C-N M120 Lecture Notes
Based very loosely on Bluman’s Elementary Statistics (2008)
B. A. Starnes
4 – 2014
Lecture
Why do we study and/or use statistics? Is it for academic
harassment ... to make sure that students in various majors run the “gauntlet”??? Is it
to appease baseball fans with meaningless reams of numbers? Is it to employ
meaningless analysis to “prove” things we already know? Is it to actually determine
if a potential cure for some ailment doesn't end up killing someone? How about
hurricane path predictions, insurance rate setting, agricultural, oil spill, and tsunami
predictions?
These are all good
questions that we hope to
answer in time. The last
questions are the best ones
and the answer to those are
all “yes”!!!
We will begin our study in
Statistics with the notion of
probability. Wackerly,
Schaffer and Mendenhall
(2002) define this as the
belief in the occurrence of a future event. This is a good definition. We will say,
more precisely, that it is man’s guess at what God is going to do.
There are three major interpretations of probability: subjective, relative frequency
and classical. Although we will study the classical interpretation in depth, the other
two are quite useful in certain situations.
For instance, we know that the law of sin that is at work in mankind because of the
constant failure of world systems to maintain peace or prevent crime. Or, one might
flip a coin 100 times and note that the number of heads is 52 and conclude that heads
will occur about half the time. These are obvious examples of
the relative frequency interpretation of probability. Here are
some examples of subjective probability.
Someone might feel that C-N has a 5% chance of winning the
NCAA DII national football championship. Mel Kiper makes
a good living off of assessing the future NFL value of college
football players. Both of these would be examples of the
subjective interpretation of probability. In this course, we will
concentrate on the classical (or theoretical) interpretation of probability. We can
discuss the rules of probability at a later time.
1S
C-N M120 Lecture Notes
Based very loosely on Bluman’s Elementary Statistics (2008)
B. A. Starnes
4 – 2014
Lecture
For now we will visualize the concepts using Venn diagrams. These
are diagrams utilized by the English mathematician John Venn to illustrate concepts
with sets. They are particularly useful for probabilities as well. We'll use one here to
set out some definitions for the course.
Here we have some illustrations of
sets A and B in the  set (omega
set). We can think of the  set as a
dart board in which the thrown dart
can never miss. Sometimes the Ω
set is also known as the sample
space, universal set, or population.
We will also assume that the dart
can hit each point in Ω with equal
likelihood ... that is, it is thrown
randomly in . The area of a specified set (A for example) divided by the total area ()
is the probability of the set being hit with the dart. Notice that in real applications, the
concept of area can be easily replaced with number within the boundaries each set, and
the idea of throwing a dart replaced with a random selection. In the graph above, the light
blue area represents (A ∪ B)c, read “the complement of, the quantity, A union B, the
quantity”. The medium blue represents A\B and B\A respectively, read “A remove B and
B remove A” respectively. The darker blue represents A ∩ B, read “A intersect B”. Note
finally that A ∪ Ac =  and A ∩ Ac = , the “null” set.
Now any study of classical probability must include the rules of probability. As Andy
Griffith once told Opie, “I like to have fun like everybody else, but we have to obey
the rules!” The first rule is …
P(E) (“the probability of event E”) ∈ [0,1].
Now this means that all probabilities must occur in the [0, 1] interval. The closer to
certainty we are for event E, the closer the probability is to 1. Do we/you know of
something having a probability of 0 or 1? Of course not! These probabilities belong
solely to God. Man's realistic probabilities lie on the open interval (0, 1). But in the
course of our study we will write probabilities of 0 or 1 for events that are highly
unlikely or very nearly certain respectively. Here's a brief video to illustrate this.
http://www.youtube.com/watch?v=KX5jNnDMfxA&feature=related
Next we wish to assert some relation rules which will help us ascertain various
probabilities. The Complementation Rule states that for an event A,
P(A) = 1 – P(Ac).
That should make sense based on the notion of the complementary event. Here is an
example.
2S
C-N M120 Lecture Notes
Based very loosely on Bluman’s Elementary Statistics (2008)
B. A. Starnes
4 – 2014
Ex.
The probability of rain tomorrow in Jefferson City is given to be 70% (or .7).
