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

Educational Research

Chapter 12Inferential Statistics

Gay, Mills, and Airasian

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

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

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

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

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

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

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 studiedObjectives 3.1, 3.2, & 3.4

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

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


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

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 acceptedObjectives 5.1 & 5.2

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

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


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

Tests of Significance

Two types Parametric Nonparametric


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


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


Types of Inferential Statistics

Two issues discussed Steps involved in testing for

significance Types of tests

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

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

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

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

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


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


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



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


Objective 14.1

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 alphaObjectives 15.1 & 15.2

Specific Statistical Tests

Multiple comparisons (continued) Example – examining the difference

between mean scores for Groups 1 & 2, Groups 1 & 3, and Groups 2 & 3


Objective 15.3

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 variablesObjective 16.1


Two-factor ANOVA (continued) Example – examining the mean score

differences for male and female students in an experimental or control group


Objective 16.1


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



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


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


Objective 18.2

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?


Specific Statistical Tests Chi Square (continued)


One-dimensional uses Nonparametric Tests procedures

Two-dimensional uses Crosstabs procedures


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

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mean standard error

standard error sampling

sampling error standard

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variables null

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mean objectives

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