central point 5.
33. The bar from 3.5 to 4.5, with a central point of 4, will be tallest; its height will be nine, because there are nine students taking four courses.
34. The histogram is a better choice, because income is a continuous variable.
35. A bar graph is the better choice, because this data is categorical rather than continuous.
2.3: Measures of the Location of the Data
36. Your daughter scored better than 80 percent of the students in her grade on math and better than 76 percent of the students in reading. Both scores are very good, and place her in the upper quartile, but her math score is slightly better in relation to her peers than her reading score.
37. You had an unusually long wait time, which is bad: 82 percent of patients had a shorter wait time than you, and only 18 percent had a longer wait time.
2.4: Box Plots
38. 5
39. 3
40. 7
41. The median is 86, as represented by the vertical line in the box.
42. The first quartile is 80, and the third quartile is 92, as represented by the left and right boundaries of the box.
43. IQR = 92 – 80 = 12
44. Range = 100 – 75 = 25
2.5: Measures of the Center of the Data
45. Half the runners who finished the marathon ran a time faster than 3:35:04, and half ran a time slower than 3:35:04. Your time is faster than the median time, so you did better than more than half of the runners in this race.
46. 61.5, or $61,500
47. 49.25 or $49,250
48. The median, because the mean is distorted by the high value of one house.
2.6: Skewness and the Mean, Median, and Mode
49. c
50. a
51. They will all be fairly close to each other.
2.7: Measures of the Spread of the Data
52. Mean: 15
Standard deviation: 4.3 μ=10+11+15+15+17+226=15μ
=10+11+15+15+17+226=15 s=∑[x−x¯]2n−1=945=4.3s=∑[x−x¯]2n−1=945=4.3
53. 15 + [2][4.3] = 23.6
54. 13.7 is one standard deviation below the mean of this data, because 15 – 4.3 = 10.7
55. z=95−855=2.0z=95−855=2.0
Susan’s z-score was 2.0, meaning she scored two standard deviations above the class mean for the final exam.
3.1: Terminology
56. P[B]=25 90=0.28P[B]=2590=0.28
57. Drawing a red marble is more likely.
P[R]=5080=0.62P[R]=5080=0.62
P[Y]=1580=0.19P[Y]=1580=0.19
58. P[F AND S]
59. P[E|M]
3.2: Independent and Mutually Exclusive Events
60. P[A AND B] = [0.3][0.5] = 0.15
61. P[C OR D] = 0.18 + 0.03 = 0.21
3.3: Two Basic Rules of Probability
62. No, they cannot be mutually exclusive, because they add up to more than 300. Therefore, some students must fit into two or more categories [e.g., both going to college and working full time].
63. P[A and B] = [P[B|A]][P[A]] = [0.85][0.70] = 0.595
64. No. If they were independent, P[B] would be the same as P[B|A]. We know this is not the case, because P[B] = 0.70 and P[B|A] = 0.85.
3.4: Contingency Tables
65.
| Study at least 15 hours/week | 482 | 200 | 682 |
| Study less than 15 hours/week | 125 | 193 | 318 |
| Total | 607 | 393 | 1,000 |
Table B5
66. P[honor roll|study at least 15 hours word per week] = 4821000=0.482P[honor roll|study at least 15 hours word per week] = 4821000=0.482
67. P[studies less than 15 hours word per week]=125+1931000=0.318 P[studies less than 15 hours word per week]=125+1931000=0.318
68. Let P[S] = study at least 15 hours per
week
Let P[H] = makes the honor roll
From the table, P[S] = 0.682, P[H] = 0.607, and P[S AND H] =0.482.
If
P[S] and P[H] were independent, then P[S AND H] would equal [P[S]][P[H]].
However, [P[S]][P[H]] = [0.682][0.607] = 0.414, while P[S AND H] = 0.482.
Therefore, P[S] and P[H] are not independent.
3.5: Tree and Venn Diagrams
69.
Figure B2
70.