What is the probability that it doesn't rain?
P(R) = 1 – P(Rc), so
P(R) - 1 = - P(Rc), and
1 - P(R) = P(Rc) = 1 - .7 = .3.
The probability that it doesn't rain in Jefferson
City tomorrow is 30% or .3. []
Lecture
The General Addition Rule for 2
events is given by...
P(A ∪ B) = P(A) + P(B) – P(A ∩ B).
We also call this “the Pirate Rule” because it's
acronym is “GAR” ... matey! The General
Addition Rule for n events is an extension of this
that the reader should attempt to figure out.
Ex.
Consider the situation in which someone is selected at random from
the famous Carson-Newman Marching Eagle band represented by this contingency
table.
C-N Band
Front
Brass
Wind
Percussion
Total
Female
14
12
24
10
60
Male
0
14
16
10
40
Total
14
26
40
20
100
Notice that the contingency
table is, in fact, , the
universal set for this
experiment. Observe the
following probabilities.
P(Male) = 40/100 = .4.
P(Female) =
P(Malec) = 1 - .4 = .6.
P(Brass) = 26/100 = .26.
P(Brass ∪ Male) = P(Brass) + P(Male) – P(Brass∩Male) = .26 + .4 - .14 = .52.
Let's see, there 40 males and 12 females who play brass. Yes, 52 of the C-N band
members are either male or play brass! []
Lecture
It's important to see in the previous example that the events Male and
Female are considered to be disjoint ... that is they cannot occur simultaneously.
3S
C-N M120 Lecture Notes
Based very loosely on Bluman’s Elementary Statistics (2008)
B. A. Starnes
4 – 2014
Lecture
Whenever we deal with a contingency table, we'll always consider
events on the same side disjoint. In this case
P (Male ∩ Female) = 0.
The conditional probability of event A given (or conditioned on) event B has already
occurred is written
P(A| B) = P(A ∩ B)/P(B).
We can use this type of probability for various things ... not the least of which is joint
probability! Now we can express
P(A ∩ B) = P(A)P(B|A) = P(B)P(A|B).
The reader should contemplate this last equation. This brings us to the notion of
dependence and independence. 2 events are considered to be independent if the
outcome of one does not affect the outcome of the other. That is
P(A|B) = P(A) (or vice versa).
If this does not occur, the events are considered to be dependent. Another way to
view this mathematically is
P(A|B) = P(A ∩ B)/P(B) = P(A) =>
P(A ∩ B) = P(A)P(B).
So then, there are two definitions of independence. Another way of looking at these
concepts is to consider information. Check out this commercial from AT&T.
The gentleman arriving late to the party asks if the invitation is information he
“would like to know”. If the prior occurrence of B leads to greater information on
P(A), then that is indeed information you would like to know and the two events are
dependent. If the prior occurrence of B gives no more information on P(A), then
that's information that is unhelpful and the two events are considered independent.
Ex.
Consider again the C-N Marching Band example from above. Observe the
following:
P(Male|Brass) = 14/26 = .5385 (approximately).
P(Brass|Male) = 14/40 = .35.
Note that they are not equal. Also observe the relationships between Brass and Male.
P(Brass ∩ Male) = .14 = P(Brass)P(Male|Brass) = .26(14/26).
P(Male|Brass) = .5385 ≠ P(Male) = .4.
Therefore these two events are considered to be dependent. Can you find any pairs of
independent events in ? []
Lecture Let's have a quick review of various symbols and their meanings.
1. ∩ - “intersection”, “and”
2. ∪ - “union”, “or”
3. | - “given”
4. Ec - “not E”, “E complement”
5. P(E) - “the probability of event E”
4S
C-N M120 Lecture Notes
Based very loosely on Bluman’s Elementary Statistics (2008)
B. A. Starnes
4 – 2014
Lecture
Practice saying (aloud) these things in order to develop the
relationships in your mind. Then, using the file given on the CMS content page try
using the Bucket Fill tool to identify some compound events for yourself.
