1
Chapter 3: Introduction to Probability (Part 6)
Relationship Between Two (Dichotomous) Variables
Example
Population: U.S. Senators in 1993 (N = 100)
Variables: 1. Gender (Male or Female)
2. Party (Democrat or Republican)
Raw data
No. Gender
1
Male
2
Fem

1
Chapter 1: Introduction to Statistics (Part 4)
Types of Variables
It should be noted that other instructors or textbooks often
classify variables in a different way.
Quantitative variables
1. Continuous (also known as measurement)
Examples: Weight, Heig

1
Chapter 1: Introduction to Statistics (Part 3)
Simple Random Sampling (revisited)
Recall the 3 by 5 inch index cards in a barrel mechanism for
obtaining a simple random sample that was discussed in Part 1.
One important detail that was omitted is the fa

1
Chapter 3: Introduction to Probability (Part 2)
Mathematical functions are often used to represent probability
distributions. These functions are of two distinct types:
1. Discrete functions
e.g. Binomial distributions
2. Continuous functions e.g. Norm

1
Chapter 1: Introduction to Statistics (Part 2)
Example No. 2
Question: Do men and women tend to weigh the same?
(What men and women are we talking about? What do we
mean by tend to weigh the same?)
Statistical Perspective
Population: Students enrolled a

1
Chapter 2: Summarizing Information for
Single Variables (Part 5)
Dichotomous Variables
Consider a dichotomous variable D (defined with regard to
some population) taking on the values 0 and 1, with
p = Pr(D = 1) and
1 p = Pr(D = 0).
Example
Population: R

1
Chapter 2: Summarizing Information for
Single Variables (Part 2)
In addition to representing quantitative data with tables and
graphs, it is also useful to summarize the data by computing
quantities called parameters (if the data is considered as
popula

1
gChapter 2: Summarizing Information for
Single Variables (Part 1)
Summarizing Quantitative Data with Tables and Graphs
Example: In a class of 20 students, the values of the variable
Course Grade were as follows (imagine the students are listed
in alphab

1
Chapter 3: Introduction to Probability (Part 4)
We have learned how to use tables of the standard normal
distribution. In order to compute probabilities for normal
distributions having any mean and standard deviation we
need to learn some results about

1
Chapter 2: Summarizing Information for
Single Variables (Part 3)
It is useful to have terms that describe the shapes of data
distributions as represented by histograms (or dot plots).
Terms used to describe distributional shapes (see handout)
1. Symmetr

1
Chapter 2: Summarizing Information for
Single Variables (Part 4)
Measures of Position
z scores
Also called a standard score or standardized score, the
z score represents the number of standard deviations that a
given value of y falls above (or below) th

1
Chapter 3: Introduction to Probability (Part 7)
Simpsons Paradox
Example: Is there gender bias with respect to graduate
school admissions?
All Students
Male
Female
Total
Admit Not Admit
233
324
88
194
321
518
Total
557
282
839
Pr(Admit | Male) = 233 / 5

1
Chapter 1: Introduction to Statistics (Part 5)
Overview of the Course
Chapter 1: Introduction to Statistics
Chapter 2: Summarizing Information for Single Variables
Chapter 3: Introduction to Probability
Chapter 4: Sampling Distributions
Chapter 5: Infer

CONCEPTUAL TOOLS
EX:
P ROBABILITY
JOINT PDF, f(x,y)
Example 2
By: Neil E. Cotter
A joint probability density function is defined as follows:
k
f (x, y) =
0
x2 + y2 1
otherwise
a)
Sketch the shape of f(x,y). (You may assume k = 1 for this sketch.)
b)
Calc

1
Chapter 3: Introduction to Probability (Part 5)
The Normal Approximation to the Binomial Distribution
Under certain conditions, a binomial distribution can be well
approximated by a normal distribution. Specifically, if
Y ~ B( p, n), then (approximately

1
Chapter 3: Introduction to Probability (Part 1)
Consider again the population of N = 20 students in a class,
for which we have the following relative frequency
distribution with respect to the variable Y = Grade Point:
y
Pr(Y = y)
0
.05
1
.15
2
.35
3
.3

1
Chapter 3: Introduction to Probability (Part 3)
Consideration of the binomial distribution has shown us how
discrete functions can represent probability distributions. We
will now consider the normal distribution, in which a
continuous function is used

1
Chapter 1: Introduction to Statistics (Part 1)
What is Statistics?
Example No. 1
Political polls: Predicting the outcome of an election
Question: How could a political pollster predict the outcome
of a race for governor between candidates Smith and Jone