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- Overview
- Why Study Probability?
- We study probability because the relationships between
samples and populations is usually stated in terms of
probability. Probability plays a central in inferential
statistics.
- Probability and Inferential Statistics
- The goal of inferential statistics is to use the limited
information we have in a sample to draw general conclusions
about the population. Probability allows us to start with
a population and predict what kind of sample is likely
to be obtained from it. Inferential statistics allows
us to say how probable the sample is to have come from
a particular population.
- Probability and Frequency Distributions
- We are usually concerned about probability when we have
a population frequency distribution: We want to know how
probable it is to obtain a specific sample. For this reason
we discuss Normal
and Binomial
distributions, the two most commonly used in statistics.
- Introduction to Probability
- Definition
- Probability is defined for a specific outcome
in a situation where several different outcomes are possible.
If the possible outcomes are denoted A, B, C, D,
etc., then the probability of A is defined as:
- Examples
- Tossing Coins: When you toss a balanced coin,
the outcome is either heads or tails. Thus, there are
a total of 2 possible outcomes. The probability of tossing
a head is
- Selecting Cards: There are 52 cards in an ordinary
deck of cards. Thus, there are a total of 52 possible
outcomes.
- The probability of drawing a Heart (there
are 13 hearts) is:
p(Heart) = 13/52 = 1/4 = .25 = 25%
- The probability of drawing an Ace (there
are 4 aces) is:
p(Ace) = 4/52 = 1/13 = .0769 = 7.69%
- The probability of drawing a Green card (there
are 0 green cards) is:
p(Green) = 0/52 = .00 = 0%
- The probability of drawing a card (there
are 52 cards) is:
p(Card) = 52/52 = 1.00 = 100%
- Definition
- Random Sampling: An independent random sample
must satisfy two requirements:
- Each individual in the population must have an equal
chance if being selected.
- If more than one individual is selected for the
sample, there must be constant probability
for each and every selection. (Sample with replacement)
- Probability and the Normal
Distribution
- Definition
- The normal distribution is defined by a complicated
equation that we don't need to know or understand.
- Why use Normal Distributions?
- What is important is to understand that the normal distribution
is used very frequently because:
- It can be shown that many characteristics of interest,
such as IQ, height and weight of people, etc., have
a normal population distribution.
- It can be shown mathematically that this shape is
guaranteed in certain situations that will
be important to us in inferential statistics.
- Characteristics:
- The normal distribution:
- is symmetrical (the left side is a mirror image
of the right side).
- has 50% of the scores below the mean and 50% above.
(Mean = Median)
- has most scores are in the middle. Few scores are
at the edges.
- The Standard Normal Distribution
- We have a standard normal distribution when the scores
in a normal distribution are expressed in standardized
z-scores. Thus, the standard normal distribution
- has a mean of 0 and a standard deviation of 1, just
like any other standardized distribution.
- has a normal shape, just like any other normal distribution.
- For a standard normal distribution it can be shown that
- 34.13% of the scores are between the mean and +1.00.
- 34.13% of the scores are between the mean and -1.00.
- 13.59% of the scores are between +1.00 and +2.00.
- 13.59% of the scores are between -1.00 and -2.00.
- 2.28% of the scores are above +2.00.
- 2.28% of the scores are below -2.00.
- Answering Probability Questions with the Unit Normal
Table :
- The unit normal table provides a listing of proportions
(probabilities) corresponding to many z-scores in the
standard normal distribution. Take the following steps
to answer probability questions using this table:
- Sketch the distribution, showing the mean and standard
deviation in raw scores.
- On the sketch, locate the specific score identified
in the problem, and draw a vertical line through the
distribution at this location.
- Make sure whether you need to find out about values
greater than (to the right side of) or less
than (to the left side of) the specific score.
- Shade the appropriate portion of the distribution
(to the right or left of your line).
- Now transform the specific score into a z-score
to identify the specific z-score in the standard normal
distribution that appears in the Unit Normal Table.
- Look at the shaded portion in your sketch to determine
which column (B or C) in the table corresponds with
the proportion you are trying to find.
- Ignore the sign of your z-score and look it up in
the table, taking the appropriate value from column
B or C.
- Percentiles and the Normal Distribution
- You can use the normal distribution to determine percentiles
(and percentile ranks).
- Because a percentile (rank) of a score is the percentage
of the scores that fall at or below the score,
you will need to find the proportion of the distribution
that is to the left of the score.
- ViSta and the Normal Distribution
- With ViSta, you can get the proportion of the scores
that are below (to the left) of a given z-score by typing,
in the listener window, the function:
(normal-cdf z)
where z is replaced with the z-score value in which you
are interested.
Multiply this by 100 for the percentile. You can do this
by typing:
(* 100 (normal-cdf z))
Subtract the value returned by the function from one to
get the proportion to the right of z. This can be done
by typing:
(- 1 (normal-cdf z))
- Probability and the Binomial
Distribution
- Definition
- When a variable is measured on a scale consisting of
only two categories, the data are called binary
or binomial. In this situation the researcher often knows
the population probabilities associated with the two categories.
When this is the case, the data have a known population
distribution, called the binomial distribution.
- Distribution Shape
- There are a whole family of different binomial population
distributions. The exact shape of a member of the family
depends on:
- N, which is the number of observations or
individuals in a sample.
- P, which is the probability of one of the
two events (Q=1-P is the probability of the other
event).
Some examples of specific binomial distributions are given
here.
- Normality of Binomials
- When the product of N and P and the product of N and
Q are both greater than or equal to 10, the binomial distribution
is nearly perfectly normal. Under these circumstances:
- The population mean is NP
- The population standard deviation is SQRT(NPQ)
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