Educational Research Chapter 12 Inferential Statistics Gay, Mills, and Airasian.
Post on 24-Dec-2015
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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
Objectives 4.1 & 6.1
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
Objectives 7.1 & 7.2
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
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
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
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
Objectives 13.1 & 13.2
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
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
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
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
Computation of the test statistic SPSS-Windows syntax
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
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
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
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
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
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
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