### Stats Refresher 2.1

```Data Analysis and Interpretation

Null hypothesis testing is used to
determine whether mean differences
among groups in an experiment are
greater than the differences that are
expected simply because of error
variation (chance).


The first step in null hypothesis testing
is to assume that the groups do not
differ — that is, that the independent
variable did not have an effect (i.e., the
null hypothesis — H0).
Probability theory is used to estimate the
likelihood of the experiment’s observed
outcome, assuming the null hypothesis
is true.

A statistically significant outcome is one
that has a small likelihood of occurring if
the null hypothesis is true.
◦ We reject the null hypothesis, and conclude that
the independent variable did have an effect on
the dependent variable.
◦ A statistically significant outcome indicates that
the difference between means obtained in an
experiment is larger than would be expected if
error variation alone (i.e., chance) were
responsible for the outcome.

How small does the probability have to
be in order to decide that a finding is
statistically significant?
◦ Consensus among members of the scientific
community is that outcomes associated with
probabilities of less than 5 times out of 100 (p
< .05) if the null hypothesis were true are
judged to be statistically significant.
◦ This is called alpha (α) or the level of
significance.

What does a statistically significant
outcome tell us?
◦ An outcome with a probability just below .05
(and thus statistically significant) has about a
50/50 chance of being repeated in an exact
replication of the experiment.
◦ As the probability of the outcome of the
experiment decreases (e.g., p = .025, p = .01,
p = .005), the likelihood of observing a
statistically significant outcome (p < .05) in an
exact replication increases.
◦ APA recommends reporting the exact
probability of the outcome.

What do we conclude when a finding is
not statistically significant?
◦ We do not reject the null hypothesis if there is
no difference between groups.
◦ However, we don’t necessarily accept the null
hypothesis either — that is, we don’t conclude
that the independent variable did not have an
effect.
◦ We cannot make a conclusion about the effect
of the independent variable. Some factor in the
experiment may have prevented us from
observing an effect of the independent variable
(e.g., too few participants).

Because decisions about the outcome of
an experiment are based on
probabilities, Type I or Type II errors
may occur.

A Type I error occurs when the null
hypothesis is rejected, but the null
hypothesis is true.
◦ That is, we claim that the independent
variable is statistically significant (because we
observed an outcome with p < .05) when
there really is no effect of the independent
variable.
◦ The probability of a Type I error is alpha — or
the level of significance (p = .05).

A Type II error occurs when the null
hypothesis is false, but it is not rejected.
◦ That is, we claim that the independent
variable is not statistically significant (because
we observed an outcome with p > .05) when
there really is an effect of the independent
variable that our experiment missed.

Because of the possibility of Type I and
Type II errors, researchers are tentative
in their claims. We use words such as
“support for the hypothesis” or
“consistent with the hypothesis” rather
than stating that a hypothesis has been
“proven.”


The appropriate inferential statistical test
when comparing two means obtained from
different groups of participants is a t -test
for independent groups.
The appropriate test when comparing two
means obtained from the same participants
(or matched groups) is a repeated measures
(within-subjects) t-test.

Research Design
◦ Analysis of Independent Variable using two
conditions
 Experimental
 Control
◦ Same group of subjects is used
◦ Each subject receives the experimental and control
◦ Subjects may be also be matched according to
certain characteristics


Statistic is based on the difference between
the scores of correlated subjects
Score compared to a difference of 0
◦ Null hypothesis assumes no difference
◦ Population mean is equal to 0

T critical obtained in same manner as t test
for single samples
t obt
D obt

sD
t obt 
D obt
SS D
N N  1
N
 D


2
SS D 

D
2
N



Used to analyze data from experiments that
use more than two groups or conditions
F is a ratio of two independent variance
estimates
Since F is a ratio of variance estimates, it will
never be negative

F test allows us to make one overall
comparison that tells whether there is a
significant difference between the means of
the groups

F distribution
◦ F distribution is positively skewed
◦ Median F value equals one
◦ F distribution is a family of curves based on the
degrees of freedom (df)


In the independent groups design, there are
as many groups as there are levels of the
independent variable
Hypothesis testing:
◦ Nondirectional
◦ H0 states that there is no difference between
conditions

ANOVA partitions total variability of data (SST)
into the variability that exists within each
group (SSW) and the variability between
groups (SSB)
◦ SS= Sum of Squares


SSB and SSW are both used as
independent estimates of the H0
population variance
F ratio
2
Fobt 
betw een - groups variance estim ate ( s B )
2
w ithin - groups variance estim ate ( sW )

The ANOVA Summary Table provides the
information for estimating the sources of variance:
between groups and within groups.
Source Sum of Squares (SS)
p
Between Groups
Within Groups


54.55
df
Variance Estimate F-test
3
18.18
37.20 16
2.33
7.80
.002
The F-test is the Between group variance estimate
is divided by the within group variance
estimate(18.18 ÷ 2.33 = 7.80).
This F-test is statistically significant because .002
< .05.
```