Educational Research

Chapter 13Inferential Statistics

Gay, Mills, and Airasian10th Edition

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 Sampling distribution tries to imagine that

you would take multiple samples from your population to get multiple means and standard deviations so that you can calculate your inferential statistics based on the most representative numbers for your population.

As you get more samples, you get better information.

Standard Error Error that occurs at random because

you used a sample (and not the whole population). The more you know about the information from a true sampling distribution (more Ss in your sample or more samples from your population) the lower your standard error.

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 studied

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)

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

Way of thinking: .10 … there is a 10% probability that this even happened by chance.

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 (there really isn’t anything going on but you decide there is)

Type II error - the null hypothesis is false and it is accepted (there really is something going on but you decide there isn’t)

Type I and Type II Errors

Power: ability of a significance test to reject a null hypothesis that is false (avoid Type II error)

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 Meaning… as you move from 10% possibility

of it being chance to a 1% possibility of it being chance, you are less likely to be incorrect when you say that something did not happen by chance.

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 You decide BEFORE you start your study if you

think that you will do a 1-tailed or 2-tailed test Ample justification needed for using one-tailed

tests (this may be previous studies, etc.)

Degrees of Freedom

The more things you want to measure (IV) means that you are more likely to make an error in your model. Thus, you want to use degrees of freedom in your calculation (and not the actual number) to correct for this chance of making an error. This can happen in samples as well. Thus, you usually see things like df=N-1 (where n=sample)

Used when entering statistical tables to establish the critical values of the test statistics

Tests of Significance Two types

Parametric: A type of statistical test that has certain “assumptions” that must be met before your can use it.

Nonparametric: Good for ordinal or nominal scale data, data the the “parametric assumptions” have been violated, or when you don’t know if it is a normal distribution or not.

Tests of Significance Four assumptions of parametric tests

Normal distribution of the dependent variable Interval or ratio data Independence of subjects: selection one S does

not effect the selection of another S (met by random sampling)

Homogeneity of variance: Variance of the data from the sample should be equal.

Advantages of parametric tests More statistically powerful More versatile

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

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 (Table 13.12 in your text) Identify the sampling distribution Identify the test statistic Compute the test statistic

Steps in Statistical Testing Identify the criteria for significance

Decide what would be considered statistically significant for you.

This may be done by deciding on your alpha level.

Compare the computed test statistic to the criteria for significance If using SPSS-Windows, compare the

probability level of the observed test statistic to the alpha level

Steps in Statistical Testing

Accept or reject the null hypothesis Accept

The observed probability level of the observed statistic is larger than alpha

Reject The observed probability level of the

observed statistic is smaller than alpha

Specific Statistical Tests T tests

Comparison of two means Example - examining the difference

between the mean pretest scores for an experimental and control group

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

Look at page 341-342 for an overview of how this is calculated.

Look at page 347 to see how to interpret it.

Specific Statistical Tests Multiple comparisons

Omnibus ANOVA results More than two sets of means are calculated (e.g.,

pretest, during test, posttest). 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 (planned… you expect a difference

b/w only 2 sets of means before hand) Post-hoc comparisons (unplanned… you are trying to

figure out where the differences are) Scheffe (type of post-hoc comparison) Tukey HSD (type of post-hoc comparison)

Specific Statistical Tests

Multiple comparisons (continued) Example – examining the difference

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

Specific Statistical Tests

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

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

Specific Statistical Tests

Analysis of covariance (ANCOVA) Comparison of two or more means

with statistical control of an extraneous variable

Specific Statistical Tests

Multiple regression Correlational technique which uses

multiple predictor variables to predict a single criterion variable

Does not use variance or standard deviation to determine significance.

You look for the relationship among things.

Relationship b/w gender, personality type, and previous scores on college GPA

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

Specific Statistical Tests Chi Square

A nonparametric test in which observed proportions are compared to expected proportions

Looks at frequency counts, percentages, or proportions

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?