lecture 6 probability and normal curve

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descriptive stats

Used to describe characteristics
(central tendency & dispersion) of population

inferential stats

Use sample data to...
(1) test hypothesis about single variable
or association between two variables
AND
(2) Determine whether we can generalize
results back to population

population

set of elements researcher
wants to understand (ie. adolescents in U.S.)

sample

subset of population that is actually
studied

We use probability theory to...

Identify type(s) of samples that would most
likely be drawn from given population
distribution
 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.

probability

defined as
proportion, fraction, or percent of all possible
outcomes

Random Sampling

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 Distributions

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
population from sample data

Properties of Normal Distribution

Bell-shaped
 Perfectly symmetrical
 Highest frequency in middle of curve
 Frequencies tapering off as move toward
either extreme

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
68-95-99 RULE

Why are properties of normal distribution important?

Many naturally occurring (empirical)
 PROBABILITY of random observation having
value between x1 and x2 equals
PROPORTION of cases between x1 and x2
 Normal distribution is guaranteed in certain
circumstances

Due to these properties we can...

Calculate how many standard deviations
from sample mean given observation is
AND
 Make accurate predictions about probability
of variable taking on a specific value or an
event occurring

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
normally distributed...
 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 inferential stats?

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
parameters
 Because sampling procedures involve certain
amount of error

Example: