1. Educational Research
Chapter 12
Inferential Statistics
Gay, Mills, and Airasian

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

16. Tests of Significance
 Two types
 Parametric
 Nonparametric

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