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Use sample data to...
(1) test hypothesis about single variable
or association between two variables
(2) Determine whether we can generalize
results back to population
We use probability theory to...
Identify type(s) of samples that would most
likely be drawn from given population
Make theoretical connections between given
population & corresponding sample
Make inferences (draw conclusions) from
sample back to given population
behavioral stats graph
population--probability theory-->sample,sample (inf. stats)--generalization-->pop.
To make inferences about population from
sample using probability theory, select
elements by this
Each element in population has an EQUAL
and KNOWN chance of being selected
Probabilities must stay constant from one
selection to next (replacement)
Empirical vs. Theoretical Probability
Until now, we have been working with
EMPIRICAL distributions - based on real data
THEORETICAL distributions have many of
same properties as EMPIRICAL distributions
But constructed using probability theory
rather than actual observations
The Normal Distribution
Many real-life phenomena approximate
NORMAL distribution (ie. height, athletic
ability, social attitudes, etc)
THEORETICAL because it is based upon
what distribution of scores on variable from
infinitely large population would look like
Use what we know about this THEORETICAL
distribution to make inferences about
population from sample data
Properties of Normal Distribution
Highest frequency in middle of curve
Frequencies tapering off as move toward
Always True for Normal Distribution
Symmetric (not skewed)
Mean, median, mode are equivalent & fall in
center of curve
Specific proportion of cases (area under
the normal curve) fall within 1 SD of mean,
2 SD of mean, 3 SD of mean
In theoretical normal distribution...
68.26% of cases fall within 1 standard
deviation of mean
95.44% of cases fall within 2 standard
deviations of mean
99.74% of cases fall within 3 standard
deviations of mean
Why are properties of normal
Many naturally occurring (empirical)
distributions follow normal curve
PROBABILITY of random observation having
value between x1 and x2 equals
PROPORTION of cases between x1 and x2
Normal distribution is guaranteed in certain
Due to these properties we can...
Calculate how many standard deviations
from sample mean given observation is
Make accurate predictions about probability
of variable taking on a specific value or an
What is a z-score good for?
Just standardized score that indicates the distance, in
SD units, of particular value from mean
In other words, z equals number of standard
deviations observation is from mean
When observation is standardized & variable is
We can use properties of normal curve to PREDICT
probability of random observation taking on any
range of values
Steps to Determine Probability
Step #1: Translate raw scores to z-scores (standardize)/Step #2: Interpret z-scores & visualize area
under curve/Step #3: Determine area under normal curve that falls
between values of interest/Step#4:Look up z-score
What does this have to do with
Take sample data & ask how extreme a zscore
to expect before proclaiming that
observed value is not due to chance, but to
some other factor
This is especially important when we are
using sample statistics to estimate population
Because sampling procedures involve certain
amount of error
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