Probability
Quantitative Methods in HPELS
440:210
Agenda
Introduction
 Probability and the Normal Distribution
 Probability and the Binomial Distribution
 Inferential Statistics

Introduction

Recall:
statistics: Sample statistic 
PROBABILITY  population parameter
 Inferential

Marbles Example
Assume:
Assume:
N = 100 marbles
N = 100 marbles
50 black, 50 white
90 black, 10 white
What is the probability of
drawing a black marble?
What is the probability of
drawing a black marble?
Introduction


Using information about a population to
predict the sample is the opposite of
INFERENTIAL statistics
Consider the following examples
While blindfolded, you
choose n=4 marbles from
one of the two jars
Which jar did you
PROBABLY choose your
sample?
Introduction

What is probability?


The chance of any particular outcome
occurring as a fraction/proportion of all
possible outcomes
Example:

If a hat is filled with four pieces of paper
lettered A, B, C and D, what is the
probability of pulling the letter A?
 p = # of “A” outcomes / # of total outcomes
 p = 1 / 4 = 0.25 or 25%
Introduction


This definition of probability assumes
that the samples are obtained
RANDOMLY
A random sample has two requirements:
Each outcome has equal chance of being
selected
2. Probability is constant (selection with
replacement)
1.
What is probability of drawing Jack of Diamonds from 52 card deck? Ace of spades?
What is probability of drawing Jack of Spades if you do not replace the first selection?
Agenda
Introduction
 Probability and the Normal Distribution
 Probability and the Binomial Distribution
 Inferential Statistics

Probability  Normal Distribution

Recall  Normal Distribution:
 Symmetrical
 Unified

mean, median and mode
Normal distribution can be defined:
 Mathematically
(Figure 6.3, p 168)
 Standard deviations (Figure 6.4, 168)
With either definition, the predictability of the Normal Distribution allows you
to answer PROBABILITY QUESTIONS
Probability Questions
Example 6.2
 Assume the following about adult height:

µ
= 68 inches
  = 6 inches

Probability Question:
 What
is the probability of selecting an adult
with a height greater than 80 inches?
 p (X > 80) = ?
Probability Questions


Example 6.2:
Process:
Draw a sketch:
2. Compute Z-score:
3. Use normal distribution to determine
probability
1.
Step 1: Draw a sketch for p(X>80)
Step 2: Compute Z-score:
Z=X-µ/
Z = 80 – 68/6
Z = 12/6 = 2.00
Step 3: Determine probability
There is a 2.28% probability that
you would select a person with a
height greater than 80 inches.
Probability Questions
What if Z-score is not 0.0, 1.0 or 2.0?
 Normal Table  Figure 6.6, p 170

Column A: Z-score
Column C: Tail = smaller side
Column B: Body = larger side
Column D: 0.50 – p(Z)
Using the Normal Table

Several applications:
Determining a probability from a specific Zscore
2. Determining a Z-score from a specific
probability or probabilities
3. Determining a probability between two Zscores
4. Determining a raw score from a specific
probability or Z-score
1.
Determining a probability from a specific Z-score

Process:
Draw a sketch
2. Locate the probability from normal table
1.

Examples: Figure 6.7, p 171
p(X > 1.00) = ?
p(X < 1.50) = ?
p(X < -0.50) = ?
Tail or Body?
Tail or Body?
p(X > 0.50) = ?
p = 15.87%
p = 93.32%
Tail or Body?
p = 30.85%
Using the Normal Table

Several applications:
Determining a probability from a specific Zscore
2. Determining a Z-score from a specific
probability or probabilities
3. Determining a probability between two Zscores
4. Determining a raw score from a specific
probability or Z-score
1.
Determining a Z-score from a specific probability

Process:
Draw a sketch
2. Locate Z-score from normal table
1.

Examples: Figure 6.8a and b, p 173
What Z-score is associated with a
raw score that has 90% of the
population below and 10% above?
20%
20%
(0.200)
(0.200)
What two Z-scores are associated
with raw scores that have 60% of the
population located between them and
40% located on the ends?
Column B (body)  p = 0.900
Column C (tail)  p = 0.200
Z = 1.28
Z = 0.84 and -0.84
Column C (tail)  p = 0.100
Column D (0.500 – p(Z))  0.300
Z = 1.28
Z = 0.84 and – 0.84
30%
30%
(0.300) (0.300)
Using the Normal Table

Several applications:
Determining a probability from a specific Zscore
2. Determining a Z-score from a specific
probability or probabilities
3. Determining a probability between two Zscores
4. Determining a raw score from a specific
probability or Z-score
1.
Determining a probability between two Z-scores

Process:
Draw a sketch
2. Calculate Z-scores
3. Locate probabilities normal table
4. Calculate probability that falls between Zscores
1.

