© 2014 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved

HLTH 300 Biostatistics for Public Health Practice,

Raul Cruz-Cano, Ph.D2/24/2014, Spring 2014

Fox/Levin/Forde, Elementary Statistics in Social Research, 12e

Chapter 5: Probability and the Normal Curve

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Announcements

• Okay, let’s have a review before the exams…but we need to reduce the number of exams

(4 Exams+ 4 Reviews = 8 Sessions = .5 of our meetings!)• See new syllabus• Rule for the reviews:

– Specific questions, not “blanket” questions. – Not obligatory to attend.

• For the exams you are allowed to bring copies of Appendix C and Appendix D

© 2014 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved

Calculate probabilities and understand the rules of probability

Understand the concept of a probability distribution

List the characteristics of the normal curve

Understand the area under the normal curve

Calculate and use z scores

CHAPTER OBJECTIVES

5.1

5.2

5.3

5.4

5.5

Calculate probabilities and understand the rules of probability

Learning ObjectivesAfter this lecture, you should be able to complete the following Learning Outcomes

5.1

Probability5.1

P = 1The outcome

is certain

P = .5The outcome is as likely to

happen as not

happen

P = 0The outcome

is Impossible

The relative likelihood of occurrence of any given outcome

Converse Rule: The probability that something will not occur

The Rules of Probability5.1

Addition Rule: The probability of obtaining one of several different and distinct outcomes (mutually exclusive)

Multiplication Rule: The probability of obtaining two or more outcomes in combination (Independent)

Number of times the outcome or event can occur(F)Total number of times any outcome or event can occur

P

F 1 FP P

A or B A BP P P

A and B A X BP P P

The Rules of Probability5.1

Addition Rule: The probability of obtaining one of several different outcomes (not mutually exclusive)

A or B A BP P P -P (A and B)

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ExampleA= Intoxicated with C02B= Intoxicated with N02

1. What is P(A)? P(B)?2. What is P(Ā)?3. What is P(A and B)?4. What is P(A or B)

Probability: Example5.1

Let’s flip a coin two times:Probability of heads on the first flip:

Probability of heads on the second flip:

Probability of getting heads on both flip:

Heads or Tails?

1 .52

1 .52

.5 .5 .25

Probability: Example5.1

What is the probability that one flip in two flips will land on heads?

Heads or Tails?

HT TH H T T H

.50 .50 .50 .50

.25 .25

.50

P P P P P P

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Example

In the U.S. the probability of a driver being uninsured is .1231. Two drivers crash:

1. P(Both uninsured)2. P(Both Insured)

2. Five Drivers Crash, P(all uninsured)?

Understand the concept of a probability distribution

Learning ObjectivesAfter this lecture, you should be able to complete the following Learning Outcomes

5.2

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Probability Distributions5.2

Mean = μStandard

Deviation = σ

Directly analogous to a frequency distribution• Except it is based on probability theory

5.2

Figure 5.1

X = # of heads P0 .251 .502 .25Total 1.00

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Examples

Problem 6:Standard 6-sided die:X: Outcome of rolla. P(X=2)b. P(X=3 or X=4)c. P(X=Odd Number)d. P(X=Anything but 5)

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ExamplesProblem 14X: A random politician from the samplea. P(X=Republican)b. P(X=Democrat that support

euthanasia)c. P(X=Does not support euthanasia)d. P(X=Republican that does not

support euthanasia)

Pol. Party Support

1 R No2 D No3 D Yes4 R No5 R No6 D No7 D Yes8 D Yes9 R No10 D Yes

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Examples

Problem 20Lottery ticket 2 numbers and a letter, e.g. 3 7

Pa. P(match 1st digit)b. P(match 2nd digit)c. P(Not match 1st digit)d. P(match 1st and 2nd digit)e. P(match letter)f. P(Perfect match)

List the characteristics of the normal curve

Learning ObjectivesAfter this lecture, you should be able to complete the following Learning Outcomes

5.3

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5.3

• Smooth• Symmetrical• Unimodal• Mean = Median = Mode• Infinite in Both Directions• Probability Distribution• Mean = μ; Standard Deviation = σ• Areas Under the Curve = 100%

Characteristics of the Normal Curve

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Characteristics of the Normal Curve

If you know the mean and the std. deviation of a normally distributed variable then you can find many probabilities:

P(X < x )P(X > x )P(x1 <X < x2)

If you know the mean and the std. deviation of the grades of exams for a class (normally distributed) then you can find many probabilities:X: Grade of a random studentP(X < 90 )P(X > 85 )P(75 < X < 95

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5.3

The normal curve is a theoretical idealMany many variables do conform to the normal

curve… few students get low grades, few students get great grades, most get around average

Some variables do not conform to the normal curve

• Many distributions are skewed, multi-modal, and symmetrical but not bell-shaped

• Assuming normality when it does not exist can impact the validity of our conclusions

The Reality of the Normal Curve

Understand the area under the normal curve

Learning ObjectivesAfter this lecture, you should be able to complete the following Learning Outcomes

5.4

5.4

Figure 5.5Like a very smooth histogram

5.4

Figure 5.6

5.4

Figure 5.7

5.4

Figure 5.8

Calculate and use z scores

Learning ObjectivesAfter this lecture, you should be able to complete the following Learning Outcomes

5.5

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5.5

It is possible to determine the area under the curve for any sigma distance from the mean

This distance is called a z-score• Indicates direction and distance that any raw score deviates

from the mean in sigma units

Standard Scores and the Normal Curve

Xz

mean of a distribution standard deviation of a distribution standard scorez

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ExampleX: Salaries in a companyMean= $20,000Std. Dev=$1,500Distribution = NormalP(20,000 < X < 22,000)=?

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P(20,000 < X < 22,000) =40.82%

Exercises:1. P(X < 22,000) 2. P(X < 22,000) 3. P(18,000 < X < 22,000)4. P(X<18,000)5. P(X>18,000)

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5.5

When the normal curve is used in conjunction with z scores and Table A in Appendix C, we can determine the probability of obtaining any raw score (X) in a distribution

• The converse, addition, and multiplication rules still apply

We can also reverse this process to calculate score values from particular portions of area or percentages

Finding Probability under the Normal Curve

X z

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X:911 Response timeMean= 5.6 minutesStd. Dev= 1.8 minutesDistribution = NormalP(X<x)=75%

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Examples

Problem 27X: SAT ScoresNormally distributedMean=500Std. Dev. = 100a. P(500< X <600)b. P(400 < X < 600)e. P(X>600)f. P(X<300)

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Homework

Chapter 5: Problems 13, 16 and 30

© 2014 by Pearson Higher Education, IncUpper Saddle River, New Jersey 07458 • All Rights Reserved

Probabilities can be calculated using the converse, addition, and multiplication rules

The probability distribution is analogous to a frequency distribution and includes a mean and

standard deviation

The normal curve is a theoretical ideal and therefore cannot be applied to all distributions

100% of the data falls under the normal curve, with 50% of the data falling to either side of the mean

By converting raw scores to z scores, we can determine the probability of randomly selecting an

individual with that score from the population

CHAPTER SUMMARY

5.1

5.2

5.3

5.4

5.5