Probability

Probability

Quantitative Methods in HPELS

440:210

Agenda

Introduction Probability and the Normal Distribution Probability and the Binomial Distribution Inferential Statistics

Introduction

Recall: Inferential statistics: Sample statistic

PROBABILITY population parameter Marbles Example

Assume:

N = 100 marbles

50 black, 50 white

What is the probability of drawing a black marble?

Assume:

N = 100 marbles

90 black, 10 white

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:1. Each outcome has equal chance of being

selected2. Probability is constant (selection with

replacement)

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


Probability Normal Distribution Recall Normal Distribution:

SymmetricalUnified 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) = ?


Example 6.2: Process:

1. Draw a sketch:

2. Compute Z-score:

3. Use normal distribution to determine probability

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.


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:

1. Determining a probability from a specific Z-score

2. Determining a Z-score from a specific probability or probabilities

3. Determining a probability between two Z-scores

4. Determining a raw score from a specific probability or Z-score

Determining a probability from a specific Z-score

Process:1. Draw a sketch

2. Locate the probability from normal table Examples: Figure 6.7, p 171

p(X > 1.00) = ?

Tail or Body?

p = 15.87%

p(X < 1.50) = ?

Tail or Body?

p = 93.32%

p(X < -0.50) = ?

p(X > 0.50) = ?

Tail or Body?

p = 30.85%

Determining a Z-score from a specific probability


2. Locate Z-score from normal table 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?

Column B (body) p = 0.900

Z = 1.28

Column C (tail) p = 0.100

Z = 1.28

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 C (tail) p = 0.200

Z = 0.84 and -0.84

Column D (0.500 – p(Z)) 0.300

Z = 0.84 and – 0.84

20%

(0.200)

20%

(0.200)

30%

(0.300)

30%

(0.300)

Determining a probability between two Z-scores


2. Calculate Z-scores

3. Locate probabilities normal table

4. Calculate probability that falls between Z-scores

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:

Z = X - µ / Z = X - µ /

Z = 55 – 58/10 Z = 65 – 58/10

Z = -0.30 Z = 0.70

Step 2: Locate probabilities

Z = -0.30 (column D) = 0.1179

Z = 0.70 (column D) = 0.2580

Step 4: Calculate probabilities between Z-scores

p = 0.1179 + 0.2580 = 0.3759

Determining a raw score from a specific probability or Z-score

Process:1. Draw sketch

2. Locate Z-score from normal table

3. Calculate raw score from Z-score equation 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

p = 0.150 (column D)

Z = 1.04

Step 3: Calculate raw score from Z-score equation

Z = X - µ / X = µ + Z

X = 500 + 1.04(100)

X = 604

Agenda


Probability Binomial Distribution

Binomial distribution?Literally means “two names”Variable measured with scale consisting of:

Two categories or Two possible outcomes

Examples:Coin flipGender

Probability Questions Binomial Distribution

Binomial distribution is predictable Probability questions are possible Statistical notation:

A and B: Denote the two categories/outcomesp = p(A) = probability of A occurringq = p(B) = probability of B occurring

Example 6.13, p 185

Heads

Tails

p = p(A) = ½ = 0.50

q = p(B) = ½ = 0.50

If you flipped the coin twice (n=2), how many combinations are possible?

Heads Heads

Heads Tails

HeadsTails

Tails Tails

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

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


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) = ?


2. Confirm normality of binomial distribution

3. Calculate population µ and : µ = pn = √npq

4. Calculate Z-scores for upper and lower real limits

5. Locate probabilities in normal table

6. Calculate probability between real limits


Step 1: Draw a sketch

Step 3: Calculate µ and

µ = pn = √npq

µ = 0.25(48) = √48*0.25*0.75

µ = 12 = 3

Step 2: Confirm normality

pn = 0.25(48) = 12 > 10

qn = 0.75(48) = 36 > 12

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

Step 5: Locate probabilities

Z = 0.50 (column C) = 0.3085

Z = 0.83 (column C) = 0.2033

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

Example extended What is the probability of selecting more than

14 males in a sample (n=48)? p(A>14) p(A>14.5) = ?


2. Calculate Z-score for upper real limit

3. 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


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

Probability

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