Algebra
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.
Identify the population, sample, parameter, statistic, variable, and data for this example. a. population b. sample c. parameter d. statistic e. variable f. data
2. What kind of data is “amount of money spent on produce per visit”?
qualitative b. quantitative-continuous c. quantitative-discrete
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:
population b. sample c. parameter d. statistic e. variable
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 aweek. 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?
cluster b. stratified c. simple random d. systematic
“Number of visits per week” is what kind of data?
qualitative b. quantitative-continuous c. quantitative-discrete
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: parameter b. statistic c. variable d. data
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:
MAT 106 Midterm Review 2a. parameter b. statistic c. variable d. data
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.
What proportion of nurses have five or more years of experience?
What proportion of nurses have ten or fewer years of experience?
Describe how you might draw a random sample of 30 students from a lecture class of 200students. 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 workforceof150. 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? a. cluster b. stratified c. simple random. systematic
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?
Two researchers studying vaccination rates independently draw samples of 50 children, ages3–18 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
MAT 106 Midterm Review 3finds 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 state wide tests had increased by ten percentage points over the previous year. Does this improvement prove that a longer school day improves student achievement?
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?
Why is non-response a problem in surveys?
Compute the mean of the following numbers, and report your answer using one more decimal place than is present in the original data: 14, 5, 18, 23, 6. 22. A psychologist is interested in whether the size of tableware (bowls, plates, etc.) influence show 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 table ware that it 20 percent smaller than normal. He records how much food is consumed by each group. Identify the following components of this study. a. population b. sample c. experimental units d. explanatory variable e. treatment f. response variable
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 toad minister the survey. What are some sources of bias with this survey?
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:
62, 64, 65, 65, 68, 70, 72, 72, 74, 75, 75, 75, 76,78, 78
MAT 106 Midterm Review 481, 83, 83, 84, 85, 87, 88, 92, 95, 98, 98, 100, 100, 740
Do you see any outliers in this data? If so, how would you address the situation?
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:
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; 5
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?
What will be the widths and central points of the other bars?
Which bar in this histogram will be the tallest, and what will be its height?
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?
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?
What is the first quartile for this data?
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. Use the following information to answer the next four exercises. This box plot represents scores on the final exam for a physics class.
MAT 106 Midterm Review 541. What is the median for this data, and how do you know?
What are the first and third quartiles for this data, and how do you know?
What is the interquartile range for this data?
What is the range for this data?
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
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?
In a left-skewed distribution, which is greater?
the mean b. the media c. the mode
In a right-skewed distribution, which is greater?
the mean b. the median c. the mode
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
Compute the mean and standard deviation for this data; use the sample formula for the standard deviation.
MAT 106 Midterm Review 653. What number is two standard deviations above the mean of this data?
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 az-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 on emarble at random for each trial, and replace it before the next trial. 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. 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. 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 science
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. If P(A) = 0.3 and P(B) = 0.5, find P(A and B). 61. C and D are mutually exclusive events. If P(C) = 0.18 and P(D) = 0.03, find P(C or D). 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: P(A) = P(B) = P(hitting the center on one shot) = 0.70 and P(B|A) = P(hitting the center on a second shot, given that she hit it on the first) = 0.85
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 four exercises. The following contingency table displays the number of students who report studying at least 15hours per week, and how many made the honor roll in the past semester.
MAT 106 Midterm Review 765. 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?
Are the events “study at least 15 hours per week” and “makes the honor roll” independent?Justify your answer numerically. 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
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. 71. If X = student’s major, then what is the domain of X?
If Y = the number of classes taken in the previous semester, what is the domain of Y?
If Z = the amount of money spent on books in the previous semester, what is the domain of Z?74. Why are X, Y, and Z in the previous example random variables?
After collecting data, you find that for one case, z = –7. Is this a possible value for Z?
What are the two essential characteristics of a discrete probability distribution?
Use this discrete probability distribution table below 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:
Define the random variable X for this example. 78. What is P(x > 2)?
What is the probability that a patron will check out at least one book?
x 0 1 2 3 4
P(x) 0.20 0.45 0.20 0.10 0.05
MAT 106 Midterm Review 880. What is the probability a patron will take out no more than three books?
If the table listed P(x) as 0.15, how would you know that there was a mistake?
What is the average number of books taken out by a patron?
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, and the human resources and sales department 60 applicants. 83. If X = the number of applications for a job, use this information to fill in table below.
What is the mean number of applicants?
