1. Probability and the Normal Curve
Chapter 5
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2. Introduction to Part II
In Part I, we learned to categorize data to see basic patterns and trends.
Measures of central tendency and variability allowed more descriptions.
In Part II, we’ll move towards using statistics to assist us in decision making.
Using the concept of probability
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Rules of Probability
Probability: the relative likelihood of occurrence of any given outcome or event
Probability
Rules of probability:
Converse Rule
Addition Rule
Multiplication Rule
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Converse Rule
Converse rule: determines the probability that something will not occur
Subtract the probability that something will occur from 1 to find the probability that the event will not occur.
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Addition Rule
Addition rule: determine cumulative probability by adding probabilities.
The probability of obtaining any one of several different and distinct outcomes equals the sum of their separate probabilities.
Outcomes must be mutually exclusive.
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Multiplication Rule
Sometimes we want to know the probability of successive outcomes.
Multiplication rule: combination of independent outcomes equals the product of their separate probabilities
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7. Probability Distributions
Probability distributions analogous to frequency distributions
Probability distributions based on probability theory
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Probability Example
Let’s flip a coin two times.
probability of heads the first time is .5
probability of heads on the second flip is .5
Using the multiplication rule, probability of getting heads on both flips is .25
Similarly, probability of getting no heads on two flips is .25.
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9. Probability Example (cont.)
What is the probability that one flip in two flips will land on heads?
When you add the probabilities of all of the options, they must equal 1.
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10. Mean of a Probability Distribution
Probability distribution has a mean – known as an expected value.
Greek letter mu used to indicate the mean in a probability distribution
Don’t confuse mu with X bar in a frequency distribution).
The Greek letter mu is shown as: μ
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11. Standard Deviation of a Probability Distribution
Probability distribution also has a standard deviation
Symbolized by the Greek letter sigma, σ
Variance in probability distributions is shown by σ2
s2 for observed data; σ2 for theoretical distribution
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12. The Normal Curve as a Probability Distribution
The normal curve is perfectly symmetrical.
Precepts of the normal curve allow for a number of assumptions based on probability theory.
But, what are the properties of the normal curve?
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13. Characteristics of the Normal Curve
Sometimes referred to as a bell-shaped curve – perfectly symmetrical.
Normal curve is unimodal.
Mean, median, mode coincide.
Both continue infinitely ever closer but without touching the baseline.
The normal curve is a theoretical ideal because it is a probability distribution.
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14. The Model and the Reality of the Normal Curve
The normal curve is a theoretical ideal.
Some variables do not conform to the normal curve.
Many distributions are skewed, multi-modal, and symmetrical but not bell-shaped.
Consider wealth – more “haves” than “have nots”
Often, radical departures from normality
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15. The Area under the Normal Curve
100% of the cases in a normal distribution fall under normal curve
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18. The Area under the Normal Curve
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Using Table A
Method exists to determine distance from the mean for standard deviations that are not whole numbers.
Table A in Appendix C
Sigma distances labeled
Values for one side of the normal curve given because of symmetry
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20. Standard Scores and the Normal Curve
It’s possible to determine area under the curve for any sigma difference from the mean.
This distance is called a z score or standard score
z score – indicates direction and degree that any raw score deviates from the mean in sigma units
z scores by obtaining the deviation
Where:
µ=mean of a distribution
σ=standard deviation of a distribution
z=standard score
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21. Finding Probability under the Normal Curve
Ideas of the normal curve can be used in conjunction with z scores.
Normal cure is a probability distribution.
Probabilities under normal curve always out of 100%
Use Table A to find the probability
Earlier rules (converse, addition and multiplication) still apply
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22. Finding Scores from Probability Based on the Normal Curve
Using the formula:
it is possible to calculate score values
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Summary
Introduction to probability
Foundation for decision making in statistics
Normal curve is a theoretical ideal
Use of the normal curve assists in understanding standard deviation
z scores used to determine area between and beyond a given sigma distance from the mean
Also able to calculate score values for a given z
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