Student problem set A
A1. Consider selecting someone at random from Virginia Tech's Marching
Virginians with the following contingency table:
VT Band
Front
Brass
Wind
Percussion
Total
Female
Male
27
55
70
8
160
3
75
45
37
160
Total
30
130
115
45
320
Find the following ...
a) P(Front|Female),
b) P(Female|Front),
c) P(Front ∩ Female),
d) P(Front ∪ Female),
e) P(Female),
f) P(Wind).
g) Are the events Brass and Male independent? Why?
A2.
In a regular 52 card deck (this is your  set), find the following …
a) P(Red)
b) P(Ace)
c) P(Heart)
d) P(Face)
e) P(Red | Ace)
f) P(Red | Heart)
g) P(Heart | Red)
h) P(Face | Black).
A3.
John and Beth
Dickerson plan to have three
children. Assuming they are
successful, find the  set for
the outcomes of this family (use a tree diagram if necessary). What is the probability
they will have two boys and a girl? What is the probability they will have two boys
and a girl, in that order, given that they have two boys and a girl?
A4. Using the image in the course content graphically illustrate the following sets:
a) A ∩ Bc.
b) Ac ∪ Bc.
c) Ac ∩ Bc.
A5. If P(A | B) = .35 = P(B | A), and P(B) = .3, find P(A ∩ B) and P(A ∩ Bc).
5S
C-N M120 Lecture Notes
Based very loosely on Bluman’s Elementary Statistics (2008)
B. A. Starnes
4 – 2014
A6. If a Christmas secret passes through three people to a fourth
person, and each of the first three independently have a 95%
probability of telling the secret accurately …
a) what is the probability that the secret is successfully
transmitted to the fourth person?
b) what is the probability that at least one person incorrectly
transmits the information to the fourth person?
A7. Using the information in problem A6, what is the probability
that the fourth person successfully gets the information given that the
first person correctly transmits the information to the second one?
Student Solution set A
A1.
a) P(Front|Female) = 27/160 = .16875, b) P(Female|Front) =27/30 = .9, c)
P(Front ∩ Female) = 27/320 = .0844, d) P(Front ∪ Female) = .5094, e) P(Female) =
160/320 = .5, f) P(Wind) = 115/320 = .3594, g) No, P(Brass|Male) = .46875 ≠ .40625
= P(Brass).
A3.
 = {GGG, GGB, GBG, BGG, GBB, BGB, BBG, BBB}.
P(2 Boys and Girl) = 3/8 = .375. P(BBG|2 Boys and Girl) = 1/3 = .3333. []
Lecture
function
Next, let's define the notion of the random variable X. If there is a
X:  -> A (⊂ ℝ),
then this is the random variable X. Here,  is exactly what we defined before (the
sample space, or the universal set)
Okay, Okay, what's this all about? Well, think of X as a big cat (like a Kentucky
Wildcat! Where did that come from???). The cat is sitting on the classroom podium
and JUMPS out on someone or something in the  space (the class of students). And,
then, turns and reports back a value ... like age ... or length ... or hair color ... or place
of origin. We can classify the random variable
according to what it reports back. Random
variables (rvs) that report back a numerical
quantity are called quantitative r.v.s. Random
variables (rvs) that report back a non-numerical
value are referred to as qualitative r.v.s (like hair
color). Moreover, quantitative r.v.s for which the
range is an interval are called continuous r.v.s,
and those for which the range is a finite or
countable set are called discrete r.v.s. Another
way to recognize discrete rvs is if there is
separation between the possible values. The
following chart should help.
6S
C-N M120 Lecture Notes
Based very loosely on Bluman’s Elementary Statistics (2008)
B. A. Starnes
4 – 2014
Lecture
RV chart
/
\
Qualitative Quantitative
/
\
Discrete
Continuous
A general mathematical convention is to use letters at the last part of the Roman (or
Latin) Alphabet for random variables (X, Y, Z), while letters at the beginning (a, b, c,
d) are normally reserved for constant representation.
Ex.
Height is a quantitative continuous rv. It is numerical and can take values on an
interval. Birthplace is a qualitative rv since we can't order the values. Number of
siblings is a quantitative discrete rv since it is numerical, but a person cannot have 4.3
siblings. []
Lecture
So now that we have an idea about the r. v. X, we want to introduce
the idea of a distribution. Here's the dealio! A distribution is the random variable's
numerical occurrence pattern. Think about this. What would one need to do to
describe the occurrence pattern? Maybe one would want to say how and where the
r.v. occurs most. Maybe one would want to indicate an expectation of the
occurrences (like give an average value). Maybe one would want to describe how far
apart the occurrences are. These measures are called the shape, location and spread of
the distribution, respectively.