Figure B3
Practice Test 2
4.1: Probability Distribution Function [PDF] for a Discrete Random Variable
Use the following information to answer the next five exercises. You conduct a survey among a random sample of students at a particular university. The data collected includes their major, the number of classes they took the previous semester, and amount of money they spent on books purchased for classes in the previous semester.
1. If X = student’s major, then what is the domain of X?
2. If Y = the number of classes taken in the previous semester, what is the domain of Y?
3. If Z = the amount of money spent on books in the previous semester, what is the domain of Z?
4. Why are X, Y, and Z in the previous example random variables?
5. After collecting data, you find that for one case, z = –7. Is this a possible value for Z?
6. What are the two essential characteristics of a discrete probability distribution?
Use this discrete probability distribution represented in this table to answer the following six questions. The university library records the number of books checked out by each patron over the course of one day, with the following result:
| 0 | 0.20 |
| 1 | 0.45 |
| 2 | 0.20 |
| 3 | 0.10 |
| 4 | 0.05 |
Table B6
7. Define the random variable X for this example.
8. What is P[x > 2]?
9. What is the probability that a patron will check out at least one book?
10. What is the probability a patron will take out no more than three books?
11. If the table listed P[x] as 0.15, how would you know that there was a mistake?
12. What is the average number of books taken out by a patron?
4.2: Mean or Expected Value and Standard Deviation
Use the following information to answer the next four exercises. Three jobs are open in a company: one in the accounting department, one in the human resources department, and one in the sales department. The accounting job receives 30 applicants, the human resources department receives 40 applicants, and the human resources and sales department receives 60 applicants.
13. If X = the number of applications for a job, use this information to fill in Table B7.
14. What is the mean number of applicants per department?
15. What is the PDF for X?
16. Add a fourth column to the table, for [x – μ]2P[x].
17. What is the standard deviation of X?
4.3: Binomial Distribution
18. In a binomial experiment, if p = 0.65, what does q equal?
19. What are the required characteristics of a binomial experiment?
20. Joe conducts an experiment to see how many times he has to flip a coin before he gets four heads in a row. Does this qualify as a binomial experiment?
Use the following information to answer the next three exercises. In a particularly community, 65 percent of households include at least one person who has graduated from college. You randomly sample 100 households in this community. Let X = the number of households including at least one college graduate.
21. Describe the probability distribution of X.
22. What is the mean of X?
23. What is the standard deviation of X?
Use the following information to answer the next four exercises. Joe is the star of his school’s baseball team. His batting average is 0.400, meaning that for every ten times he comes to bat [an at-bat], four of those times he gets a hit. You decide to track his batting performance his next 20 at-bats.
24. Define the random variable X in this experiment.
25. Assuming Joe’s probability of getting a hit is independent and identical across all 20 at-bats, describe the distribution of X.
26. Given this information, what number of hits do you predict Joe will get?
27. What is the standard deviation of X?
4.4: Geometric Distribution
28. What are the three major characteristics of a geometric experiment?
29. You decide to conduct a geometric experiment by flipping a coin until it comes up heads. This takes five trials. Represent the outcomes of this trial, using H for heads and T for tails.
30. You are conducting a geometric experiment by drawing cards from a normal 52-card pack, with replacement, until you draw the Queen of Hearts. What is the domain of X for this experiment?
31. You are conducting a geometric experiment by drawing cards from a normal 52-card deck, without replacement, until you draw a red card. What is the domain of X for this experiment?
Use the following information to answer the next three exercises. In a particular university, 27 percent of students are engineering majors. You decide to select students at random until you choose one that is an engineering major. Let X = the number of students you select until you find one that is an engineering major.
32. What is the probability distribution of X?
33. What is the mean of X?
34. What is the standard deviation of X?
4.5: Hypergeometric Distribution
35. You draw a random sample of ten students to participate in a survey, from a group of 30, consisting of 16 boys and 14 girls. You are interested in the probability that seven of the students chosen will be boys. Does this qualify as a hypergeometric experiment? List the conditions and whether or not they are met.
36. You draw five cards, without replacement, from a normal 52-card deck of playing cards, and are interested in the probability that two of the cards are spades. What are the group of interest, size of the group of interest, and sample size for this example?
