Download Chapter 13 - Weber State University

Chapter 13
Using Inferential Statistics
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.
Basic Concepts

Sampling Distribution

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
The distribution of every possible sample taken from a
population
The critical values of a statistic are the sampling
distribution for that statistic
Sampling Error


The difference between a sample mean and the
population mean
The standard error of the mean is a measure of
sampling error
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

Degrees of Freedom


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The number of scores in sample with a known mean
that are free to vary and is defined as n-1
Used to find the appropriate tabled critical value of a
statistic
Parametric vs. Nonparametric Statistics


Parametric statistics make assumptions about the
nature of an underlying population
Nonparametric statistics make no assumptions about
the nature of an underlying population
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.
Relationship Between Population
and Samples When a Treatment
Had No Effect
Population

Sample 1
M1
Sample 2
M2
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Relationship Between Population
and Samples When a Treatment
Had An Effect
Control
group
population
c
Treatment
group
population
t
Treatment
group
sample
Control
group
sample
Mc
Mt
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.
Statistical Errors
True State of Affairs
Decision
Reject Ho
Do not
reject Ho
Ho True
Ho False
Type I
error
Correct
decision
Correct
decision
Type II
error
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.
Parametric Statistics

Assumptions

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Scores are sampled randomly from the population
The sampling distribution of the mean is normal
Within-groups variances are homogeneous

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Serious violation of one or more assumption(s) may bias a
statistical analysis
Two-Sample Tests


t test for independent samples used when subjects
were randomly assigned to your two groups
t test for independent samples used when samples are
not independent (e.g., repeated measure)
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

z test for the difference between two proportions is
used to determine if two proportions differ significantly

Beyond Two Samples

The Analysis of Variance is used when you have more
than two groups in an experiment

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The F-ratio is the statistic computed in an Analysis of Variance
and is compared to critical values of F
A significant overall F may require further planned or
unplanned (post hoc) follow-up analyses
The analysis of variance may be used with unequal sample
size (weighted or unweighted means analysis)
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

Factorial Designs

The Analysis of Variance is also used to analyze data
from multifactor designs

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Main effects and interactions can be evaluated
If an interaction is significant, main effects are not normally
interpreted
Versions of the Analysis of Variance are available for mixed
designs and other specialized designs
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.
Nonparametric Statistics


Used if data violate assumptions of parametric
statistics or if you have ordinal or nominal data
Chi-square

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Used when your dependent variable is a dichotomous
decision (e.g., yes or no)
Chi-square for contingency tables is used when you
have more than one variable to analyze
Mann-Whitney U-Test


Used when data scaled on at least an ordinal scale
Good nonparametric alternative to the t-test when
assumptions are violated
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

The Wilcoxon Signed Ranks Test

Used for a single-factor design with correlated samples
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.
Special Topics

Power of a statistical test

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Power refers to a statistic’s ability to detect differences
between groups
Power is affected by
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The alpha level chosen
Sample size
Whether a one-tailed or two-tailed test is used
Effect size
Power can be determined statistically
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

Statistical vs. Practical Significance

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A statistically significant effect is not likely due to
chance
Statistical significance does not mean that a difference
is important
A finding may have practical significance if the finding
has practical applications
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

The meaning of statistical significance


The alpha level adopted (e.g., p < .05) tells you
the likelihood of making a type I error
A finding found to be significant at p < .01 is NOT
more significant than one at p < .05
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

Data Transformations

Data may need to be transformed with a data
transformation

Adding or subtracting a constant from each datum does not
change the shape of the original frequency distribution

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Multiplying by a constant does change the distribution and the
mean and standard deviation
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The mean and standard deviation do not change
This is a linear transformation
You may need to transform data if the data do not
meet assumptions of statistical tests

Data must be rechecked for other problems
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.

Alternatives to Inferential Statistics

Some research designs preclude using inferential
statistics (e.g., single-subject design)

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Reliability of data may be checked using replication
 You should be able to repeat (replicate) a reliable finding
Replication need not be limited to single-subject
designs
© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved.