Download Math 110 Exam I

Math 110 Exam I
Past Exam Problems
1) Give an example of a continuous random variable.
2) IQ scores have a mean 100 and a standard deviation 16. Is an IQ score of 150 unusual?
3) Describe the sampling method used for each of the following:
a) From a list of 1000 ID numbers, choose a starting point by chance and then select every 10th ID
number.
b) 25 students are randomly selected from each grade level at a high school and surveyed about
their study habits.
4) The following data points are the test scores for a statistics class. Compute each of the following:
a)
P60
b) percentile of 13
c)
Q3
50 18 13 80 14 38 11 13 39 41
5) Are the following events a) independent b) disjoint? Justify your answer.
i) A: Selecting a person under 24 years old.
B: Selecting a person earns less than $20,000.
ii)
A: Selecting a male college student.
B: Selecting an adult man who works part-time.
6) At a small four-year college, all freshmen and sophomores are required to enroll in exactly one of the
following math classes: Math I, Math II or Math III. 200 freshmen and 450 sophomores are enrolled in
Math I, 350 freshmen and 400 sophomores are enrolled in Math II, and 250 freshmen and 300
sophomores are enrolled in Math III. a) Find the probability that a randomly chosen person is either a
freshman or enrolled in Math II. b) Find the probability that a randomly chosen person is taking Math I
given that he is a sophomore. c) When two students are chosen randomly, find the probability that they
are both freshmen and enrolled in Math I.
7) Through a manufacturing error, three cans marked “regular soda” were accidentally filled with diet
soda and placed into a 12-pack. When 2 cans are randomly chosen from the 12-pack without
replacement, find the probability of selecting a) two diet soda cans b) at least one diet soda c)exactly
one diet can.
8) A student experiences difficulties with malfunctioning alarm clocks. Instead of using one alarm clock,
he decides to use four. a) What is the probability that at least one alarm clock works correctly if each
clock has a 95% chance of working correctly? b) Find the probability that all of the clocks work
correctly. c) Find the probability that exactly three clocks work correctly.
9) A) When 5 babies are born, how many different gender sequences are possible? B) When 5 dice are
rolled, how many different outcomes are possible?
10) Compute the sample standard deviation: x = 1, 5, 8, 10 and the saple z-score of 10.
11)
Class
1.3-1.8
1.9-2.4
2.5-3.0
Frequency
2
12
6
a) Construct a relative frequency histogram
b) Compute the mean.
12) A batch of 50 calculators contains 5 defective calculators. When 3 calculators are selected at random
without replacement, what is the probability that a) all 3 calculators are defective b) at least one
calculator is defective, c) exactly two calculators are defective.
13) A magazine editor must choose 4 short stories for this month’s issue from 30 submissions. In how
many ways can the editor choose this month’s stories?
14) When five cards are drawn without replacement from a standard deck of 52 cards, find the probability
of obtaining exactly two kings.
15) Find the probability of selecting six winning numbers from 1 to 51.
16)
5, 9, 11, 14, 15, 17, 19, 23
a) Construct a frequency table using 5 as a starting point and class width of 7.
b) Construct a relative frequency histogram.
17) The following table data are characteristics of voting-age population regarding a recent election.
A Voted
B Males
B
Females
230
370
A
Did not vote
400
240
a) When a person is chosen at random, find the probability that the person did not vote, given that the
person is female.
b) When a person is chosen at random, find the probability that the person is female or voted.
c) When two people are chosen without replacement, find the probability of choosing two males
who voted.
18) 65 percent of residents in Greenville, PA are women. When 3 people are selected at random, find the
probability that a) at least one woman is selected b) none of the women is selected.
19) A) How many different area code 909 telephone numbers are possible? (you may assume that the first
digit of any phone number cannot be 0 or 1) B) Suppose a computer requires eight characters for a
password. The first three characters must be letters and the remaining five characters must be
numbers. How many passwords are possible?
20) Use the following frequency table to compute the mean:
Class
Frequency
0-4
6
5-9
7
21) A case of 10 bottles of wines contains 7 Merlot and 3 Cabernet. 3 bottles are chosen randomly WITH
replacement .
a) Find the probability that all 3 are Merlot. b) Find the probability that at least one is Merlot.
22) (2 points each) The average annual salary of Mt. SAC professors is $23,000.
a) Identify the population of the study.
b) Is $23,000 a parameter or statistic? Justify your answer.
c) Is $23,000 quantitative or categorical? Justify your answer.
23) Identify the sampling method used:: An social scientist is studying the effect of education on
salary and conducts a survey of 200 selected workers from each of the following categories:
less than a high school degree; high school degree; more than high school degree.
24) A researcher wants to investigate a correlation between meth use as a teen, and having marital
problems as an adult.
a) Indentify the population.
b) Which type of study is more appropriate: observational or experimental? Justify your answer
25) A study finds that the average cost of a gym membership in the US is $34 a month. Is $34 a parameter
or statistic? Justify your answer.
26) ( 3 points) Locate the mean, median and mode for the graph shown below:
Median ___
Mode_______ Mean_______
27) In how many ways can 8 people be arranged for a photograph session?
28) Identify the sampling technique used in the following study: In an effort to determine customer
satisfaction, Delta Airline randomly select 59 flights during certain week and survey some passengers
on those flights.
29) Suppose 40 cars start at a car race. How many ways can the top three cars finish the race?
30) A machine has a probability 0.03 of producing a defective golf ball. When 3 golf balls are manufactured
by this machine, find the probability that none of the golf balls is defective.
31) When three people are selected randomly, find the probability that their birthdays are all different.
32) The mean credit card debt for college students in the U.S. is $2100. Is 2100 a parameter or statistic?
Justify your answer.
33) How many different outcomes are possible when 3 dice and 2 coins are tossed?
34) Determine whether the following events are a) disjoint b) independent or dependent
A) Randomly selecting a male driver who weighs less than 140 pounds
B) Randomly selecting a male driver who has more than three speeding tickets.
35) (2 points each) Fill in the blank
a) If the distribution of a random variable is skewed to the right, the value of the mean is ______________
than the value of the median.
b) A ______________sample is the one in which respondents themselves decide whether to be included in