What is the PDF for X?
Add a fourth column to the table, for −
2(). 87. What is the standard deviation of X?
In a binomial experiment, if p = 0.65, what does q equal?
What are the required characteristics of a binomial experiment?
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 atleast 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. 91. Describe the probability distribution of X. 92. What is the mean of X?
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. 94. Define the random variable X in this experiment. 95. Assuming Joe’s probability of getting a hit is independent and identical across all 20 at-bats, describe the distribution of X. 96. Given this information, what number of hits do you predict Joe will get?
What is the standard deviation of X?
What are the three major characteristics of a geometric experiment?
MAT 106 Midterm Review 999. 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. 100. You are conducting a geometric experiment by drawing cards from a normal 52-cardpack, with replacement, until you draw the Queen of Hearts. What is the domain of X for this experiment?
You are conducting a geometric experiment by drawing cards from a normal 52-carddeck, 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. 102. What is the probability distribution of X?
What is the mean of X?
What is the standard deviation of X?
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. 106. 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?
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. 108. If X = the number of drivers, and the average numbers of drivers per hour is four, how would you express this distribution?
What is the domain of X?
What are the mean and standard deviation of X?
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?
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?
For a continuous random variable, why are P(x < c) and P(x ≤ c) equivalent statements?114. For a continuous probability function, P(x < 5) = 0.35. What is P(x > 5), and how do you know?115. Describe how you would draw the continuous probability distribution described by the function f (x) = 1/10 for 0 ≤ x ≤ 10 . What type of a distribution is this?
For the continuous probability distribution described by the function f (x) = 1/10 for 0≤x≤10,what is the P(0 < x < 4)?
For the continuous probability distribution described by the function f (x) = 1/10 for 0≤x≤10, what is the P(2 < x < 5)?
MAT 106 Midterm Review 10Use 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. 118. If X equals the number of minutes a person waits, what is the distribution of X?
Write the probability density function for this distribution. 120. What is the mean and standard deviation for waiting time?
What is the probability that a patient waits less than ten minutes?
The distribution of the variable X, representing the average time to failure for an auto mobile battery, can be written as: X ~ Exp(m). Describe this distribution in words. 123. If the value of m for an exponential distribution is ten, what are the mean and standard deviation for the distribution?
Write the probability density function for a variable distributed as: X ~ Exp(0.2).
MAT 106 Midterm Review 11Midterm Review Key
a. 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
c
d
d
c
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
a
9.
0.75
0.55
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, then30percentof 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
MAT 106 Midterm Review 1216. 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. 21. 13.2
a. 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
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.
MAT 106 Midterm Review 1326. 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. 27. The value 740 is an outlier, because the exams were graded on a scale of 0 to 100, and740isfaroutside 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.
Most scores on this exam were in the range of 70–89, with a few scoring in the 60–69range, and a few in the 90–100 range. 30. RF = 7/35 = 0.2
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. 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. 38. 5
3
7
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
Range = 100 – 75 = 25
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
49.25 or $49,250
The median, because the mean is distorted by the high value of one house.
MAT 106 Midterm Review 1449. c
a
They will all be fairly close to each other. 52. Mean: 15 Standard deviation: 4.3
15 + (2)(4.3) = 23.6
13.7 is one standard deviation below the mean of this data, because 15 – 4.3 = 10.7
Susan’s z-score was 2.0, meaning she scored two standard deviations above the class mean for the final exam. 56. P(B) =
25
90 = 0.28
Drawing a red marble is more likely. P(R) =
50
80 = 0.62, P(Y) =
15
80 = 0.19
P(F and S)
P(E|M)
P(A and B) = (0.3)(0.5) = 0.15
P(C or D) = 0.18 + 0.03 = 0.21
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
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. 65.
P(honor roll|study at least 15 hours word per week) = 482 = 0.482
P(studies less than 15 hours word per week) = 125 + 193 = 0.318
Let P(S) = study at least 15 hours per week Let P(H) = makes the honor roll From the table,
MAT 106 Midterm Review 15P(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. 69. 70.