4.6: Poisson Distribution
37. What are the key characteristics of the Poisson distribution?
Use the following information to answer the next three exercises. The number of drivers to arrive at a toll booth in an hour can be modeled by the Poisson distribution.
38. If X = the number of drivers, and the average numbers of drivers per hour is four, how would you express this distribution?
39. What is the domain of X?
40. What are the mean and standard deviation of X?
5.1: Continuous Probability Functions
41. You conduct a survey of students to see how many books they purchased the previous semester, the total amount they paid for those books, the number they sold after the semester was over, and the amount of money they received for the books they sold. Which variables in this survey are discrete, and which are continuous?
42. With continuous random variables, we never calculate the probability that X has a particular value, but always speak in terms of the probability that X has a value within a particular range. Why is this?
43. For a continuous random variable, why are P[x < c] and P[x ≤ c] equivalent statements?
44. For a continuous probability function, P[x < 5] = 0.35. What is P[x > 5], and how do you know?
45. Describe how you would draw the continuous probability distribution described by the function f[x]=110f[x]=110 for 0≤x≤100≤x≤10. What type of a distribution is this?
46. For the continuous probability distribution described by the function f[ x]=110f[x]=110 for 0≤x≤100≤x≤10, what is the P[0 < x < 4]?
5.2: The Uniform Distribution
47. For the continuous probability distribution described by the function f[x]=110f[x]=110 for 0≤x≤10 0≤x≤10, what is the P[2 < x < 5]?
Use the following information to answer the next four exercises. The number of minutes that a patient waits at a medical clinic to see a doctor is represented by a uniform distribution between zero and 30 minutes, inclusive.
48. If X equals the number of minutes a person waits, what is the distribution of X?
49. Write the probability density function for this distribution.
50. What is the mean and standard deviation for waiting time?
51. What is the probability that a patient waits less than ten minutes?
5.3: The Exponential Distribution
52. The distribution of the variable X, representing the average time to failure for an automobile battery, can be written as: X ~ Exp[m]. Describe this distribution in words.
53. If the value of m for an exponential distribution is ten, what are the mean and standard deviation for the distribution?
54. Write the probability density function for a variable distributed as: X ~ Exp[0.2].
6.1: The Standard Normal Distribution
55. Translate this statement about the distribution of a random variable X into words: X ~ [100, 15].
56. If the variable X has the standard normal distribution, express this symbolically.
Use the following information for the next six exercises. According to the World Health Organization, distribution of height in centimeters for girls aged five years and no months has the distribution: X ~ N[109, 4.5].
57. What is the z-score for a height of 112 inches?
58. What is the z-score for a height of 100 centimeters?
59. Find the z-score for a height of 105 centimeters and explain what that means In the context of the population.
60. What height corresponds to a z-score of 1.5 in this population?
61. Using the empirical rule, we expect about 68 percent of the values in a normal distribution to lie within one standard deviation above or below the mean. What does this mean, in terms of a specific range of values, for this distribution?
62. Using the empirical rule, about what percent of heights in this distribution do you expect to be between 95.5 cm and 122.5 cm?
6.2: Using the Normal Distribution
Use the following information to answer the next four exercises. The distributor of lotto tickets claims that 20 percent of the tickets are winners. You draw a sample of 500 tickets to test this proposition.
63. Can you use the normal approximation to the binomial for your calculations? Why or why not.
64. What are the expected mean and standard deviation for your sample, assuming the distributor’s claim is true?
65. What is the probability that your sample will have a mean greater than 100?
66. If the z-score for your sample result is –2.00, explain what this means, using the empirical rule.
7.1: The Central Limit Theorem for Sample Means [Averages]
67. What does the central limit theorem state with regard to the distribution of sample means?
68. The distribution of results from flipping a fair coin is uniform: heads and tails are equally likely on any flip, and over a large number of trials, you expect about the same number of heads and tails. Yet if you conduct a study by flipping 30 coins and recording the number of heads, and repeat this 100 times, the distribution of the mean number of heads will be approximately normal. How is this possible?
