2. Topics Discussed in this Chapter
Concepts underlying inferential statistics
Types of inferential statistics
Parametric
T tests
ANOVA
One-way
Factorial
Post-hoc comparisons
Multiple regression
ANCOVA
Nonparametric
Chi square
3. Important Perspectives
Inferential statistics
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
4. Underlying Concepts
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
5. Sampling Distributions
A distribution of sample statistics
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
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
6. Standard Error
Sampling error – the expected random or chance
variation of means in sampling distributions
The calculation of standard errors to estimate
sampling error
Standard error of the mean
Formula
Dependency on sample size with n in the denominator
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
7. Null and Alternative Hypotheses
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
8. Null and Alternative Hypotheses
Comparisons between groups
Null: no difference between the mean scores of
the groups
Alternative: differences between the mean scores
of the groups
Relationships between variables
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
9. Null and Alternative Hypotheses
Acceptance of the null Rejection of the null
hypothesis hypothesis
The difference between
The difference between
groups is so large it can
groups is too small to
be attributed to
attribute it to anything but something other than
chance chance (e.g.,
The relationship between experimental treatment)
variables is too small to The relationship between
attribute it to anything but variables is so large it
chance can be attributed to
something other than
chance (e.g., a real
relationship)
Objectives 3.3 & 4.2
10. Tests of Significance
Statistical analyses to help decide whether to
accept or reject the null hypothesis
Alpha level
An established probability level which serves as
the criterion to determine whether to accept or
reject the null hypothesis
Common levels in education
.01
.05
.10
Objectives 4.1 & 6.1
11. Tests of Significance
Specific tests are used in specific
situations based on the number of
samples and the statistics of interest
One-sample tests of the mean, variance,
proportions, correlations, etc.
Two-sample tests of means, variances,
proportions, correlations, etc.
Objective 4.1
12. Type I and Type II Errors
Correct decisions
The null hypothesis is true and it is accepted
The null hypothesis is false and it is rejected
Incorrect decisions
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
13. Type I and Type II Errors
Reciprocal relationship between Type I and
Type II errors
Control of Type I errors using alpha level
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
14. One-Tailed and Two-Tailed Tests
One-tailed – an anticipated outcome in a specific
direction
Treatment group is significantly higher than the control group
Treatment group is significantly lower than the control group
Two-tailed – anticipated outcome not directional
Treatment and control groups are equal
Ample justification needed for using one-tailed tests
Objectives 7.1 & 7.2
15. Degrees of Freedom
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
17. Tests of Significance
Four assumptions of parametric tests
Normal distribution of the dependent variable
Interval or ratio data
Independence of subjects
Homogeneity of variance
Advantages of parametric tests
More statistically powerful
More versatile
Objectives 8.1 & 8.2
18. Tests of Significance
Assumptions of nonparametric tests
No assumptions about the shape of the
distribution of the dependent variable
Ordinal or categorical data
Disadvantages of nonparametric tests
Less statistically powerful
Require large samples
Cannot answer some research questions
Objectives 8.3 & 8.4
19. Types of Inferential Statistics
Two issues discussed
Steps involved in testing for significance
Types of tests
20. Steps in Statistical Testing
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
21. Steps in Statistical Testing
Identify the criteria for significance
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
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
22. Steps in Statistical Testing
Accept or reject the null hypothesis
Accept
The observed test statistic is smaller than the critical
value
The observed probability level of the observed statistic is
smaller than alpha
Reject
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
23. Two Important Issues
Types of samples
Independent samples
Two or more distinct groups are measured on a
single variable
Groups are independent of one another
Dependent samples
One group measured on two or more variables
Objective 10.1
24. Two Important Issues
Gain scores
Subtracting the pretest scores from the posttest
scores
Serious problems with this analysis
Each subject does not have the same opportunity for
“gain”
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
25. Specific Statistical Tests
T test for independent samples
Comparison of two means from independent
samples
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
26. Specific Statistical Tests
T test for dependent samples
Comparison of two means from dependent
samples
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
27. Specific Statistical Tests
Simple analysis of variance (ANOVA)
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
28. Specific Statistical Tests
Multiple comparisons
Omnibus ANOVA results
Significant difference indicates whether a difference
exists across all pairs of scores
Need to know which specific pairs are different
Types of tests
A priori contrasts
Post-hoc comparisons
Scheffe
Tukey HSD
Duncan’s Multiple Range
Conservative or liberal control of alpha
Objectives 15.1 & 15.2
29. Specific Statistical Tests
Multiple comparisons (continued)
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
30. Specific Statistical Tests
Two-factor ANOVA
Also known as factorial ANOVA
Comparison of means when two
independent variables are being examined
Effects
Two main effects – one for each independent
variable
One interaction effect for the simultaneous
interaction of the two independent variables
Objective 16.1
31. Specific Statistical Tests
Two-factor ANOVA (continued)
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
32. Specific Statistical Tests
Analysis of covariance (ANCOVA)
Comparison of two or more means with statistical
control of an extraneous variable
Use of a covariate
Advantages
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
33. Specific Statistical Tests
Multiple regression
Correlational technique which uses
multiple predictor variables to predict a
single criterion variable
Characteristics
Increased predictability with additional variables
Regression coefficients
Regression equations
Objective 18.1
34. Specific Statistical Tests
Multiple regression (continued)
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
35. Specific Statistical Tests
Chi Square
A nonparametric test in which observed proportions are
compared to expected proportions
Types
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
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
36. Specific Statistical Tests
Chi Square (continued)
Computation of the test statistic
SPSS-Windows syntax
One-dimensional uses Nonparametric Tests
procedures
Two-dimensional uses Crosstabs procedures
Objectives 19.1 & 19.2