### Inferential Statistics

```Inferential Statistics
Chapter 13
Inferential Statistics

Inferential stats are used to determine whether we
can make statements that the results found in the
present experiment reflect a true difference in the
entire population of interest and not just the sample
used in the experiment.
 Therefore inferential statistics allow us to make
predictions about the entire population based on the
findings of sample groups.
 Inferential statistics give a probability that the
difference between the two means from the sample
used in the experiment represents a true difference
based on the manipulation of the IV, and not random
error.
Null and Research Hypotheses
Null hypothesis  states simply that the
population means (after conducting the
experiment) are equal and that any observed
differences are due to random error.
 Alternative hypothesis  states that the
population means are not equal and therefore
the treatment or independent variable had an
effect.
 Statistical significance  indicates that there
is a low probability that the difference between
the obtained sample was due to random error.
 Alpha level -pre-determined probability level
used to make a decision about statistical
significance.

Probability and Sampling distributions
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Probability  likelihood of the occurrence or some
event or outcome.
Statistical Significance—is a matter of probability.
Sampling distribution  probability distributions
based on many different samples taken over and over
and shows the frequency of different sample
outcomes from many separate random samples.
Sampling distribution- is based on the assumption
that the null hypothesis is true.
Critical Values are obtained from Sampling
Distributions and they are calculations of probability
based on sample size and degrees of freedom
Sample Size
 The
sample size also has an effect on
determining statistical significance.
 The more samples you collect, the more
likely you are to obtain an accurate
estimate of the true population value
 Thus, as your sample size increases, you
can be more confident that your outcome
is actually different from the expectations
of the null hyp
Differential Statistics
 T-tests
and F-tests are differential statistics
because they detect differences between
groups.
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The sampling distribution of all possible t values
has a mean of 0 and a standard deviation of 1
It reflects all the possible outcomes we could expect
if we compared the means of two groups and the
null hypothesis is correct
T-test

The calculated t value is a ratio of two aspects of
the data
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The difference between the group means
The variability within groups
Group difference  difference between your obtained means
• Under the null hypothesis you expect this difference to be 0.
• The value of t increases as the difference betweent your
obtained sample means increases
Within-group variability  the amt of variability of scores
T-test Formula
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t=
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group difference
within-group variability
The numerator of the formula is the difference between the
means of the two groups
The denominator is the variance (s2) of each group divided
by the number of Ss in the group, which are added together
The square root of the variance divided by the number of
subjects = standard deviation
Finally, we calculate our obtained t value by dividing the
mean difference by the SD
You would then compare your obtained t to those listed in
the t-table of critical values to determine if it is significant or
not
One Tailed d vs. Two-Tailed Tests
A one-tailed test is conducted if you are
interested only in whether the obtained
value of the statistic falls in one tail of the
sampling distribution for that statistic.
--This is usually the case when your
research hypothesis is directional.
---Group one will score higher than group
two.
---The critical region in a one-tailed test
contains 5% of the total area under the
curve (alpha = .05)
Two Tailed Test

Two-tailed test if you wanted to know whether the
new therapy was either better or worse than the
standard method.
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You need to check whether your obtained statistic falls
into either tail of the distribution
There are two critical region in a two-tailed test
• To keep the probability at .05, the total percentage of cases
found in the two tails of the distribution must equal 5%
• Thus each critical region must contain 2.5% of the cases
• So the scores required to reach statistical significance must be
more extreme than was necessary for the one-tailed test
When to use a one vs. two tailed?

Major implication - for a given alpha level, you must
obtain a greater difference between the means of
your two treatment groups to reach statistical
significance if you use a two-tailed test than if you
used a one-tailed test
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The one-tailed test is more likely to detect a real
difference if one is present (that is, it is more powerful)
However, using a one-tailed test means giving up any
info about the reliability of a difference in the other,
untested direction
The general rule of thumb is: Always use a twotailed test unless there are compelling
F-test

The analysis of variance or F test is an extension of
the t test
 When a study has only one IV, F and t are virtually
identical—the F = t-squared
 ANOVA is used when there are more than two levels
of an independent variable
 The F statistic is a ratio of two types of variance:
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Systematic variance  the deviation of the group means from the
grand mean or the mean score of all individual groups
Error variance  the deviation of the individual scores in each goup
from their respective group means
The larger the F value, the more likely the score is
significant
Effect Size
size  quantifies the size of the
difference between groups
 If we have two grps, the effect size is the
difference between the groups expressed in
standard deviation units.
 Therefore, the effect size is between O and
1. The effect size indicates the strength of
the relationship. The closer to one, the
stronger the relationship.
 The advantage of the effect size is that it is
not a function of the sample size
 Effect
Type one and Type two errors
I error  occurs when the
researcher says that a relationship exists
when in fact it does not
 Type
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You have falsely rejected the null hyp
II error  occurs when the
researcher says that a relationship does
not exist, when in fact it does
 Type
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You have falsely accepted the null hyp
True State of Affairs

Reject Null
Null is true
Null is False
Type I error
Correct Decision
alpha
1-beta
C
Accept
Null
Correct Decision
1-alpha
Type II error
beta
Probability of Type II error
•
If we set a low alpha level to decrease the chances of a
Type I error (accepting a hypothesis that is true when it
is not (e.g., p<.01), we increase the chances of a Type
II error
•
True differences are more likely to be detected if the
sample size is large.
•
If the effect size is large, a Type II error is unlikely
Interpreting non-significant results
Negative or nonsignificant results are
difficult to interpret
 There are several causes for nonsignificant
results:

• The instruction could be hard to understand
• Have a weak manipulation of the indep var
• Using an unreliable or insensitive dep measure
• Sample size is too small.
Choosing a Sample Size: Power Analysis

Sample size can be based on what is typical in that
particular area of research
 Sample size can also be based on a desired
probability of correctly rejecting the null hyp

This probability is called the power of the statistical test 
the sensitivity of the statistical procedure to detect