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Educational Research Chapter 13 Inferential Statistics Gay, Mills, and Airasian 10th 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 Two-tailed – anticipated outcome not directional Treatment group is significantly higher than the control group Treatment group is significantly lower than the control group 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?