Download MATH-138: Objectives

MATH 138
Statistics Classroom Handouts
Unit 2
Name: _________________________
Course Number: ________________
Instructor: _____________________
Rev. 11/2015
MATH-138: Objectives
Intro Stats, by Richard D. De Veaux, Paul F. Velleman, and David E. Bock
Unit II
The student will be able to:
Chapter 9: Understanding Randomness
1.
2.
3.
Explain what it means to be random.
Accurately model a situation through simulation.
Discuss the results of a simulation study and draw conclusions about the questions
being investigated.
Chapter 10: Sample Surveys
4.
5.
6.
7.
Classify samples, populations, parameters and statistics.
Identify and explain the different sampling techniques: census, simple random
samples, stratified, cluster, and systematic.
Identify and explain possible sampling mistakes: convenience, voluntary
response, bad sampling frame, and under coverage.
Recognize different types of bias.
Chapter 11: Experiments & Observational Studies
8.
Recognize the difference between an experiment and an observational study.
Chapter 12: From Randomness to Probability
9.
10.
11.
12.
State the definition of trial, outcome, sample space, event and P(A).
Apply the Law of Large Numbers.
Recognize when events are disjoint and when events are independent.
State the basic definitions and apply the rules of probability for disjoint
and independent events.
Chapter 13: Probability Rules!
13.
14.
15.
Apply the General Addition and General Multiplication rules.
Compute conditional probabilities and use the rule to test for independence.
Use a tree diagram to understand conditional probabilities and to reverse
conditioning.
Chapter 14: Random Variables and Probability Models
16.
17.
18.
19.
20.
21.
22.
Distinguish between discrete and continuous random variables.
Find the probability model for a discrete random variable.
Find and interpret in context the mean (expected value) and the standard deviation
of a random variable.
Determine if a situation involves Bernoulli trials.
Know the appropriate conditions for using a Binomial model.
Calculate binomial probabilities.
Find and interpret in context the mean and standard deviation of a Binomial
model.
Unit 2: Technology Instructions
Technology Instructions for Unit 2
Unit 2
Chapter 9: Understanding Randomness
1) Are the random numbers generated by your calculator truly random?
a) Reset your memory.
b) Generate some random numbers using RandInt(0, 9, 5).
c) Compare your random numbers to your neighbor. What do you observe? Why?
2) 57 students are in a lottery for the spacious triple dorm room. 20 were from the varsity team
and all three winners were from this team. How likely is this? Was it rigged?
(De Veaux et al., 2014)
Plan → Simulation
– Components: Selection of the students.
– Outcomes: Generate numbers from 1 to 57. 1-20 will represent the team members.
– Trial: Pick the first three distinct numbers.
– Response Variable: Yes if all three are 1-20
a) Run this simulation with 10 trials.
b) Based on your simulation, estimate the
probability of all three winners being from the
varsity team.
c) Based on your simulation, estimate the
probability of at least one winner being from
the varsity team.
Trial Components
1
2
3
4
5
6
7
8
9
10
Outcome
Unit 2
3) Many couples want to have both a boy and girl. If they decide to continue to have children
until they have one child of each sex, what would the average family size be? (De Veaux et
al, 2014)
Assume that boys are girls are equally likely. Run a simulation with 20 trials
a) How are you using random numbers to simulate
the events of having a boy and having a girl?
b) Based on your simulation, what would the
average family size be?
c) Based on your simulation, what would be the
probability of a couple having exactly one boy
and one girl?
d) Based on your simulation, what would be the
probability of a couple having 4 or more
children?
4) You take a quiz with 6 multiple choice questions.
After studying, you estimate that you have a 70%
chance of getting any question right.
What are you chances of getting them all right?
Run at least 20 trials.
Trial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Components
Outcome
Trial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Components
Outcome
Unit 2
Extra Practice / Homework
5) You are taking a multiple choice quiz which consists of 6 questions. Each question has five
possible answers to choose from of which only one is correct. Answer the following
questions
a) What is your chance of choosing the correct answer for any particular problem? (express
as a decimal rounded to two decimal places.)
b) Explain how you could use your calculator to generate and use random numbers to
simulate guessing on the each question of the quiz.
c) Conduct a simulation with 20 trials indicating what random numbers were used and how
many correct answers you got for each trial.
d) Use your simulation to estimate for the
probability of getting exactly one correct answer
on the quiz if you guessed at all questions.
e) Use your simulation to estimate for the
probability of getting a score of less than 33.33%
(fewer than 2 of 6 correct) on the quiz.
Trial
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Components
Outcome
Unit 2
Chapter 10 – Sample Surveys
The Three Basic Ideas of Sampling:
(1) Examine a Part of the Whole
(2) Randomize
(3) It’s the Sample Size
1) A random sample of 1000 community college students was asked the following question:
“On average, how much time per day do you spend on social media of any kind?”
a) What is the sample?
b) What is the population?
c) What is the sample statistic?
d) What is the population parameter?
2) Identify the type of sampling as census, simple random sample, stratified, cluster, multistage, or systematic.
a) Use the company e-mail directory to contact 150 employees from among those employed
for less than five years, 150 from among those employed 5 – 10 years, and 150 from among
those employed for more than 10 years.
b) Use the company e-mail directory to contact every 50th employee on the list.
c) Select several divisions of the company at random. Within each division, draw a simple
random sample of employees to contact.
d) Select several divisions of the company at random and then contact every member of the
division.
e) Randomly select 450 employees from the company e-mail directory.
f) Require every employee to be contacted.
Unit 2
General Vocabulary
• The ________________consists of all items of interest (people, places, firms, etc.)
• A
is a (hopefully representative and random) subset of the
population.
• The ________________is the set of items belonging to the population from which
samples will be drawn.
• A numerical value/characteristic of a population is called a _______________
these are usually unknown.
• A numerical value/characteristic of a sample is called a ________________.
Sampling Methods
• A special sample which consists of the entire population: __________________.
• In a ________________members from the population are selected in such a way that
each individual member has an equal chance of being selected.
• A __________________ of n subjects is selected in such a way that every
possible sample of size n has the same chance of being chosen.
• Sometimes the population is first sliced into homogeneous groups, called strata,
before the sample is selected. Then simple random sampling is used within each
stratum before the results are combined. This common sampling design is called
__________________.
• The sampling method in which the population is split into similar heterogeneous
groups then one or a few of these are selected at random and a census is performed
within each of them: __________________.
• Sampling schemes that combine several methods are called__________________.
• In a __________________ every kth item of the sampling frame is selected,
starting from a first element.
• In a __________________ , a large group of individuals is invited to respond, and all
who do respond are counted.
• In ____________________ , we simply include the individuals who are
convenient.
Bias
• The bias in which some portion of the population is not sampled at all or has a
smaller representation in the sample than it has in the population:
__________________.
• The bias introduced by inviting free responses from a large group of people and
which tends to favor those with strong opinions: __________________.
• The bias introduced when people cannot or will not respond to a survey:
__________________.
• The bias introduced by anything in the survey design that influences the
responses: __________________.
Unit 2
Extra Practice / Homework
3) A tax auditor has a pile of federal tax returns and she has been directed to randomly select 20
of these returns for a special audit.
Describe how systematic sampling could be used.
4) A recent radio station asked listeners to go to their website and vote for their favorite singing
star. A responder would be randomly chosen and would receive two tickets to the
performance of their choice.
What type of sampling is used for the vote?
5) A community college is selected at random from all US community colleges and the GPAs
and ages of all students from that college are examined.
a) What type of sampling was done?
b) What is the population of interest for this study?
c) If the average GPA is computed for this sample would it be considered a statistic or a
parameter?
d) Suppose instead the researchers wanted to implement a Simple Random Sample of
community college students, how might they do this?
6) Explain the difference between cluster sampling and stratified sampling.
Unit 2
Chapter 11 – Experiments and Observational Studies
1) What is the difference between an experimental study and an observational study?
2) What type of study is needed to establish causality?
3) What is the difference between a prospective study and a retrospective study?
Unit 2
Chapter 12 – From Randomness to Probability
1) You roll a single die. What is the probability that you roll a five (call this A)?
a) What represents the trial?
b) What is the outcome?
c) What is the sample space?
d) What is P(A)?
e) What is the P(you roll a 5 or 6)
f) What is the P(you roll a 5 and a 6)
2) Now suppose you roll your die twice.
a) Are the trials independent? Why or why not?
b) What the P(you roll a 5 and then a 6)
c) How is this use of “and” different from the use in example 1?
