### 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

### 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.

### 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

Use what we know about this THEORETICAL

distribution to make inferences about

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)

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

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