Practice Exam 1
Math 338
Instruction for the Actual Exam
You have 75 minutes to complete the exam. The exam is closed-book and note. A copy of
the formula sheet, which was posted on Canvas, is supplied.
You may also use a calculator, but using a computer or other electronic devices is
prohibited.
Please read the questions carefully and supply sufficient details supporting your answers.
When applicable, use tables, tree diagrams, or other means to provide details, and list the
formula you used for computations.
Good Luck
Discrete Probability Events
1)
Let 𝑋 have a binomial distribution. Select any expression that is equal to 𝑃(9 ≤ 𝑋 ≤ 14).
a) P(X≤14)-P(X≤9)
b) P(X≤14)-P(X<9)
c) P(X≤14)-P(X≤8)
d) P(X≤14.5)-P(X≤8.5)
(b, c, d)
Dot Plot
2)
Which one of the following describes a dot plot?
a) Visual display of numerical data organized into equal intervals.
b) Representing the distribution of data using median, quartiles, and extremes.
c) Visual display of data where each value is represented by a dot (or an x).
d) Display of change in data over time
(c)
Valid Lottery
3)
A new lottery randomly selects winners form players who buy a ticket and pays them $100 with
probability 𝑝1 , $10 with probability 𝑝2 , or $1 with probability 𝑝3 . Losers get no money.
The distribution of amount won can be shown as follows:
Table 1. Winning Distribution of a New Lottery
𝒊
𝟏
𝟐
𝟑
𝟒
𝒘𝒊
$100
$10
$1
$0
𝒑𝒊
𝑝1
𝑝2
𝑝3
𝑝4
Which of the following distribution choices are a valid distribution for the amount
won by a player.
Table 2. Distribution Choices for the New Lottery
𝑫𝒊𝒔𝒕𝒓𝒊𝒃𝒖𝒕𝒊𝒐𝒏 𝑪𝒉𝒐𝒊𝒄𝒆𝒔
𝒑𝟏
𝒑𝟐
𝒑𝟑
𝒑𝟒
𝑨
0.1
0.05
0.49
0.35
𝑩
0.05
0.25
0.05
0.75
𝑪
0.29
0.36
0.15
0.2
𝑫
0.01
0.49 −0.05 0.55
𝑬
0.17
0.33
0.14
0.36
a) Distribution Choice A
b) Distribution Choice B
c) Distribution Choice C
d) Distribution Choice D
e) Distribution Choice E
(c, e)
City Government (Sampling Method).
4)
The government of a town needs to determine if the city's residents will support the
construction of a new town hall. The government decides to conduct a survey of a sample of the
city's residents. Which one of the following procedures would be most appropriate for obtaining
a sample of the town's residents?
a) Survey a random sample of persons within each geographic region of the city.
b) Survey every 8th person who walks into city hall on a given day.
c) Survey the first 300 people listed in the town's telephone directory.
d) Survey a random sample of employees at the old city hall.
(a)
Art Magazine Writer (Sampling Method)
5)
A writer for an art magazine randomly selects and interviews fifty male and fifty female artists.
What sampling technique is used?
a) cluster
b) convenience
c) systematic
d) simple random
e) stratified
(e)
Mean Calculations
6)
Find the missing data if the mean is 39.5.
49, 49, ? , 34, 32, 41
32
Female Senators (Z-Score)
7)
Assume 𝜇 = 69.4 and 𝜎 = 8.6 for the age of female senators in the US senate.
What is the z-score for the age of a female senator who is 56 years old.
(56-69.4)/8.6=-1.56
Regular Gas Prices (Five Number Summary)
The price of regular gas in some randomly selected gas station in Irvine are:
2.86, 3.40, 3.24, 3.40, 3.80, 3.40, 3.24, 3.50, 3.80, 3.30.
8)
Find the mean and the five-number summary for the observed data.
(a) Mean:__________
(d) Range:___________
(b) Median:_________
(e) Q1:___________
(c) Mode:___________
(f)
Q3:___________
9)
Find if there are any outliers.
a) 10 sorted observations: 2.86, 3.24, 3.24, 3.3, 3.4, 3.4, 3.4, 3.5, 3.8, 3.8
Min = 2.86; Max = 3.8; Med=(3.4+3.4)/2=3.4
Q1=3.24, Q3=3.5
2.86…….3.24…3.4..3.5……3.8
b) IQR = 3.5-3.24= 0.26, 1.5IQR=1.5(0.26)=0.39
LF = 3.24-0.39 = 2.85 Min > LF => No outlier
UF = 3.5 + 0.39 = 3.89 Max < UF => No Outlier
New Lottery:
Assume that the distribution of the amount won in the new lottery is:
Table 3. Winning Distribution
𝒊
𝟏
𝟐
𝟑
𝟒
𝒙𝒊
$100
$10
$1
$0
𝒑𝒊
0.05
0.10
0.25
0.60
Use this information to answer the next two questions.
10) Calculate the mean winning amount. Round the response to 3 decimal place, if needed.
6.25 (with margin: 0)
11) What is the best pricing approach for the entity that sells this lottery tickets? Select an
appropriate response:
a) Price the ticket to be slightly more than the mean winning amount.
b) Price the ticket to be slightly less than the mean winning amount.
c) Let people bid on the tickets.
d) Stop selling, people are always winning.
