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Educational Research
Chapter 12
Inferential Statistics
Gay, Mills, and Airasian
Topics Discussed in this Chapter
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Concepts underlying inferential statistics
Types of inferential statistics
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Parametric
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T tests
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ANOVA
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One-way
Factorial
Post-hoc comparisons
Multiple regression
ANCOVA
Nonparametric
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Chi square
Important Perspectives
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Inferential statistics
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Allow researchers to generalize to a population of
individuals based on information obtained from a
sample of those individuals
Assess whether the results obtained from a
sample are the same as those that would have
been calculated for the entire population
Probabilistic nature of inferential analyses
Underlying Concepts
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Sampling distributions
Standard error
Null and alternative hypotheses
Tests of significance
Type I and Type II errors
One-tailed and two-tailed tests
Degrees of freedom
Tests of significance
Sampling Distributions
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A distribution of sample statistics
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A distribution of mean scores
A distribution of the differences between two mean scores
A distribution of the ratio of two variances
Known statistical properties of sampling distributions
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The mean of the sampling distribution of means is an
excellent estimate of the population mean
The standard error of the mean is an excellent estimate of
the “standard deviation” of the sampling distribution of the
mean
Objectives 1.1 & 1.2
Standard Error
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Sampling error – the expected random or chance
variation of means in sampling distributions
The calculation of standard errors to estimate
sampling error
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Standard error of the mean
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Formula
Dependency on sample size with n in the denominator
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The larger the sample, the smaller the standard error of the mean
Standard error of the differences between two means
Objectives 1.2, 1.3, & 1.4
Null and Alternative Hypotheses
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The null hypothesis represents a
statistical tool important to inferential
tests of significance
The alternative hypothesis usually
represents the research hypothesis
related to the study
Null and Alternative Hypotheses
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Comparisons between groups
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Null: no difference between the mean scores of
the groups
Alternative: differences between the mean scores
of the groups
Relationships between variables
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Null: no relationship exists between the variables
being studied
Alternative: a relationship exists between the
variables being studied
Objectives 3.1, 3.2, & 3.4
Null and Alternative Hypotheses
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Acceptance of the null
hypothesis
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The difference between
groups is too small to
attribute it to anything
but chance
The relationship between
variables is too small to
attribute it to anything
but chance
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Rejection of the null
hypothesis
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The difference between
groups is so large it can
be attributed to
something other than
chance (e.g.,
experimental treatment)
The relationship between
variables is so large it
can be attributed to
something other than
chance (e.g., a real
relationship)
Objectives 3.3 & 4.2
Tests of Significance
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Statistical analyses to help decide whether to
accept or reject the null hypothesis
Alpha level
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An established probability level which serves as
the criterion to determine whether to accept or
reject the null hypothesis
Common levels in education
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.01
.05
.10
Objectives 4.1 & 6.1
Tests of Significance
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Specific tests are used in specific
situations based on the number of
samples and the statistics of interest
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One-sample tests of the mean, variance,
proportions, correlations, etc.
Two-sample tests of means, variances,
proportions, correlations, etc.
Objective 4.1
Type I and Type II Errors
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Correct decisions
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The null hypothesis is true and it is accepted
The null hypothesis is false and it is rejected
Incorrect decisions
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Type I error - the null hypothesis is true and it is
rejected
Type II error - the null hypothesis is false and it is
accepted
Objectives 5.1 & 5.2
Type I and Type II Errors
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Reciprocal relationship between Type I and
Type II errors
Control of Type I errors using alpha level
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As alpha becomes smaller (.10, .05, .01, .001,
etc.) there is less chance of a Type I error
Value and contextual based nature of
concerns related to Type I and Type II errors
Objective 5.3
One-Tailed and Two-Tailed Tests
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One-tailed – an anticipated outcome in a specific
direction
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Two-tailed – anticipated outcome not directional
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Treatment group is significantly higher than the control
group
Treatment group is significantly lower than the control group
Treatment and control groups are equal
Ample justification needed for using one-tailed tests
Objectives 7.1 & 7.2
Degrees of Freedom
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Statistical artifacts that affect the
computational formulas used in tests of
significance
Used when entering statistical tables to
establish the critical values of the test
statistics
Tests of Significance
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Two types
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Parametric
Nonparametric
Tests of Significance
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Four assumptions of parametric tests
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Normal distribution of the dependent variable
Interval or ratio data
Independence of subjects
Homogeneity of variance
Advantages of parametric tests
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More statistically powerful
More versatile
Objectives 8.1 & 8.2
Tests of Significance
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Assumptions of nonparametric tests
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No assumptions about the shape of the
distribution of the dependent variable
Ordinal or categorical data
Disadvantages of nonparametric tests
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Less statistically powerful
Require large samples
Cannot answer some research questions
Objectives 8.3 & 8.4
Types of Inferential Statistics
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Two issues discussed
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Steps involved in testing for significance
Types of tests
Steps in Statistical Testing
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State the null and alternative
hypotheses
Set alpha level
Identify the appropriate test of
significance
Identify the sampling distribution