3) Are the following events are independent? If so, find the P(A and B)
a) A = Rolling a six with one die and then B = rolling a six a second time with the same
die.
b) A = Selecting an Ace (there are 4 in 52 cards) and then B = selecting another Ace
without replacement
c) A = Selecting a Face Card (there are 12 in 52 cards) and then B = selecting another Face
Card with replacement
Unit 2
4) There are 35 scores on a test: 5 As, 9 Bs, 14 Cs, 4 Ds and 3 Fs.
What is the probability a certain student passes with at least a C?
5) You roll a fair die three times. What is the probability that:
a) you roll all 6s?
b) you roll all odd numbers?
c) you roll at least one 5?
6) In a large stat class, the professor reports that 55% of the students enrolled have never taken a
calculus course, 32% have taken only one semester of calculus, and the rest have taken two
or more semesters of calculus. The professor randomly assigns students to groups of three to
work on a project. (De Veaux et al., 2009)
What is the probability that the first groupmate you meet has studied
a) 2+ semesters of Calculus?
b) Some Calculus?
c) No more than one semester of Calculus?
What is the probability that, of your two groupmates
d) Neither has studied Calculus?
e) Both have studied at least one semester of Calculus?
f) At least one has had more than one semester of Calculus?
Unit 2
Extra Practice / Homework
7) Consider the following distribution of blood types in the U.S.
Blood type percent
A
40
B
11
O
45
AB
???
a) What percent of the U.S. population have type AB?
b) What percent of the U.S. population does not have Type B blood?
c) If one donor is randomly selected, what is the probability that the person has type A or B?
d) If one donor is randomly selected, what is the probability that the person has type AB and
type B? Show your work.
e) If you examine one person, are the events that the person is Type A and that the person is
Type B disjoint, independent, or neither?
f) If you examine two people, are the events that the first is Type A and the second Type B
disjoint, independent, or neither?
g) If you examine two people, what is the probability that the first person is Type A and the
second person is not Type O?
h) Among three potential donors, what is the probability that all three have Type A blood?
i) Among three potential donors, what is the probability that at least one person has type A
blood? Show your work.
Unit 2
Chapter 13 —Probability Rules!
1) A Survey asked students if they were currently in a relationship and if they were involved in
sports? (De Veaux et al., 2014)
Results
• 33% are in a relationship.
• 25% are involved in sports.
• 11% answered “yes” to both.
Venn Diagram:
Find the following probabilities:
a) P(Relationship or Sports) =
b) P(Relationship but not Sports) =
c) P(neither in a relationship nor Sports) =
2) Records show that 80% of the students preparing for the high school equivalency exam need
help in math, 70% need help in English and 55% need work in both. If a student is randomly
chosen find the probability that the person needs help in math or English.
General Addition Rule
Unit 2
3) A junk box in your room contains a dozen batteries, five of which are totally dead. You start
picking batteries one at a time and test them. (De Veaux et al., 2014)
Are you sampling with or without replacement?
Find the probability that:
a) The first two you choose are both good.
General Multiplication Rule
b) The first 4 you pick are all good.
c) At least one of the first 3 works.
d) You have to pick 5 batteries before you get a good one.
4) Does it pay to ask for a raise? A national survey showed that, of the women surveyed, 24%
had asked for a raise and of the women who asked for a raise, 45% received the raise. If a
woman is randomly selected from this group find:
a) P(woman asked for a raise)
b) P(woman received a raise given she asked for one)
c) P(woman asked and received a raise)
Unit 2
5) In its monthly report, the animal shelter states that it currently has 24 dogs and 18 cats
available for adoption. 8 of the dogs are male and 6 of the cats are male.
If you choose to adopt two pets, what is the probability that:
a) They are both cats?
b) Exactly one is a cat
c) At least one is a cat?
Find the probability that a randomly selected animal is:
d) a male, given that it is a cat.
Conditional Probability Formula
e) a cat given that it is a male.
f) a female, given that it is a dog.
g)Are the events of selecting a male and selecting a cat independent?
Independence Test
Unit 2
6) Recall a Survey asked students if they were currently in a relationship and if they were
involved in sports?
• 33% are in a relationship.
• 25% are involved in sports.
• 11% answered “yes” to both.
a) What is the probability that a randomly selected student said they were in a relationship
given that they were involved in sports?
b) Are the events of a student being in a relationship or being involved in sports
independent?