At this point we will plan on using the terms random variable and distribution
interchangeably. Are they the same word? No, but they do have similar adjectives.
The quantification of these notions of shape, location and spread are called
parameters of the r.v. All of these are important, so we use a graph to illustrate each
facet in a consolidated picture. The function on the right is called the probability
density function (or pdf), OR in the discrete case the probability mass function (or
pmf). The pmf graph appears as “points in space”, while the pdf graph appears as a
smooth curve. Here's a picture of a pdf (and cdf ... to the left … ask your teacher
about that) to help get us started.
7S
C-N M120 Lecture Notes
Based very loosely on Bluman’s Elementary Statistics (2008)
B. A. Starnes
4 – 2014
Ex.
Take, for instance, the situation in which we track the movement of a vehicle
through an intersection. In this case Ω = {Left, Straight, Right}. Define X: Ω → ℕ. as
X(Left) = -1, X(Straight) = 0, and X(Right) = 1 (note that A = {-1, 0, 1}).
X is a quantitative and discrete r.v. Further, define
P(-1) = P(0) = P(1) = 1/3.
This is an example of an r.v. with a Discrete
Uniform distribution. This is a famous
distribution which is also considered to be
symmetric. Note too that
P(vehicle turns) = ca.= .67.
Here, we would also say that X is a
symmetric r.v. []
James 1
17Every good gift and every perfect gift is from above, and cometh down from the Father
of lights, with whom is no variableness, neither shadow of turning.
Ex.
Next suppose that we have another intersection with Y being the direction taken
by the motorist, the same set A, but these probabilities.
Y
P(Y)
-1
.2
0
.7
1
.1
What is the probability of a motorist turning here? What is the probability that the
motorist turns left (wrong)? Is the spread of this rv greater, less or about the same as that
of X? []
Student problem set B
B1. Classify the following r.v.s: Birthplace, Letter grade in this class, ACT Score,
Weight, Hair Color, Class Rank, Number of Siblings, Age.
B2. Describe the shape of the distributions affiliated with the same r.v.s listed in B1
for the current class.
B3. Give the shape, location and spread of the discrete uniform distribution (X) in the
intersection problem above.
B4. Give the shape location and spread of the variable Y from the second intersection
problem above.
Student Solution Set
B1.
Birthplace
ACT Score
Weight
Hair Color
Age
Qualitative
Quantitative Discrete
Quantitative Continuous
Qualitative
Quantitative Continuous.[]
8S
C-N M120 Lecture Notes
Based very loosely on Bluman’s Elementary Statistics (2008)
B. A. Starnes
4 – 2014
Lecture
For the large remainder of the course,we will focus on quantitative
r.v.s ... which have dim = 1. These quantitative r.v.s, then, have distributions that we
can study in C-N MATH 120. In this case we can use a variety of graphs to give us
insight into not only the shape, but the location and spread of the r.v. as well. In some
instances, particularly those in which the r.v. is either continuous or close to it, we
utilize a stem 'n' leaf plot. This is somewhat similar to the histogram, except that it
gives much more information. The idea is to divide the data values into stems of a
multiples of a power of 10, and then list the next lower power of 10 values into the
appropriate stem. A histogram, on the other hand, is a stem 'n' leaf plot without the
leaves!
Ex.
Suppose we have a class of 25 students having the following
commuting times (door to door) for getting to M201 (in min). Consider this a sample
of 25 observations of the r.v. X ... the commuting time of a 2007 C-N student.
Data (min): 23, 2, 10, 15, 2, 5, 5, 6, 15, 6, 10, 6, 2, 45, 35, 2, 5, 5, 25, 20, 5, 5, 5, 2, 6.
And here is the data displayed in a stem 'n' leaf plot.