69. The mean of a normally-distributed population is 50, and the standard deviation is four. If you draw 100 samples of size 40 from this population, describe what you would expect to see in terms of the sampling distribution of the sample mean.
70. X is a random variable with a mean of 25 and a standard deviation of two. Write the distribution for the sample mean of samples of size 100 drawn from this population.
71. Your friend is doing an experiment drawing samples of size 50 from a population with a mean of 117 and a standard deviation of 16. This sample size is large enough to allow use of the central limit theorem, so he says the standard deviation of the sampling distribution of sample means will also be 16. Explain why this is wrong, and calculate the correct value.
72. You are reading a research article that refers to “the standard error of the mean.” What does this mean, and how is it calculated?
Use the following information to answer the next six exercises. You repeatedly draw samples of n = 100 from a population with a mean of 75 and a standard deviation of 4.5.
73. What is the expected distribution of the sample means?
74. One of your friends tries to convince you that the standard error of the mean should be 4.5. Explain what error your friend made.
75. What is the z-score for a sample mean of 76?
76. What is the z-score for a sample mean of 74.7?
77. What sample mean corresponds to a z-score of 1.5?
78. If you decrease the sample size to 50, will the standard error of the mean be smaller or larger? What would be its value?
Use the following information to answer the next two questions. We use the empirical rule to analyze data for samples of size 60 drawn from a population with a mean of 70 and a standard deviation of 9.
79. What range of values would you expect to include 68 percent of the sample means?
80. If you increased the sample size to 100, what range would you expect to contain 68 percent of the sample means, applying the empirical rule?
7.2: The Central Limit Theorem for Sums
81. How does the central limit theorem apply to sums of random variables?
82. Explain how the rules applying the central limit theorem to sample means, and to sums of a random variable, are similar.
83. If you repeatedly draw samples of size 50 from a population with a mean of 80 and a standard deviation of four, and calculate the sum of each sample, what is the expected distribution of these sums?
Use the following information to answer the next four exercises. You draw one sample of size 40 from a population with a mean of 125 and a standard deviation of seven.
84. Compute the sum. What is the probability that the sum for your sample will be less than 5,000?
85. If you drew samples of this size repeatedly, computing the sum each time, what range of values would you expect to contain 95 percent of the sample sums?
86. What value is one standard deviation below the mean?
87. What value corresponds to a z-score of 2.2?
7.3: Using the Central Limit Theorem
88. What does the law of large numbers say about the relationship between the sample mean and the population mean?
89. Applying the law of large numbers, which sample mean would expect to be closer to the population mean, a sample of size ten or a sample of size 100?
Use this information for the next three questions. A manufacturer makes screws with a mean diameter of 0.15 cm [centimeters] and a range of 0.10 cm to 0.20 cm; within that range, the distribution is uniform.
90. If X = the diameter of one screw, what is the distribution of X?
91. Suppose you repeatedly draw samples of size 100 and calculate their mean. Applying the central limit theorem, what is the distribution of these sample means?
92. Suppose you repeatedly draw samples of 60 and calculate their sum. Applying the central limit theorem, what is the distribution of these sample sums?
Practice Test 2 Solutions
Probability Distribution Function [PDF] for a Discrete Random Variable
1. The domain of X = {English, Mathematics,….], i.e., a list of all the majors offered at the university, plus “undeclared.”
2. The domain of Y = {0, 1, 2, …}, i.e., the integers from 0 to the upper limit of classes allowed by the university.
3. The domain of Z = any amount of money from 0 upwards.
4. Because they can take any value within their domain, and their value for any particular case is not known until the survey is completed.
5. No, because the domain of Z includes only positive numbers [you can’t spend a negative amount of money]. Possibly the value –7 is a data entry error, or a special code to indicated that the student did not answer the question.