7) 56% of the students at a college live on campus; 62% of the students have a meal program;
and, 42% live on campus and have a meal program. Is “living on campus” and “having a
meal program” independent? Explain your reasoning.
Unit 2
8) A study was undertaken at a certain college to determine what relationship, if any, exists
between mathematics ability and interest in mathematics. The ability and interest for 150
students were determined, with the results in the following table.
INTEREST
ABILITY
Low
Average
High
Below grade level
40
8
11
Grade level
15
17
19
Above grade level
5
10
25
If one of the participants in the study is randomly chosen, what is the probability:
a) of selecting a person who has low interest in mathematics?
b) of selecting a person with grade level ability?
c) that the person has high interest or below grade level ability?
d) that the person has low interest and has grade level ability?
e) that the person has a high interest given that the person has grade level ability?
f) that the person has above grade level ability given that the person has a high
interest?
g) that the person has grade level ability although he came from the low interest group?
h) that the person has low interest if you already know he has grade level ability?
i) Are the events in h independent? Are they disjoint?
Unit 2
9) Reversing the Conditioning -- An airline offers discounted “advance-purchase” fares to
customers who buy tickets more than 30 days before travel and charges “regular” fares for
tickets purchased during those last 30 days. The company has noticed that 60% of its
customers take advantage of the advance-purchase fares. The “no-show” rate among people
who paid regular fares is 30%, but only 5% of customers with advance-purchase tickets are
no shows. (De Veaux et al., 2014)
a) Represent the following situation using a tree diagram.
b) What percent of all ticket holders are no-shows?
c) What’s the probability that a customer who didn’t show had an advance-purchase ticket?
d) What’s the probability that a customer with an advance-purchase ticket did not show?
e) What’s the probability that a customer has a regular fare ticket and showed for the flight?
f) Is being a no-show independent of the type of ticket a passenger holds?
Unit 2
Extra Practice / Homework
10) In a stat class, there are 18 juniors and 10 seniors; 6 of the seniors are females and, 12 of the
juniors are males.
a) Complete the following contingency table for this
scenario.
If a student is randomly chosen from this group:
b) What is P(student is a junior)?
Male Female Total
Junior
Senior
Total
c) What is P(student is a male and a junior)?
d) What is P(student is a male or a senior)?
e) What is the probability the student is a male if you know the student is a junior?
f) Are the events being a junior and being a male disjoint? (You must supply a reason.)
g) Are the events being a junior and being a male independent events? (You must supply a
reason and show your formula used.)
11) Suppose you randomly select three students for a special assignment, selecting them one at a
time. What is the probability that your first two selections are juniors and the third is a
senior?
12) HIV Testing: In July 2005, the journal Annals of Internal Medicine published a report on the
reliability of HIV testing. Results of a large study suggested that among people with HIV,
99.7% of the tests were (correctly) positive, while for people without HIV, 98.5% of the tests
were (correctly) negative. A clinic serving an at-risk population offers free HIV testing,
believing that 15% of the patients may actually carry HIV.
(Hint: a tree diagram may be helpful)
a) What is the probability that a person will test positive for HIV?
b) What is the probability that a person will test negative for HIV?
c) What is the probability that a person testing negative is truly free of HIV?
22
Unit 2
Chapter 14 – Random Variable and Probability Models
1) A game costs $5 to play. You draw a card from a deck of cards. If you draw the ace of hearts,
you win $100. For any other ace, you get $10 and for any other heart you get $5. If you draw
anything else, you lose. Find the expected winnings or losses for this game.
Using the TI Much of what you have to do for discrete random data is very much like what we
did to get the descriptive statistics for data involving frequencies only here we will use decimals
(probabilities).
2) A recent study involving households and the number of children in the household showed
that 18% of the households had no children, 39% had one, 24% had two, 14% had three, 4%
has four and 1% had five.
Find the expected value, mean number of children per household, and the standard deviation
of this distribution.
Place the values of the variable in L1: 0,1,2,3,4,5 and the associated probabilities in L2: .18,
.39, .24, .14, .04 and .01 STATCalc1:1-VarStats L1, L2
3) A day trader buys an option on a stock that will return $100 profit if it goes up today and
loses $200 if it goes down. If the trader thinks there is a 75% chance that the stock will go
up, what is his expected value of the option?
