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

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 hypothesis
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
Rejection of the null hypothesis
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

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
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