The Normal Distribution James H. Steiger. Types of Probability Distributions There are two fundamental types of probability distributions Discrete Continuous.

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Presentation transcript:

The Normal Distribution James H. Steiger

Types of Probability Distributions There are two fundamental types of probability distributions Discrete Continuous

Discrete Probability Distributions These are used to model a situation where The number of events (things that can happen) is countable (i.e., can be placed in correspondence with integers) Probabilities sum to 1 Each event has a probability

A Discrete Probability Function

Continuous Probability Distributions These are used to model situations where the number of things that can happen is not countable Probability of a particular outcome cannot be defined Interval probabilities can be defined: Probability of an interval is the area under the probability density curve between the endpoints of the interval

Probability Density Function

The Normal Distribution Family

The Standard Normal Distribution

The Normal Curve Table Z F(Z)

Some Typical “Routine” Problems

1.What percentage of people have IQ scores between 100 and 115? 2.What percentage of people have IQ scores greater than 130? 3.What percentage of people have IQ scores between 70 and 85? 4. What is the percentile rank of an IQ score of 122.5? 5. What IQ score is at the 99 th percentile?

General Strategy for Normal Curve Problems ALWAYS draw the picture Estimate the answer from the picture, remembering: Symmetry Total area is 1.0 Area to the left or right of center is.50 Convert to Z-score form Compute interval probability by subtraction

Problem #1

Problem #2

Problem #2 (ctd)

The Normal Curve Table Z F(Z)

Problem #3

The Normal Curve Table Z F(Z)

Problem #4

The Normal Curve Table Z F(Z)

Problem #5

Find Z-score corresponding to 99 th percentile. Convert to a raw score.

Relative Tail Probability Problems These problems involve analyzing relative probabilities of extreme events in two populations. Such problems: Are seldom found in textbooks Are often relevant to real world problems Sometimes produce results that are counterintuitive

Relative Tail Probability #1 Group A has a mean of 100 and a standard deviation of 15. Group B has a mean of 100 and a standard deviation of 17. What is the relative likelihood of finding a person with a score above 145 in Group B, relative to Group A?

The Normal Curve Table Z Area Above Z Answer: About 3 to 1.

Relative Tail Probability #2 Group A has a mean of 100 and a standard deviation of 15. Group B has a mean of 100 and a standard deviation of 17. What is the relative likelihood of finding a person with a score above 160 in Group B, relative to Group A?

The Normal Curve Table Z Area Above Z Answer: About 6.57 to 1.