23
Unit 2
4) You roll a die. If it comes up a 6, you win $100. If not, you get to roll again. If you get a 6
the second time you win $50, if not, you lose.
a) Create a probability model for this game
b) Find the expected amount you’ll win
c) How much would you be willing to pay to play this game?
d) Find the standard deviation of the amount you might win
5) Your company bids for two contracts. You believe the probability that you get contract #1 is
0.8. If you get contract #1, the probability that you get contract #2 is 0.2, and if you do not
get contract #1, the probability that you get #2 will be 0.3.
1) Are the two contracts independent? Explain
2) Find the probability that you get both contracts
3) Find the probability that you get no contract
4) Let X be the number of contracts you get. Find the probability model for X
5) Find the expected value and standard deviation of X
24
Unit 2
Building the Binomial Model
What are Bernoulli trials?
Give some examples of Bernoulli trials:
6) Consider rolling a die 5 times and counting the number of 6’s rolled.
a) Are these Bernoulli trials?
b) Let’s call rolling a 6 “success”
c) Find the P(the first roll is a “success”)
d) Find the P(the 3rd roll is a “success”)… remember the trial are independent!
e) Find the P(exactly 6 successes)
f) Find the P(the first success occurs on the 5th roll)
g) Find the P(we get exactly 1 success)
The Binomial Model
Formula:
Mean: ______ Standard Deviation: ______
Calculator Functions:
P(X=x)
DISTRBinompdf(n, p, x)
P(X≤x)
DISTRBinomcdf(n, p, x)
25
Unit 2
The Binomial Distribution -- How to use your calculator to solve problems
Input parameters to calculator functions:
n=number of trials
p=probability of success on each trial
x=number of successes
binompdf(n,p,x)
– computes the probability of getting exactly x successes in n trials, if the probability of
success is p.
– For example, binompdf(10, .25, 3), gives you the probability of getting exactly 3
successes in 10 trials, if the probability of success is .25
binomcdf(n,p,x)
– computes the probability of getting x or fewer successes in n trials, if the probability of
success is p.
– For example, binomcdf(10, .25, 3), gives you the probability of getting 3 or fewer
successes (that is, 0, 1, 2, or 3 successes) in 10 trials, if the probability of success is p.
----------------------------------------------------------------------------------------------------------How to use these functions:
Let’s say we have Binomial distribution Binom(n,p), with probability of success p, and we’re
performing n trials.
Description
Diagram
Calculator
P(5 successes)
0 1 2 3 4 [5] 6 7 8 9, …
binompdf(n,p 5)
P(5 or less)
P(no more than 5)
P(at most 5)
0 1 2 3 4 5] 6 7 8 9, …
same as above
same as above
binomcdf(n,p,5)
P(less than 5)
0 1 2 3 4] 5 6 7 8 9, …
binomcdf(n,p,4)
P(more than 5)
0 1 2 3 4 5 [6 7 8 9, …
1 – P(5 or less)
= 1 – binomcdf(n,p,5)
P(5 or more)
0 1 2 3 4 [5 6 7 8 9, …
1 – P(less than 5)
= 1 – binomcdf(n,p,4)
P(no less than 5)
P(at least 5)
same as above
same as above
P(between 5 and 8)
(inclusive)
0 1 2 3 4 [5 6 7 8] 9, …
binomcdf(n,p,8)
- binomcdf(n,p,4)
26
Unit 2
Example Using the TI-83 for the Binomial
– Suppose a light bulb company has a 20% defective rate. Consider taking a sample of 6
bulbs.
– What is the probability of getting exactly 1 defective bulb in that group of 6?
o (Even though a defect isn't pleasant at times, it is considered a success in this
experiment since that is where our focus is!)
o If we computed this probability long hand we would do 6 nCr 1 (.20)1 (.80)5 OR
2nd Distr…(#0) for binompdf(…6…","… .2…","…1…")"..enter to get .393216
o binompdf gives you the probability at a particular x. The pdf must be followed by
n (total number of trials), p (probability of a success), x (number of successes you
are interested in) binmocdf, which we use next, will compute cumulative
probabilities.