X (10 min)
X (min)
0
2,2,2,2,2,5,5,5,5,5,5,5,6,6,6,6
1
0,0,5,5
2
0,3,5
3
5
4
5
Do you see what's going on in terms of the powers of 10? The left column deals with
units of 101 while the right deals with units of 100. This can be done with other
powers of 10 as you shall see. Since the “heights” of the stacks go “down” from left
to right, we classify the shape of this variable as right skewed. Turn your head to the
right to see this?
This can also be generated using the TI calculator, SAS or R. []
9S
C-N M120 Lecture Notes
Based very loosely on Bluman’s Elementary Statistics (2008)
B. A. Starnes
4 – 2014
Lecture
There are obviously ways to quantify what we're seeing in the
distributions in terms of shape, center and spread ... particularly the last two. As
mentioned earlier, these are called the parameters of the r.v. (or distribution). We
shall now consider the following parameters of location: mean, median, mode. The
mean of a r.v. X is its expectation. That is, it's where we expect the r.v. to occur in
general. Sometimes the mean is written E(X) for “expectation of X”. Yet another
way to express the mean is to say the arithmetic average of X. The mean in statistics
is the same thing mathematically as the first raw moment in physics. It is a function
of distance and mass. As a rule, the mean of a random variable is assigned the value
 and is the balance point of the distribution.
The mean give a fair approximation for the location of a r.v., but the drawback is that
it is susceptible to extreme values. A more robust (less susceptible) parameter is the
median. The median is the probabilistic midpoint of the distribution. It is the value
for which the r.v. X has an equal probability (or likelihood) of occurring above or
below. The median is also called the 50th%ile or Q2. Similarly Q1, and Q3, are
represent the 25th%ile (First Quartile) and 75%ile (Third Quartile) for X respectively.
The median is estimated by taking the “middle value(s)” in the ordered data set. Can
you see why this would be a more robust parameter?
Finally, the mode of a r.v. is simply the value of X corresponding to the highest
point(s) of the pdf or pmf. It is estimated by finding the value in a sample data set
that occurs most often. An interesting note is that the mean, median and mode often
occur according to the skew of the r.v.
Ex.
Consider the r.v. X having the following graph ...
Notice that the r.v. is quantitative continuous and left skewed. Consequently, the
mean, median and mode will occur from left to right. The student should also attempt
to locate the quartiles on the graph. []
Lecture
Measures of spread for a r.v. X are not as easily quantified.
Nevertheless, they are useful. We shall consider the following parameters of spread:
variance, standard deviation, range. Like the expectation (or mean) the variance is a
moment. In the case of the mean we are looking at an expected location for the r.v.
X.
10S
C-N M120 Lecture Notes
Based very loosely on Bluman’s Elementary Statistics (2008)
B. A. Starnes
4 – 2014
Lecture
Here, we are looking at an expected squared missed distance from ,
although we can't get at that just yet. If we simply added up the missed distances
from  together, we'd always get 0, so we have to do the squared missed distances to
get a meaningful value. The expectation of this squared missed distance is called the
variance, and denoted 2. This is all well and good, but something is lost in the
translation ... namely the unit of X. It's hard to see the variance on a graph of X. So
we can take the square root of the variance and have (essentially) what we wanted to
begin with ... the average missed distance ... sort of. This is called the standard
deviation of X and is designated with . This is probably the most useful measure of
spread. The range of X is simply that ... the distance between the smallest and largest
possible values of X. Note that this is the length of A, the other range ... back when
we first defined r.v.s on p 6.
Ex.
Consider again, the r.v. from the last example. Notice that we have
depicted here three of the spread parameters.
The variance would make little to no sense here. []
Lecture
So is it enough to just look at pictures and estimate the parameters of
the r.v.s??? Of course not! Again, from the (hopefully) random sample of
observations (data set) of the r.v. we estimate the aforementioned parameters with
these estimators ... er ... statistics. Since they are obtained from a data set, we use the
word “sample” in front of them. Here is a recap.
Sample Shape:
Left skewed, Right skewed, Symmetric
Sample Mean:
Sample Median:
Sample Mode:
Sample Quartiles:
Xn = ( i=1n xi)/n
Estimates: 
middle value in the ordered data set
Estimates: X.5)
The most recurrent value in the data set
The medians of the halves (ordered data set) Estimates: X.75,X.25
Sample Variance:
Sample Std Devn:
Sample Range:
s2 =  i=1n (xi- Xn)2/(n-1)
s = √(sample variance (s2))
max(xi) – min(xi)
Estimates: 2
Estimates: 
We divide by n – 1 in the variance estimate to account for using the data to estimate
one parameter already and thereby losing a degree of freedom.
