This chapter discusses inferential statistics and their underlying concepts. It covers key topics like types of inferential statistics (parametric vs nonparametric), important perspectives like generalizing from samples to populations, conceptual underpinnings such as null/alternative hypotheses and errors. Specific statistical techniques are explained like t-tests, ANOVA, regression as well as important considerations like sample distributions, standard error, degrees of freedom, and the steps to conduct statistical tests.

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