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**preview**shows half of the first page. to view the full**1 pages of the document.**CH6 – PROBABILITY THEORY AND THE NORMAL PROBABILITY DISTRIBUTION

PROBABILITY THEORY – analysis & understanding of chance occurrences

o Physical nature cyclical, follow strict scientific laws predictable

o Human behavior less predictable

Social/behavioral scientists req. incl. degrees of accuracy

o Many statistic predication based on normal curve

What is a Probability?

A PROBABILITY (p) – a specification of how frequently a particular event of

interest is likely to occur over a large number of trails

o Probability of Success – probability of this interesting event occurring

o Probability of Failure – probability of this interesting not event

occurring

o Observe “outcomes” of large # of trails, identify all possible outcomes

Probability can be considered as proportions

Basic Rules of Probability Theory

Rule 1: probability in range between 0 and 1

o 0 = event cannot happen, 1 = event must happen

Rule 2: Addition rule for Mutually Exclusive events – the probability of

many mutually exclusive (alternative) events is equal to the sum of the

probabilities of the individual events

o P(A or B) = P(A) + P(B) iff A & B are independent (scales)

Rule 3: Adjust for joint occurrence (General addition rule)

o Joint Occurrence – an event that double0counts success or joins two

aspects of success

o P(A or B) = P(A) + P(B) – P(A and B) (doesn’t scale)

Rule 4: Multiplication rule for Compound of Independent events –

probability of a compound of independent events is equal to the multiple

of the probabilities of the separate parts of the events

o Compound events – multiple-part events

o P(A and B) = P(A)P(B) iff A & B are independent (scales)

Rule 5: Account for replacement with compound events

Using the Normal Curve as a Probability Distribution

Z-score – how many standard deviation away from the mean a raw (X-

score) lies

Specific AUC of normal distribution = probability of occurrence of any

single score falling in its domain

For normal distribution: the follow are equivalent

o 1. Proportion of cases between two scores

o 2. The AUC between these two scores

o 3. Probability of randomly selecting a case between these scores

WAYS TO INTERPRET P

o 1. A distributional interpretation that describes the results in relations

to the distribution of scores in a population or sample

o 2. A graphical interpretation that describes the proportion of areas

under a normal curve (assuming the distribution is normal shape)

o 3. A probabilistic interpretation that describes the probability of a

single random drawing of a subject from this population

PARTITIONING AREAS UNDER THE NORMAL CURVE – to identify part of

the curve and compute the proportion (p) of the total curve this part

represents

o Using statistical table & Z-score

o Req. variable is normally distributed in the population

o Req. variable is interval/ratio measurement level

= proportion of area under the curve

= total sample size

Computing percentiles for normally distributed populations

o Percentile Ranks – the percentage of a sample or population that falls

at or below a specific value of a variable

If normally distributed find value’s corresponding z-value to

quickly compute percentile ranks

o Percentile ranks possible for non-normal distributions

The Normal Curve as a Tool for Proportional Thinking

Equal differences btwn scores doesn’t always indicate the same thing

o Raw scores by themselves misleading

o use standard deviation as unit of measure w/ normal distributions for

accurate insight into significance of a raw score

The Gambler’s Fallacy: Independence of Probability Events

Common statistical mistake w/ probability: misunderstand independence

of part of compound events

o Gamblers fallacy – Assume on losing streak, luck will eventually turn

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