11S
C-N M120 Lecture Notes
Based very loosely on Bluman’s Elementary Statistics (2008)
B. A. Starnes
4 – 2014
Ex.
Suppose we have a class of 25 students having the following
commuting times (door to door) for getting to M201 (in min).
Data (min): 23, 2, 10, 15, 2, 5, 5, 6, 15, 6, 10, 6, 2, 45, 35, 2, 5, 5, 25, 20, 5, 5, 5, 2, 6.
Find the aforementioned shape, location and spread estimators for these bad boys.
Earlier we estimated the shape to be right skewed from the stem 'n' leaf plot.
X25 = 10.68min, X.5 = 6min, mode = 5min.
(From stem 'n' leaf plot) First Quartile = 5min, Third Quartile = 15min.
s = 11.0707min, s2 = 122.56min2, IQR = 10min, Range = 43min.
And finally, one quick way to summarize all three features of a r.v. X is to give what
is known as the 5 number summary. This is simply the min(xi), Q1, Q2, Q3, max(xi) ...
in that order. The 5 number summary, (2, 5, 6, 15, 45) min, is useful in creating the
boxplot (below). The boxplot reveals the shape, location and spread of the r.v. X.
The 5 number summary can also be found easily using the TI calculator, SAS or R. []
Student problem set C
C1. Here we have a data set on the r.v. X = winter temperature in C in Jefferson City. Hourly readings (in C) were taken on a randomly selected blustery winter day. f represents the frequency for each reading
X(C)
f
­­­­­­­­­­­
7
3
8
3
9
5
10 6
11
7
­­­­­­­­­­­
total
24
Find the sample mean, sample median, sample mode, s2, 5 number summary, s, and a shape description of this r.v. 12S
C-N M120 Lecture Notes
Based very loosely on Bluman’s Elementary Statistics (2008)
B. A. Starnes
4 – 2014
C2. Consider the following data set of
observations from the r.v. X,
attendance at
C-N home football games (Roy
Harmon Field at Burke-Tarr Stadium).
Attendance(fan):
5130, 4292, 3315, 4093, 4714.
Find the sample mean, sample
median, s, SSE, 5 number summary
and a shape description for this r.v. Do a boxplot to confirm the shape.
C3. Another way of thinking of  for a r.v. X is as its expectation. So for a discrete
r.v. we have ...  = E(X) =   xP(x). Suppose it costs $3 to play a game in which a
player rolls two dice. If the player rolls a 2 or 12, she wins $20. If she rolls a 7, she
wins $5. If X is the r.v. representing the amount gained by playing the game find  =
E(X). Is it wise to play this game?
C4. Lett X be the previously defined random variable
in problem C3. Find the variance of X.
C5. Let X be the previously defined random variable in
problem C3. Describe the shape of X.
C6. Let X be = number of saves/week got by the Indy Mustangs baseball team during
the 2011 BRBA season. Thus far we have the following data set.
X(save): 5, 9, 3, 4, 5, 5, 2, 4, 5
Find the sample mean and sample standard deviation for X.
C7. Let X be previously defined random variable in problem C6. Give the 5 number
summary for X and describe the distribution.
Student Solution Set C
C1. 9.4583 C, 10 C, 11 C, 1.9119 C2, (7 C, 8.5 C, 10 C, 11C, 11C), 1.383 C, Quantitative Continuous and Left Skewed.
C2. 4308.8f, 4292f, 684.297f, 1873050f2, (3315f, 3704f, 4292f, 4922f, 5130f),
Quantitative Discrete and Left Skewed.
C3. $17 x (1/18) + $2 x (1/6) - $3 x (7/9) = -$1.06. []
13S
C-N M120 Lecture Notes
Based very loosely on Bluman’s Elementary Statistics (2008)
B. A. Starnes
4 – 2014
Lecture
This should be a good primer for C-N Math 201.
14S