The domain of X = {English, Mathematics,….], i.e., a list of all the majors offered at the university,plus “undeclared.” 72. The domain of Y = {0, 1, 2, …}, i.e., the integers from 0 to the upper limit of classes allowed by the university. 73. The domain of Z = any amount of money from 0 upwards. 74. Because they can take any value within their domain, and their value for any particular case is not known until the survey is completed. 75. 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. 76. The probabilities must sum to 1.0, and the probabilities of each event must be between0and1,inclusive. 77. Let X = the number of books checked out by a patron. 78. P(x > 2) = 0.10 + 0.05 = 0.15
P(x ≥ 0) = 1 – 0.20 = 0.80
P(x ≤ 3) = 1 – 0.05 = 0.95
MAT 106 Midterm Review 1681. The probabilities would sum to 1.10, and the total probability in a distribution must always equal 1.0. 82. x= 0(0.20) + 1(0.45) + 2(0.20) + 3(0.10) + 4(0.05) = 1.35
83.
x= 9.90 + 13.20 + 19.80 = 42.90
P(x = 30) = 0.33 P(x = 40) = 0.33 P(x = 60) = 0.33
86.
= 54.91 + 2.78 + 96.49 = 12.42
q = 1 – 0.65 = 0.35
1. There are a fixed number of trials. 2. There are only two possible outcomes, and they add up to 1. 3. The trials are independent and conducted under identical conditions. 90. No, because there are not a fixed number of trials
X ~ B(100, 0.65)
μ = np = 100(0.65) = 65
= = 100 ∗ 0.65 ∗ 0.35 = 4.77
X = Joe gets a hit in one at-bat (in one occasion of his coming to bat)
X ~ B(20, 0.4)
μ = np = 20(0.4) = 8
= = 20 ∗ 0.40 ∗ 0.60 = 2.19
(1) A series of Bernoulli trials are conducted until one is a success, and then the experiment stops. (2) At least one trial is conducted, but there is no upper limit to the number of trials. (3) The probability of success or failure is the same for each trial. 99. T T T T H
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. 101. 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 17draws. 102. X ~ G(0.24)
µ = 1/p = 1/0.27 = 3.70
MAT 106 Midterm Review 17104. σ =
1−p
p
2 =
1−0.27
0.27
2 = 3.16
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). 106. 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. 107. 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. 108. X ~ P(4)
The domain of X = {0, 1, 2, 3, …..) i.e., any integer from 0 upwards. 110. µ = 4, σ = 4 = 2
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. 112. 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. 113. Because P(x = c) = 0 for any continuous random variable. 114. P(x > 5) = 1 – 0.35 = 0.65, because the total probability of a continuous probability function is always 1. 115. 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 1/10 and 0. 116. P(0 < x < 4) = (4 − 0)1/10=0.4
P(2 < x < 5) = (5 − 2)1/10=0.3
X ~ U(0, 15)
f (x) =
1
−
for (a ≤ x ≤ b) so f (x) = 1/30 for (0 ≤ x ≤ 30)
µ =
+
2 =
0+30
2 = 15.0, σ=
−
2
12 =
30−0
2
12 = 8.66
P(x < 10) = 10*1/30 = 0.33
X has an exponential distribution with decay parameter m and mean and standarddeviation1/m. In this distribution, there will be a relatively large numbers of small values, with values becoming less common as they become larger. 123. µ = σ = 1/m = 1/10 = 0.1
f(x) = 0.2e−0.2x
, where x ≥ 0.
MAT Midterm Exam
106 03, 04
Shibalovich
October 6
1 hr. 30 min.
X
X
X
This test is a take-home test. Take it by yourself without help from anyone else.
Use calculator with statistical functions.
Midterm Exam
Determine whether the statement describes a population or a sample.
1) The annual salaries of all the teachers in a trade school.
A) Sample B) Population
Determine if the numerical value describes a parameter or a statistic.
2) A recent poll of all corporate executives showed that the average price of their cars is $66,700.
A) Population Parameter B) Sample Statistic
Determine and indicate the level of measurement for the data.
3) The sample of spheres categorized from softest to hardest.
A) Nominal B) Ratio C) Ordinal D) Interval
Classify the following data as discrete, continuous, or neither.
4) The number of limbs on a 2-year-old oak tree is 21.
A) Continuous B) Discrete
Identify which of these types of sampling is used.
5) A sample consists of every 40th student from a group of 496 students.
A) Convenience
B) Stratified
C) Random
D) Systematic
E) Cluster
Is the study experimental or observational?
6) A political pollster reports that his candidate has a 17% lead in the polls with 20% undecided.
A) Experimental B) Observational
Classify the following data as qualitative or quantitative.
7) A marketing group asked the following question to residents of 40 countries: “Rate your experience with European products?”
1) Below Average 2) Average 3) Above Average 4) Good to Excellent
A) Qualitative B) Quantitative
Find the mean for the given sample data.
