Trigonometry and Discrete Math
Final Exam
Practice Packet
Your preparation for the final exam should include reviewing your old tests and quizzes,
going over your notes and projects, and completing this review. Good luck!
Trigonometry and Discrete Math
Final Exam Topics 2015
Unit 3: Statistics
Chapter 3- Probability, Data Analysis, and Statistics
3.1 Sampling Methods- Convenience, Stratified, and Random
What makes something Bias?
Survey Project
Measures of Central Tendency (Mean, Median, Mode, Range)
3.2 Frequency Tables and Histograms
3.3 Scatter Plots
Correlation: Positive/Negative/None
Strength: Strong/Moderate/Weak
Line of Best Fit
3.4 Box and Whisker Plots
Finding the IQR (inner quartile range), upper quartile, lower quartile, and the median
Which measures of central tendency does a box plot not reveal? Mean and mode
3.5 Variance and Standard Deviation
Use the mean to figure out each data point’s “distance from the mean”
Smaller standard deviation means the data points were more consistent and closer to the mean
3.6 Probability (“And” vs. “Or”)
Compound probability- two separate events multiplied together
Conditional Probability- P (A|B) = P (A and B together) / P(B)
Experimental Probability- ex: probability of something coming out defective, based off a previous trial
3.7 Permutations (when order matters)
Combinations (when order does not matter)
Unit 4: Discrete Math
Chapter 4- Graph Theory and Voting Theory
4.1 Graph Theory
Euler Paths and Circuits (must go over every edge once and only once)
Hamilton Paths and Circuits (must hit every vertex once and only once)
Circuits= end where you started
Travelling Salesman Problem and Nearest Neighbor Algorithm
Minimum Spanning Trees (Kruskal’s Algorithm)
4.2 Coloring Maps and Graphs
Four Color Theorem
Creating a graph and coloring vertices to solve scheduling problems using time slots
4.3 Voting Possibilities- Plurality, Pairwise, Borda Counts, Hare (Plurality with elimination)
Voting Impossibilities- Majority, Condorcet (Head to Head), Monotonicity, Irrelevant Alternatives
How many ways can N candidates be arranged on a single ballot? N!
How many pairs are needed with N candidates? NC2
1) Find the mean, median, mode, and range of the following data sets.
a. Weights of textbooks in ounces 12, 10, 9, 15, 16, 10
b. Ages of students on the math team 14, 14, 15, 15, 16, 15, 15, 16, 15
c. Time spent on the internet in minutes per day 75, 38, 43, 120, 65, 48, 52
2) Determine if the following samples are random, stratified, or convenience.
a. Surveying every 6th person that walks into a supermarket to find out what the most popular
candy of North Jersey is.
b. Questioning only the teachers you see in one class day to find out if people of all ages prefer
minty or fruity gum.
c. Surveying every 4th student that walks into school if they believe the school should allow
students to wear headphones during hallway time.
d. Surveying every person that walks out of a convenience store only with a soda to find out what
the most popular soda flavor of your town is.
3) The data below gives the ages of 25 adults running in the town council election. Complete the frequency
table and draw the corresponding histogram
57, 61, 57, 57, 58, 57, 61, 54, 68, 51, 49, 64, 50, 48, 65, 52, 56, 46, 54, 49, 50, 47, 55, 55, 54
Interval
41 – 45
46 – 50
51 – 55
56 – 60
61 – 65
66 - 70
Tallies
Frequency
a. How many people are older than 55?
b. How many were younger than 51?
4) The data below shows the number of vehicles that pass through an intersection between 4:00 pm and
5:00 pm over a two-week period: 88 92 114 82 94 83 84 85 85 90 95 82 95 103
Find the minimum, lower quartile, median, upper quartile, maximum, and put your data on a box and
whisker diagram.
5) In the boxplot to the right, answer the following:
a. Who had a higher median?
b. Who had a larger IQR?
6) Describe the data in the scatter plot to the right as
either strong or weak and positive or negative.
7) List the steps below on how to find the standard
deviation from a set of data.
a.
b.
c.
d.
e.
_
_
_
_
_
8) Find the mean and the standard deviation of the following set of data: 14, 11, 12, 5, 2, 8, 10, 3, 4, 2
9) Find the mean and the standard deviation of the following set of data: 5, 16, 13, 6, 3, 8, 6, 7, 4, 12
10) Out of #8 and #9 (above), which set of data was more consistent? (Explain why).
11) A jar contains 12 caramels, 7 mints and 16 dark chocolates. What is the probability of selecting a dark
chocolate and then a caramel with replacement?
12) There are 27 students available to represent the upperclassmen at a fair. 13 are juniors and 14 are
seniors. What is the probability that the first one chosen will be a senior and the second one will be a
junior without replacement?
13) You have the following coins in your pocket: 5 quarters, 6 dimes, 2 nickels and 12 pennies. What is the
probability you will draw a penny and then another penny without replacement?
14) There are 53 black pens and 44 blue. On the first selection, you choose a blue pen. Based off your
experiment, what is the probability that the next pen you select is blue?