6. The probabilities must sum to 1.0, and the probabilities of each event must be between 0 and 1, inclusive.
7. Let X = the number of books checked out by a patron.
8. P[x > 2] = 0.10 + 0.05 = 0.15
9. P[x ≥ 0] = 1 – 0.20 = 0.80
10. P[x ≤ 3] = 1 – 0.05 = 0.95
11. The probabilities would sum to 1.10, and the total probability in a distribution must always equal 1.0.
12. x¯x¯ = 0[0.20] + 1[0.45] + 2[0.20] + 3[0.10] + 4[0.05] = 1.35
Mean or Expected Value and Standard Deviation
13.
| 30 | 30/130 | 6.92 |
| 40 | 40/130 | 12.31 |
| 60 | 60/130 | 27.69 |
Table B8
14. x¯x¯ = 6.92 + 12.31 + 27.69 = 46.92
15.
P[x = 30] = 0.23
P[x = 40] = 0.31
P[x = 60] = 0.46
16.
| 30 | 0.23 | 6.92 | [30 – 46.92]2[0.23] = 66.09 |
| 40 | 0.31 | 12.31 | [40 – 46.92]2[0.31] = 14.75 |
| 60 | 0.46 | 46.92 | [60 – 46.92]2[0.46] = 78.93 |
Table B9
∑x=[66.09+14.75+78.93]=12.64∑ x=[66.09+14.75+78.93]=12.64
Binomial Distribution
18. q = 1 – 0.65 = 0.35
19.
- There are a fixed number of trials.
- There are only two possible outcomes, and they add up to 1.
- The trials are independent and conducted under identical conditions.
20. No, because there are not a fixed number of trials
21. X ~ B[100, 0.65]
22. μ = np = 100[0.65] = 65
23. σx=npq=100[0.65][0.35]=4.77σx=npq=100[0.65][0.35]=4.77
24. X = Joe gets a hit in one at-bat [in one occasion of his coming to bat]
25. X ~ B[20, 0.4]
26. μ = np = 20[0.4] = 8
27.σx=npq =20[0.40][0.60]=2.19σx=npq=20[0.40][ 0.60]=2.19
4.4: Geometric Distribution
28.
- A series of Bernoulli trials are conducted until one is a success, and then the experiment stops.
- At least one trial is conducted, but there is no upper limit to the number of trials.
- The probability of success or failure is the same for each trial.
29. T T T T H
30. The domain of X = {1, 2, 3, 4, 5, ….n}. Because you are drawing with replacement, there is no upper bound to the number of draws that may be necessary.
31. The domain of X = {1, 2, 3, 4, 5, 6, 7, 8., 9, 10, 11, 12…27}. Because you are drawing without replacement, and 26 of the 52 cards are red, you have to draw a red card within the first 17 draws.
32. X ~ G[0.24]
33. μ= 1p= 10.27=3.70μ= 1p= 10.27=3.70
34. σ= 1−pp2= 1−0.270.272=3.16σ= 1−p p2= 1−0.270.272=3.16
4.5: Hypergeometric Distribution
35. Yes, because you are sampling from a population composed of two groups [boys and girls], have a group of interest [boys], and are sampling without replacement [hence, the probabilities change with each pick, and you are not performing Bernoulli trials].
36. The group of interest is the cards that are spades, the size of the group of interest is 13, and the sample size is five.
4.6: Poisson Distribution
37. A Poisson distribution models the number of events occurring in a fixed interval of time or space, when the events are independent and the average rate of the events is known.
38. X ~ P[4]
39. The domain of X = {0, 1, 2, 3, …..] i.e., any integer from 0 upwards.
40. μ=4μ=4
σ=4=2σ=4
=2
5.1: Continuous Probability Functions
41. The discrete variables are the number of books purchased, and the number of books sold after the end of the semester. The continuous variables are the amount of money spent for the books, and the amount of money received when they were sold.
42. Because for a continuous random variable, P[x = c] = 0, where c is any single value. Instead, we calculate P[c < x < d], i.e., the probability that the value of x is between the values c and d.
43. Because P[x = c] = 0 for any continuous random variable.
44. P[x > 5] = 1 – 0.35 = 0.65, because the total probability of a continuous probability function is always 1.
45. This is a uniform probability distribution. You would draw it as a rectangle with the vertical sides at 0 and 20, and the horizontal sides at 110110 and 0.
46. P[0