– What is the probability of getting at most 2 defective light bulbs.
o This means P(0) + P(1) + P(2) OR use the cumulative binomial button
o 2nd Distr…#A for binom cdf( …6…","… .2…","… 2…")"…enter to get .90112
– What is the probability of getting at least two defective light bulbs?
o At least two means two or more which is the same as adding the probabilities of 2
to 3 to 4 etc… OR 1 minus the complement of "at least two" which is 1 minus the
cdf to 1
o 1 - binomcdf (6,.2,1) = .34464
1) What is the probability of getting from two to four defective light bulbs?
a) You could do the (pdf for 2) + (pdf for 3) + (pdf for 4)
b) or be a little creative and do binomcdf(6,.2,4) - binomcdf(6, .2,1) = .34304
27
Unit 2
7) Ten customers enter a clothing store during a one-hour period. Past experience has shown
that approximately 30% of the people entering the store make a purchase.
a) What is the probability exactly 6 of the 10 make a purchase?
b) What is the probability at least one customer makes a purchase?
c) What is the probability that 8 or fewer make a purchase?
d) What is the mean and standard deviation of the number of people who may make a
purchase?
8) A burglar alarm system has 6 fail-safe components. The probability of each failing is .05.
Find the probability that:
a) Exactly 3 fail
b) Fewer than 2 fail
c) At least one fails
d) Between 2 and 5 fail
9) Thirty percent of the students at the University attended Catholic high schools. A random
sample of 50 of this year’s students has been taken.
What is the probability that:
a) Exactly 10 of the students selected attended Catholic high schools?
b) 20 or more of the students attended Catholic high schools?
c) The number of students who attended Catholic high schools was between 10 and 20
inclusive?
28
Unit 2
Extra Practice / Homework
10) A man pays $160 a year for a life insurance policy with coverage of $145,000 payable to his
survivor upon his death. The probability that he will live through the year is 0.9992
a) Supply the probability table for this problem. (In this case the outcomes X are the
possible payouts (positive or negative) the man might receive. Your textbook has similar
problems if you need to see an example.)
b) Compute the expected value and standard deviation of the model.
c) Interpret the expected value in the context of the problem.
11) It has been reported that 5% of Americans are afraid of being alone in a house at night. We
are going to randomly select 20 Americans for this exercise.
a) Find the probability that exactly 5 people in the sample we chose are afraid of being
alone in a house at night. (Show the calculator or formula values that you used.)
b) Find the probability that at most 4 people are afraid. (Show how you obtained the
answer.)
c) Find the probability that at least three people are afraid. (Show your work again.)
d) Find the mean and the standard deviation of this distribution
29
Unit 2
Unit 2 Review
1) A basketball player makes 80% of her free throws. Suppose she wakes up every morning and
starts shooting free throws.
a) How could you use the random digits 0-9 to model this scenario?
b) On an “average” morning, on which free throw will she have her first miss?
Perform 20 trials.
2) Suppose you want to sample the gas mileage of cars driven in your community. Describe a
way that you could sample according to each of the sampling methods:
a) Census
b) Simple Random
c) Systematic
d) Stratified
e) Cluster
f) Voluntary Response
g) Convenience
30
Unit 2
3) Identify the type of sampling used:
a) All Math 138 students are required to take the Final Exam to assess their mastery of the
course content areas. The results of these exams are investigated to explore areas which
may need further attention in subsequent semesters.
b) A man inherits a house full of possessions from a relative. The house contains many
Chinese bowls, paintings, and old books. He is interested in getting a sense of what the
possessions might be worth so he picks a few bowls, a few paintings, and a few books to
bring with him to show an appraiser.
c) Your coworker is interested in changing primary care physicians. She asks you and your
fellow coworkers for their opinions on doctors in the area.
d) A list of students student IDs are randomly generated and those students with matching
ID numbers receive an email survey from the College. Students who do not respond to
the survey are sent reminder emails urging them to participate.
4) Suppose that 70% of job postings on a career website require applicants to have a college
degree, 50% of job postings require that an applicant have previous experience in the field.,
and that 35% of job postings require both than applicants have a college degree and
experience in the field.
Find:
a) The probability that a random job posting will require applicants to have a degree or
experience in the field.
b) The probability that a random job posting will require neither that applicants have a
degree nor that they have experience in the field.
c) The probability that a random job posting will require applicants to have a college degree
but not experience in the field.
d) The probability that a random job posting will require applicants to have a degree given
that the posting requires applicants to have previous experience in the field.
e) Are the events that job posting will require a college degree independent and that a job
posting will require previous experience independent? Show your test
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Unit 2
5) You roll a die. If it comes up 6, you win $100. If not, you get to roll again. If it comes up 6
the second time, you win $50. If not, you lose.
a) Create the probability model and find the expected amount you’ll win.
b) How much should you pay to play this?