8) Six college students spent $135.84, $120.68, $205.63, $204.02, $183.04, and $260.47 respectively for books. Compute the mean amount spent. Round your answer to the nearest cent.
A) $277.42 B) $221.94 C) $184.95 D) $209.94
Answer the question.
9) A self-selected survey is especially prone to _______.
A) Informed Consent
B) Researcher Bias
C) Participation Bias
D) Sampling Error(s)
E) Processing Error(s)
Find the variance for the given data. Round your answer to one more decimal place than the original data.
10) 1, 5, -8, 11, 9
A) 56.7 B) 56.8 C) 59.5 D) 45.4
Find the median salary for the given sample data.
11) The salaries of ten randomly selected doctors are shown below.
$120,000 $113,000 $195,000 $214,000 $209,000
$110,000 $135,000 $764,000 $226,000 $174,000
A) $251,000 B) $195,000 C) $226,000 D) $184,500
Find the range for the given data.
12) A class of sixth grade students kept accurate records on the amount of time they spent playing video games during a one-week period. The times (in hours) are listed below:
14.3 20.4 8.0 26.5 23.2
28.5 18.0 24.6 25.3 16.7
A) 20.5 B) 6.1 C) 8.0 D) 23.2
Find the mode(s) for the given sample data.
13) -20, -32, -46, -32, -49, -32, -49
A) -37.1 B) -32 C) -46 D) -49
Use the empirical rule to solve the problem.
14) At one college, GPA’s are normally distributed with a mean of 3 and a standard deviation of 0.6. What percentage of students at the college have a GPA between 2.4 and 3.6?
A) 95.44% B) 99.74% C) 68.26% D) 84.13%
Solve the problem.
15) A state lottery involves the random selection of six different numbers between 1 and 25. If you select one six number combination, what is the probability that it will be the winning combination?
A) 1
127,512,000 B) 1
244,140,625 C) 1
720 D) 1
177,100
16) The heights of the adults in one town have a mean of 67.4 inches and a standard deviation of 3.4 inches. What can you conclude from Chebyshev’s theorem about the percentage of adults in the town whose heights are between 57.2 and 77.6 inches?
A) The percentage is at least 88.9% B) The percentage is at most 99.7%
C) The percentage is at least 99.7% D) The percentage is at most 88.9%
Find the standard deviation for the given data. Round your answer to one more decimal place than the original data.
17) 22, 29, 21, 24, 27, 28, 25, 36
A) 4.2 B) 2.8 C) 4.8 D) 1.6
Solve the problem. Round results to the nearest hundredth.
18) The mean of a set of data is 4.14 and its standard deviation is 4.34. Find the z score for a value of 11.52.
A) 2.00 B) 1.87 C) 1.70 D) 1.53
Find the indicated probability.
19) What is the probability that 6 rolls of a fair die will show exactly three fives?
A) 0.0536 B) 0.0322 C) 0.0154 D) 0.0286
Find the indicated percentile, decile, or quartile.
20) The test scores of 32 students are listed below. Find 23d percentile.
32 37 41 44 46 48 53 55
56 57 59 63 65 66 68 69
70 71 74 74 75 77 78 79
80 82 83 86 89 92 95 99
A) 7.36 B) 55 C) 8 D) 53
Find the indicated probability.
21) A sample of 100 wood and 100 graphite tennis rackets are taken from the warehouse. If 8 wood and 17 graphite are defective and one racket is randomly selected from the sample, find the probability that the racket is wood or defective.
Answer ______________________
Use the given data to construct a frequency table.
22) A medical research team studied the ages of patients who had strokes caused by stress.
The ages of 34 patients who suffered stress strokes were as follows.
29 30 36 41 45 50 57 61 28 50 36 58
60 38 36 47 40 32 58 46 61 40 55 32
61 56 45 46 62 36 38 40 50 27
Construct a frequency table for these ages. Use 8 classes beginning with a lower class limit of 25.
Age Frequency
Find the indicated probability.
23) Find the probability of correctly answering the first 2 questions on a multiple choice test if random guesses are made and each question has 6 possible answers.
Answer ______________________
Solve the problem.
24) Approximately 65% of Chemistry students do their homework independently and on time?
In a chemistry class of 45 students, what is the probability that at least 35 students do their homework on time? State your answer to four decimal places.
Answer ______________________
State the sample space of possible outcomes for given problem.
25) Flip a coin three times.
Answer ______________________