the sample and often reflects opinions of those with strong interests.
c) A binomial experiment requires that a __________ number of trials are performed, each trial has two
possible outcomes, the trials are ________________, and the probability of success remains the same.
d) The running time of a randomly selected movie is an example of ___________________ random variable.
e)
A researcher divides the population into three groups by their incomes: low, medium, and high.
Then he selects a random sample from each group. He used __________________________ sampling
method in this study.
f)
There are ______________ possible outcomes when a coin is tossed 6 times.
36)
a) A correlation between two variables does not imply one is a ___________ of the other.
b) In a ________________
study, data are collected from the past by going back in time.
c) In _________________ sampling method, every nth item is selected.
d) A _____________________ is a numerical measurement of a population.
e) A value is unusual if its z-score is more than ________ or less than _________.
f)
To apply the 68-95-99.7 principle, the distribution of the variable must be _________-___________.
g) When the effects of one factor cannot be separated from the effects of some other factors, the
effects are said to be_______________________.
h) When the researcher controls the assignment of members to different groups, the study is
__________________ study .
i)
True/false: If the distribution is skewed, we are usually better off with the mean than the median.
37) A recent study finds that 34% of adult American males are obese. Based on this study, what is the
probability that all 3 are obese when 3 adult American males are chosen randomly?
38) How many different social security numbers with the last four digits 1234 are possible?
39) A random sample of 300 Mt. SAC students majoring in math or psychology is selected. These students
were asked whether they are happy with their majors. The results are summarized in the following
table:
Happy
Unhappy
Math
100
20
Psychology
10
170
a) When one student is selected randomly from this group, find the probability this student is happy
with the choice of major given that the student is a math major.
b) When two students are chosen randomly, find the probability that they are both math majors.
40) (3 points) Choose the data set with a smaller standard deviation. Justify your answer.
A) The IQ scores of all Mt. SAC students.
B) The IQ scores of all Mt. SAC students majoring in mathematics.
41) (3 points) I
A retail store manager wants to conducts a study regarding the shopping habits of his customers. He selects
60 customers who enter his store on a Tuesday morning.
a) Identify the sampling method used in the study.
b) Is the sample representative of the population? Justify your answer.
42) ( 3 points) Determine whether the following events are dependent or independent. Justify your answer.
A : You eat fast food for dinner every night.
B: You gain weight.
43) Researchers wanted to determine if there is a correction between coffee consumption and the
occurrence of skin cancer on women. The researchers looked at 10,000 women and asked them to
report their coffee drinking habits. Is this an observational or an experimental study? (note: the groups
are coffee drinkers and non coffee drinkers). Justify your answer.
44) A study finds the average amount spent per gift for Mom on Mother’s Day is $45. Is $45 a statistic or
parameter? Justify your answer.
45) A study finds male children born to women who smoke during pregnancy run a higher risk of criminal
behaviors that last into adulthood than male children born to women who do not smoke.
a) (2 points) Determine whether this study is observational or experimental.
b) (2 points) Can we say smoking during pregnancy is responsible for higher criminal behavior?
Justify your answer.
46) 3 points) Identify the sampling method used: A random sample of 100 people from each of five different
age categories was selected.
47) From a group of 80 people, a jury of 12 people is selected. In how many ways can a jury of 12 people be
selected?
48) In a group of 10 persons, 4 have a Type A personality and 6 have a type B personality. When three
people are chosen randomly without replacement, find the probability that at least one person has a
Type A personality
49) (3 points) Determine whether the following events are independent or dependent. Justify your answer.
A: USC wins a football game.
B: You get a high score on a statistic exam
50) A question is posed on the ESPN website asked visitors to the site to say whether they thought that
marijuana should be legally available for medical purposes.
a) Identify the population.
b) Identify the sample.
c) Is the study cross sectional, prospective, or retrospective? Justify your answer.
d) Is the sample likely to be representative of the population? Justify your answer.
51) A department store mails a customer satisfaction survey to people who make credit card purchases at
the store. Surveys are mailed to 1000 o these people chosen at random, and 80 people returned the
survey.
c) Indentify the population.
d) What is the sample?
e) Is the sample likely representative of the population? Justify your answer.
52) The MLB reports that the average annual salary of the major league baseball players is $5,100,000.
a) Identify the population of the study.
b) Is $5.1 M a parameter or statistic? Justify your answer.
53) (3points) Determine whether the following variable is discrete or continuous. Justify your answer.
Lifetime of a TI 84 calculator
54) (2 points) Classify the following variable as quantitative or categorical:
Colors of baseball uniforms .
55) In a recent study, subjects were randomly assigned to two groups, and one group was given an herb
and the other group a placebo. After 6 months, the numbers of respiratory tract infections each group
had were compared.
a)
Is the study cross sectional, prospective, or retrospective? Justify your answer.
b) Is the study observational or experimental? Justify your answer.
56)
a ) True/false In an observational study, a definite cause-and-effect cannot be shown
c) In an experimental study, the subjects in the _______________ group receive a dummy treatment,
enabling the researchers to control for the placebo effect.
1)
2)
3)
4)
5)
Answers (Warning: I have minor brain damage. Some answers may not be correct.)
x = height of a man is an example of a continuous random variable.
Yes, its z-score is 3.13, which is more than 2.
A) systematic b) stratified
a) 38.5 b) P10 c) 41
i) a) They are not disjoint. There are many people under the age of 24 who make less than $20,000. b)
They are dependent. The probability that a person under 24 makes under $20,000 is much higher than
that of the general population.
ii) a) They are not disjoint. B) They are dependent as college students are more likely to work part-time
than the general population
1200
1950
6) a)
b)
450
1150
c)
Math I
200
450
Freshmen
Sophomores
7) a)
8) a)
200 199