6) Use the data below to answer question 7. This synthetic sample data (i.e. I made it up) shows
1,000 people who either smoked or didn’t smoke, and who either died of lung cancer or some
other cause of death. Suppose I randomly sample one person from this data.
Smoker
Non-Smoker
Lung Cancer Deaths
50
80
Non-Lung Cancer Deaths
150
720
a) What is P(Smoker)?
b) What is P(Non-Lung Cancer Death)?
c) What is the P(Smoker or Lung Cancer Death)
d) What is the P(Smoker and Lung Cancer Death)
e) What is P(Smoker│Lung Cancer Death)?
f) What is P(Non-Lung Cancer Death│Non-Smoker)?
g) Are smoking and lung cancer deaths independent? Explain how you know using
probability formulas.
7) Which of the following (if any) are NOT valid probability values?
a. 0.40
b. -0.20
c. 1.00
d. 0.99999
8) Which of the following (if any) are NOT valid probability values?
a. 0.00
b. 0.67
c. 0.80
d. 1.14
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Unit 2
9) Suppose a jar contains 40 red marbles, 40 blue marbles and 20 green marbles (100 marbles
total). If TWO marbles are drawn WITHOUT REPLACEMENT from the jar (that is, one
marble is drawn and NOT put back into the jar, and then another marble is drawn), what are
the following probabilities?
a) P(both are green)
b) P(neither are green)
c) P(first marble is red and the second marble is blue)
d) Given the first marble is red, what is P(second marble is red)?
e) Given the first marble is green, what is P(second marble is blue)?
10) A commuter must pass through 5 traffic lights on her way to work and will have to stop at
each one that is red. She estimates the probability model for the number of red lights she stop
for as shown below.
X=# of red
P(x)
0
0.05
1
0.25
2
0.35
3
0.15
4
0.15
5
0.05
How many red lights should she expect to stop for each day?
11) Suppose that 33% of all college students smoke cigarettes. If 15 are selected at random:
a) What is the probability that exactly 10 smoke?
b) What is the probability that at most 5 of the students smoke?
c) What is the probability that between 5 and 10 students smoke (inclusive)?
12) In the Presidential election of 1968, Richard Nixon won 43.4% of the popular vote and
Hubert Humphrey won 42.7% of the popular vote. In a Gallup poll conducted in September
of 1967, 43% of those polled favored Nixon while 28% favored Humphrey. Identify which of
these numbers are parameters and which are statistics.
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Unit 2
13) In July 2008, a science journal published a report on the reliability of Alzheimer testing.
Results of a large study suggested that among people with Alzheimer’s, 59% of the tests
conducted were (correctly) positive. For people who do not have Alzheimer’s, 68% of the
tests were (correctly) negative. A clinic serving an at-risk population offers free Alzheimer’s
testing, believing that 15% of the patients may actually have Alzheimer’s disease. Find the
following probabilities (hint: a tree diagram may be helpful)
a) P(a patient tested negative for Alzheimer’s and that patient does not have Alzheimer’s)
b) P(a patient tested negative for Azheimer’s and that patient truly has Alzheimer’s)
c) P(a patient test positive for Alzheimer’s and that patient does not have Alzheimer’s)
d) P(a patient will test negative for Alzeheimer’s)
e) Given that a person tested negative for Alzheimer’s, find the P(that person is truly Alzheimer’s
free).
14) Explain the difference between response bias and non-response bias.
15) Explain the difference between an experiment and an observational study. Which can be
used to establish causality?
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Unit 2
16) A moving target at a police academy target range can be hit 80% of the time by a particular
individual. Suppose the person takes four shots at the target. What is the probability that:
a) The first shot is a hit?
b) The second shot is a hit?
c) There are hits on all four?
d) There are misses on all four?
e) There is only one hit?
f) There are exactly two hits?
g) There is at least two hits?
h) What is the expected number of hits?
i) What is the standard deviation of hits?
17) In a given city 80% of taxi’s are blue and 20% are black. One night there is a hit-and-run
accident. A witness testifies that he saw a black taxi leaving the scene of the crime. The
defense tests the accuracy of the witnesses observations under conditions similar to the night
of the accident and finds that he correctly identifies the color of the taxi 75% of the time.
What is the probability that the taxi involved was actually blue given that the witness says it
was black? (hint: a tree diagram may be helpful)
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