Identify the test statistic
Compute the test statistic
Objectives 20.1 – 20.9
Steps in Statistical Testing
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Identify the criteria for significance
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If computing by hand, identify the critical value of the test
statistic
If using SPSS-Windows, identify the probability level of the
observed test statistic
Compare the computed test statistic to the criteria for
significance
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If computing by hand, compare the observed test statistic to
the critical value
If using SPSS-Windows, compare the probability level of the
observed test statistic to the alpha level
Objectives 20.1 – 20.9
Steps in Statistical Testing
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Accept or reject the null hypothesis
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Accept
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The observed test statistic is smaller than the critical
value
The observed probability level of the observed statistic is
smaller than alpha
Reject
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The observed test statistic is larger than the critical value
The observed probability level of the observed statistic is
smaller than alpha
Objective 20.9
Two Important Issues
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Types of samples
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Independent samples
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Two or more distinct groups are measured on a
single variable
Groups are independent of one another
Dependent samples
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One group measured on two or more variables
Objective 10.1
Two Important Issues
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Gain scores
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Subtracting the pretest scores from the posttest
scores
Serious problems with this analysis
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Each subject does not have the same opportunity for
“gain”
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A person scoring close to the top of the test doesn’t have
as much to gain as someone scoring in the middle of the
test
Low reliability
ANCOVA as an appropriate analysis
Objectives 13.1 & 13.2
Specific Statistical Tests
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T test for independent samples
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Comparison of two means from independent
samples
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Samples in which the subjects in one group are not
related to the subjects in the other group
Example - examining the difference between the
mean pretest scores for an experimental and
control group
Computation of the test statistic
SPSS-Windows syntax
Objectives 9.1 & 11.1
Specific Statistical Tests
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T test for dependent samples
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Comparison of two means from dependent
samples
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One group is selected and mean scores are compared for
two variables
Two groups are compared but the subjects in each group
are matched
Example – examining the difference between
pretest and posttest mean scores for a single class
of students
Computation of the test statistic
SPSS-Windows syntax
Objectives 9.1 & 12.1
Specific Statistical Tests
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Simple analysis of variance (ANOVA)
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Comparison of two or more means
Example – examining the difference
between posttest scores for two treatment
groups and a control group
Computation of the test statistic
SPSS-Windows syntax
Objective 14.1
Specific Statistical Tests
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Multiple comparisons
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Omnibus ANOVA results
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Significant difference indicates whether a difference
exists across all pairs of scores
Need to know which specific pairs are different
Types of tests
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A priori contrasts
Post-hoc comparisons
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Scheffe
Tukey HSD
Duncan’s Multiple Range
Conservative or liberal control of alpha
Objectives 15.1 & 15.2
Specific Statistical Tests
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Multiple comparisons (continued)
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Example – examining the difference
between mean scores for Groups 1 & 2,
Groups 1 & 3, and Groups 2 & 3
Computation of the test statistic
SPSS-Windows syntax
Objective 15.3
Specific Statistical Tests
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Two-factor ANOVA
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Also known as factorial ANOVA
Comparison of means when two
independent variables are being examined
Effects
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Two main effects – one for each independent
variable
One interaction effect for the simultaneous
interaction of the two independent variables
Objective 16.1
Specific Statistical Tests
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Two-factor ANOVA (continued)
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Example – examining the mean score
differences for male and female students in
an experimental or control group
Computation of the test statistic
SPSS-Windows syntax
Objective 16.1
Specific Statistical Tests
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Analysis of covariance (ANCOVA)
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Comparison of two or more means with statistical
control of an extraneous variable
Use of a covariate
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Advantages
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Statistically controlling for initial group differences (i.e.,
equating the groups)
Increased statistical power
Pretest is typically the covariate
Computation of the test statistic
SPSS-Windows syntax
Objectives 17.1 & 17.2
Specific Statistical Tests
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Multiple regression
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Correlational technique which uses multiple
predictor variables to predict a single
criterion variable
Characteristics
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Increased predictability with additional
variables
Regression coefficients
Regression equations
Objective 18.1
Specific Statistical Tests
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Multiple regression (continued)
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Example – predicting college freshmen’s
GPA on the basis of their ACT scores, high
school GPA, and high school rank in class
Computation of the test statistic
SPSS-Windows syntax
Objective 18.2
Specific Statistical Tests
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Chi Square
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A nonparametric test in which observed proportions are
compared to expected proportions
Types
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One-dimensional – comparing frequencies occurring in different
categories for a single group
Two-dimensional – comparing frequencies occurring in different
categories for two or more groups
Examples
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Is there a difference between the proportions of parents in
favor of or opposed to an extended school year?
Is there a difference between the proportions of husbands and
wives who are in favor of or opposed to an extended school
year?
Objectives 19.1 & 19.2
Specific Statistical Tests
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Chi Square (continued)
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Computation of the test statistic
SPSS-Windows syntax
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One-dimensional uses Nonparametric Tests
procedures
Two-dimensional uses Crosstabs procedures
Objectives 19.1 & 19.2