1950 1949
Math II
350
400
3 2
9 8
b) 1 
c)
12 11
12 11
1  0.054
b)
0.95 4
3
c)
Math III
250
300
C1 9 C1
12 C 2
4(0.95) 3 (0.05)
Use the binomial formula or observe there are 4
possible sequences for this event.
9) A) 32 B)
6 5 (make 5 boxes.
10)
x
X
1  5  8  10
6
4
(x  x)2
1
(1  6) 2  25
5
(5  6) 2  1
8
(8  6) 2  4
10
(10  6) 2  16
There are 6 possible outcomes for each die.)
  46
46
x  x 10  6

 1.0
 3.9 , the sample z-score of 10 is z 
s
3.9
3
11)
Frequency
Class Bd
Midpoint
Freq(mid)
1.3-1.8
2
1.25-1.85
1.55
3.1
1.9-2.4
12
1.85-2.45
2.15
25.8
2.5-3.0
6
2.45-3.05
2.75
16.5
  45.4
N=20
The mean is 45.4/20=2.27
12) a)
(
13)
30
14)
(4
5 4 3
)( )( )
50 49 48
C4
C 2  48 C3
)
52 C 5
(
1 (
45 44 43
)( )( )
50 49 48
c)
3(
5  4  45
)
50  49  48
or
5
C 2  45 C1
50 C 3
( without replacement, the order does not matter)
kings.
15) a)
b)
1
)
51 C 6
You must choose 2 kings and 3 non-kings. There are four kings and 48 non-
16) Use class boundaries for the histogram
Frequency Class
Boundaries
5-11
3
4.5-11.5
12-18
3
11.5-18.5
19-25
2
18.5-25.5
240
610
17) a)
b)
Class
midpoint
8
15
22
Relative
frequency
37.5%
37.5%
25%
230 229
840
c)

1240 1240 1239
18) a)for at least one, use complement: first find the probability that all three are men. Then
subtract from 1.
19)
8  10 6
1  0.353
b) This is just the probability of all three are men.
(0.35) 3
(the order matters, with replacement: use the counting principle. Make 7 boxes, the
first digit must be 2-9 so there are only 8 choices. ) b)
26 3  10 5
20)
Frequency
0-4
5-9
6
7
  13
Class
midpoint
2
7
(Class mid)(freq)
12
49
  61
The mean is 61//13 = 4.7
21)a)
 7
 
 10 
3
b)
 3
1  
 10 
3
22) ) a) all Mt, SAC professors. B) $23,000 is more likely to be a parameter. The salaries of Mt. SAC professors
are available to the public. C) quantitative: salaries can be measured numerically.
23) This is stratified: a random sample is taken from each subgroup.
24) A) all married adults B) observational; it is unethical to make people in the control group use meth.
25) Statistic: it is impossible to survey ALL the gyms in the U.S.
26) Medan: A mode: B mean C: The mode is the highest point of the graph. The mean is to the right of the
median (the distribution is skewed to the right)
27) This is permutation: the order matters (different order produces different pictures and without
replacement).
8(7)(6)…(1) = 40320 or
8
P8
28) Cluster : here the subgroups are the flights. Random samples are taken from some of the subgroups, not
all.
29)
30)
40
P3
or
(0.97)
3
(40)(39)(38)
use permutation since the order matters and without replacement
: they can be treated as independent events, so make three boxes. The probability of a golfball
not defective is 1-0.03-=0.97. Then multiply this number three times).
31)
(
365 364 363
)(
)(
)
365 365 365
(the birthday of the first person can be any day, thus there are 365 choices. The birthday
of the second person must be different from the first person, thus there are 364 choices. The third birthday
must be different from the first two. There are 363 choices)
32) this is a statistics. It must be computed from the sample as the population is too large.
33) use the counting principle: the order matters, with replacement: 6 possible outcomes for a die, 2 for a coin,
(6) 3 (2) 2
34) They are not disjoint. They are independent: The weight of a person does not affect his driving skills.
35) a) greater b) voluntary c) fixed, independent d) continuous e) stratified f) 64
36) a) cause b) retrospective c)systematic d) parameter e) 2.0, -2.0 f) bell-shaped g) confounded h) experimental
i) true (since the median is not affected by a small number of extreme values)
37) (0.34)3
38) (10)5 (make nine boxes: the first five digits have 10 possible choices each, the last four have one choice
each)
39)A)
100
120
b)
120 119

300 299
40) B has a smaller standard deviation. Math majors are all intelligent with high IQs.
Thus the variation of IQ
scores among the math majors would be smaller.
41)a) Convenience. B) No. Many customers cannot visit the store on Tuesday due to obligations such as work,
school, going to the gym.
42) They are dependent. People with unhealthy diets are more likely to be obese.
43) This is observational. It is impractical to place the subjects into coffee drinking group and noncoffee
drinking group since we cannot make people drink coffee.
44) It is a statistic. $45 cannot be a parameter since we cannot survey all Americans
45)
a) It is observational. We cannot make people smoke cigarettes.
b) No. In observational studies, effects are often confounded. For example, smokers tend to have
lower educational attainments and lower SES than nonsomkers and children from low SES families
exhibit higher criminal behavior.
46) This is stratified since SRS is taken from each subgroup.
47) First observe that the selections are made without replacement and the order does not matter.
So use combination:
80
C12
48) P(at least one type A) is 1 – P(all type B)  1 
6 5 4
 
10 9 8
49) For most people, they are independent since the outcome of an football game has no effect on
the exam performance. For me, they are dependent. I get depressed when USC football team loses
and I can’t think clearly.
49) a) all EPSN site visitors b) people who responded to the survey. C) This is a cross sectional
study. d) No. This is a voluntary response sample. People who respond are more likely to have
strong opinions on the topic.
50) A) People who make credit card purchases B) 80 people responded to the survey C) No. The
response rate is too low: it may be that the survey was too long.
51) a) all major league baseball players. b) it is more likely to be a parameter: the salary data are
stored in a database and the average salary can easily be computed.
52) It is continuous: the lifetime of a calculator can be any number.
53) It is categorical: it can be classified: red, blue, grey, etc but cannot be measured.
54)
a) The study is prospective. The study participants meet with the researchers at least twice. Once when the
study began, then again six month later.
b) It is an experimental study: the researchers randomly assign people into the herb group and the placebo
group.
55) a) true b) control
(The following formulas will be provided
2
x  mean 2  ( x  x )
z
s 
std .dev
n 1
,
mean 
 (frequency) (midpoint)
n