(a)
Real Estate Sales in NYC (Bayes)
12) In Bronx, NYC, 10.4% of real estate sales in 2020 has been commercial buildings, 59.6% of them
are selling for more than $200/sq.ft. The proportion of residential buildings that sold for more
than $200/sq.ft is 79.1%.
If a you see a sales ad for a building that is asking for more than $200/sq.ft, what is the
probability that the building is a residential building.
Music and Exam Performance:
Researcher believe that 75% of students (or more) can benefit from listening to music while
studying for an exam. To verify this, they recruited 120 high school students from different
grade levels (Freshmen to Senior) to participate in a study. The students were asked to take
a test immediately after they studied a new subject in a room for three hours. The
researchers asked students to repeat the process two times: once without listening to
music and another time while listening to their preferred music. The order of the tests (with
or without music) and the subjects were randomly assigned.
The researchers recorded the difference between with-music and without-music scores to
measure the impact of listening to music on score. In addition to the above information,
they recorded students' gender.
Use this information to answer the following questions titled Music and Exam Performance.
13) The response variable in this study is ____________,
a) Music: whether a student listened to music or not.
b) Score Change: the increase (decrease) in test score.
c) Subject: the material that the student studied.
d) Gender: the gender of the student.
e) Grade Level: the student's school grade level (9th, 10th, 11th, 12th)
(b)
14) The response variable is a _________ variable.
a) Numerical (quantitative)
(a)
b) Categorical (qualitative)
15) The explanatory variables are ___________.
a) Gender, Music, Grade Level and Score Change.
b) Gender, Music, Score Change and Subject.
c) Gender, Music, Grade Level and Subject.
d) Gender, Score Change, Grade Level and Subject.
e) Score Change, Music, Grade Level and Subject.
(c)
16) There are _________ categorical and __________ numerical explanatory variables.
a) none and 4
b) 1 and 3
c) 2 and 2
d) 3 and 1
e) 4 and none
(e)
17) If 65 of the students scored better in the test that they prepared for while listening to music, the
proportion of students who can benefit from listening to music, rounded to four decimal values,
is:
0.5417 (with margin: 0)
18) The 75% value is a parameter.
a) True
b) False
(a) TRUE
19) A case in this study is ________ and there are _________ cases in the dataset.
a) a subject
b) a test
c) a music
d) a student
(d: a student) and (g: 120)
e) 2
f) 65
g) 120
20) The population of interest is ________ and the sample used in the study is _________.
a) all high-school students
b) all tests taken by high-school students
c) 120 students who participated in the study
d) 65 students who did better with music
(a) and (c)
21) This study is an observational study.
a) True
b) False
FALSE
Statistic Class
The following table shows the probabilities for randomly selecting a student from a 25student class of statistics:
Table 4. Proportions for a Class of 25
Graduating?
Yes (Senior)
Computer Sci. (Yes) Non-Computer Sci. (No) Total Graduating
28%
4%
32%
No (Non-Senior)
52%
16%
68%
Total Computer Sci.
80%
20%
100%
Let 𝐺 represent the event that a randomly selected student in graduating, and 𝐶 represent
the event that the selected student studies Computer Science.
Use the above information to answer the following questions.
22) Match the probabilities with the events:
a) 𝑷(𝑮)=
b) 𝑷(𝑪𝒄 )=
32%
20%
28%
84%
c) 𝑷(𝑮 ∩ 𝑪)=
d) 𝑷(𝑮 ∪ 𝑪)=
23) Consider these two events:
A. Selecting a computer student if we draw from graduating students (seniors).
B. Selecting a graduating student given the student studies Computer Science.
Choose a proper notation from the following mathematical statements:
a) 𝐺
d) 𝐺 ∪ 𝐶
b) 𝐶
e) 𝐶|𝐺
c) 𝐺|𝐶
f) 𝐺 ∩ 𝐶
A : (e), B: (c)
The following histogram shows the age distribution of male patients who are diagnosed
with Alzheimer's disease. The mean age is 76.66 years, and the standard deviation is 9.51
years.
24) Approximate the above distribution with a normal distribution by completing the x-axis values.
[2 points]
25) Use the normal approximation (and empirical rule) to find the age value(s) that separates the
lower 2.5% of the patients. [1 point]
26) Do you expect the real value that separates the lower 2.5% of patients to be smaller or larger
than what you found by normal approximation? Why? [1 point]
The National Alzheimer's Coordinating Center (NACC) maintains a dataset on many
patients and their health information related to their cognitive impairment. These are three
variables in their dataset:
- Age: The patient's age at the visit, recorded in whole years.
- Sex: Determining if the patient is female or male.
- AlzD: The cognitive status of the patient, taking three values: Normal (if no cognitive
impairment is recognized), Dementia (if the patient has dementia, but no sign of
Alzheimer's is detected), or Alzheimer's (if the patient is diagnosed with having
Alzheimer's disease).
The following graph summarizes the information on these three variables.
Use the above information to answer the following questions:
27) Mark the age distribution of Normal Male patients and describe it. [1 point]
28) Can the above graph be used to determine the association between the following pairs of
variables? If yes, explain if there is an association or not. [3 point]
a) AlzD and Sex.
b) Sex and Age.
c) AlzD and Age.