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# Ch12

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### Ch12

1. 1. Educational Research Chapter 12 Inferential Statistics Gay, Mills, and Airasian
2. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 16. Tests of Significance Two types  Parametric  Nonparametric
17. 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. 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. 19. Types of Inferential Statistics Two issues discussed  Steps involved in testing for significance  Types of tests
20. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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