Example: Figure 6.10, p 176

What proportion of people drive between the
speeds of 55 and 65 mph?
Step 1: Sketch
Step 2: Calculate Z-scores:
Step 2: Locate probabilities
Z=X-µ/
Z=X-µ/
Z = -0.30 (column D) = 0.1179
Z = 55 – 58/10
Z = 65 – 58/10
Z = 0.70 (column D) = 0.2580
Z = -0.30
Z = 0.70
Step 4: Calculate probabilities between Z-scores
p = 0.1179 + 0.2580 = 0.3759
Using the Normal Table

Several applications:
Determining a probability from a specific Zscore
2. Determining a Z-score from a specific
probability or probabilities
3. Determining a probability between two Zscores
4. Determining a raw score from a specific
probability or Z-score
1.
Determining a raw score from a specific
probability or Z-score

Process:
Draw sketch
2. Locate Z-score from normal table
3. Calculate raw score from Z-score equation
1.

Example: Figure 6.13, p 178

What SAT score is needed to score in the
top 15%?
Step 1: Sketch
Step 2: Locate Z-score
Step 3: Calculate raw score from Z-score equation
p = 0.150 (column D)
Z=X-µ/ X=µ+Z
Z = 1.04
X = 500 + 1.04(100)
X = 604
Agenda
Introduction
 Probability and the Normal Distribution
 Probability and the Binomial Distribution
 Inferential Statistics

Probability  Binomial Distribution

Binomial distribution?
 Literally
means “two names”
 Variable measured with scale consisting of:
Two categories or
 Two possible outcomes


Examples:
 Coin
flip
 Gender
Probability Questions  Binomial Distribution
Binomial distribution is predictable
 Probability questions are possible
 Statistical notation:

 A and
B: Denote the two categories/outcomes
 p = p(A) = probability of A occurring
 q = p(B) = probability of B occurring

Example 6.13, p 185
Heads
Tails
p = p(A) = ½ = 0.50
If you flipped the coin twice (n=2), how
many combinations are possible?
Heads
Heads
Heads
Tails
Tails
Heads
Tails
Tails
q = p(B) = ½ = 0.50
Each outcome has an equal chance of
occurring  ¼ = 0.25
What is the probability of obtaining at
least one head in 2 coin tosses?
Figure 6.19, p 186
Normal Approximation  Binomial Distribution
Binomial distribution tends to be NORMAL
when “pn” and “qn” are large (>10)
 Parameters of a normal binomial
distribution:

 Mean:
µ = pn
 SD:  = √npq

Therefore:
Z
= X – pn / √npq
Normal Approximation  Binomial Distribution
To maximize accuracy, use REAL LIMITS
 Recall:

 Upper
and lower
 Examples: Figure 6.21, p 188
Note: The binomial distribution is a
histogram, with each bar extending
to its real limits
Note: The binomial distribution
approximates a normal distribution
under certain conditions
Normal Approximation  Binomial Distribution


Example: 6.22, p 189
Assume:
 Population:
Psychology Department
 Males (A) = ¼ of population
 Females (B) = ¾ of population

What is the probability of selecting 14 males in a
sample (n=48)?
 p(A=14)
 p(13.5<A<14.5) = ?
Normal Approximation  Binomial Distribution

Process:
Draw a sketch
2. Confirm normality of binomial distribution
3. Calculate population µ and :
1.


µ = pn
 = √npq
Calculate Z-scores for upper and lower real limits
5. Locate probabilities in normal table
6. Calculate probability between real limits
4.
Step 1: Draw a sketch
Step 2: Confirm normality
pn = 0.25(48) = 12 > 10
qn = 0.75(48) = 36 > 12
Step 3: Calculate µ and 
µ = pn
 = √npq
µ = 0.25(48)
 = √48*0.25*0.75
Step 5: Locate probabilities
µ = 12
=3
Z = 0.50 (column C) = 0.3085
Z = 0.83 (column C) = 0.2033
Step 4: Calculate real limit Z-scores
Z = X–pn/√npq
Z = X-pn/√npq
Z = 13.5-12/3
Z = 14.5-12/3
Z = 0.50
Z = 0.83
Z = 0.50 (column C) = 0.3085
Z = 0.83 (column C) = 0.2033
Step 6: Calculate probability between the real limits
p = 0.3085 – 0.2033
p = 0.1052
There is a 10.52% probability of selecting 14 males
from a sample of n=48 from this population
Normal Approximation  Binomial Distribution


Example extended
What is the probability of selecting more than
14 males in a sample (n=48)?


p(A>14)  p(A>14.5) = ?
Process:
1.
2.
3.
Draw a sketch
Calculate Z-score for upper real limit
Locate probability in normal table
Step 1: Draw a sketch
Step 2: Calculate Z-score of upper real limit
Z = X–pn/√npq
Z = 14.5 – 12 / 3
Z = 0.83
Step 3: Locate probability
Z = 0.83 (column C) = 0.2033
There is a 20.33% probability of
selecting more than 14 males in
a sample of n=48 from this
population
Agenda
Introduction
 Probability and the Normal Distribution
 Probability and the Binomial Distribution
 Inferential Statistics

Looking Ahead  Inferential Statistics

PROBABILITY links the sample to the
population  Figure 6.24, p 191
Textbook Assignment

Problems: 1, 3, 6, 8, 12, 15, 17, 27