15) Below are the counts (in thousands) of earned degrees in the United States in a recent year, classified by
level and by the sex of the degree recipient. One person is selected at random, find the following
probabilities.
a. Probability of a male.
b. Probability of a female and also has a doctorate.
c. Probability of a masters or professional.
d. Probability of a male given they have their doctorate.
e. Probability of a bachelor’s given they are female.
16) The probability that Becky studies and passes her mathematics test is 86%. The probability that Becky
just studies is 91%, find the probability that Becky passes her math test, given that she has studied.
17) The probability that Dan buys a new car is 47%. The probability that he gets a raise is 50%. The
probability that he buys a new car and gets a raise is 36%. Find the probability that Dan buys a new car,
given he gets a raise.
18) A jar contains 5 blue marbles, 8 red marbles, and 7 yellow marbles. Find the following probabilities:
a. Selecting a blue marble, replacing it, then picking a yellow marble.
b. Selecting two red marbles in a row without replacing the first.
19) How many ways can you arrange the letters of the word FACTOR?
20) How many ways can you choose two jellybeans from a bag of 15?
21) How many ways can five different textbooks be arranged on a shelf?
22) How many groups of 3 toys can a child choose to take on a vacation from a toy box containing 11 toys?
23) How many different sets of 6 questions for a test can be chosen from a file containing 22 questions?
24) What is the actual formula for a permutation without using the calculator? Combination?
25) What is the requirements for the degrees on the vertices for an Euler Path? Euler Circuit?
26) Find the shortest route starting and ending at point A for the following graph.
List the route you are taking and the total distance.
27) Make a graph which has six vertices and seven edges and is an Euler path.
28) Can an Euler path be an Euler circuit? Can an Euler circuit be an Euler path?
29) Finding a Hamilton circuit requires you to hit every ___________ and not hit any more than _________.
30) Use Kruskal’s Algorithm to keep certain edges and create a minimum spanning tree for the following
graph. List the edges you are keeping and find the total weight of your tree.
31) Use Kruskal’s Algorithm to keep certain edges and create a minimum spanning tree for the following
graph. List the edges you are keeping and find the total weight of your tree.
32) Use the following chart to set up a graph with correctly labeled vertices and edges. If a salesman needs
to make deliveries to each town, starting and ending at point A, find the shortest route he can take. Then
find the longest route. What is the difference between the shortest and longest routes (in miles)?
Towns
A
B
C
D
A
-34
33
30
B
34
-39
32
C
33
39
-34
D
30
32
34
--
33) Use the chart below to set up a graph and solve the scheduling dilemma. The following classes cannot
be scheduled at the same time because a certain student may need to take both. Figure out the least
amount of time slots are needed and also list the classes that can be put together.
Class:
Cannot be
scheduled
with:
A
D, I
B
D, I, J
C
E, F, I
D
A, B, F
E
H, I
F
I
G
J
H
E, I, J
I
A, B,
C, E,
F, H
J
B, G,
H
34) The following people want to see new movies in the theater. If there are ONLY 3 time slots, solve the
scheduling problem and list the movies that must be scheduled at the same time. Billy wants to see
Insidious and Pitch Perfect. Sarah wants to see Jurassic World and Insidious. Jose wants to see Spy and
Entourage. Michael wants to see Insidious and Spy. Sara wants to see Spy and Pitch Perfect.
How should the movies be scheduled so that they all get to see the movies that they want to see? Create
a graph and use the correct coloring scheme to help you solve this.
35) What is the least amount of colors needed to correctly color any map in the real world? (Ex. Map of
America)
36) Correctly color the map of Morris County
(use shapes, stripes, polka dots, etc. if you
do not have colors).
37) When coloring a map or graph, any regions
or vertices that share a ________ cannot be
colored the same color.
38) Create a voter profile for each of the following individual ballots.
1st
A
B
A
A
C
B
C
A
B
C
A
2nd
B
A
C
B
A
A
A
B
A
A
B
3rd
C
C
B
C
B
C
B
C
C
B
C
39) Using the voting profile that you created, how many voters voted? How many votes would a candidate
need to have majority?
40) What is the difference between majority and plurality?
Use the table to the right for the following
questions. (#41-47)
41) Who is the plurality winner?
42) How many pairs are needed for the pairwise
comparison method?
# of
Voters
1st
14
11
7
5
4
4
D
A
C
B
A
C
2nd
A
C
B
A
B
D
3rd
C
D
A
C
C
B
4th
B
B
D
D
D
A
43) Who is the winner using the pairwise comparison method?
44) How many Borda points does the first place winner receive when scoring?
45) Who is the winner by the Borda count method?
46) On one single voting ballot, how many Borda points will there be in total?
47) Who is the winner by the Hare method?
48) What is the voting method that is favored by the Condorcet criterion?
49) Which of the four criterions does not favor ANY of the voting methods?
50) List the 4 criterions and which voting methods satisfy it and which ones do not.
51) How many ways could 6 candidates running in an election be arranged on a single voter’s ballot?
52) In an election with 50 voters and 4 candidates running, what is the most amount of Borda points a
candidate could possibly receive? Least?