© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Chapter 13 Using Inferential Statistics

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Basic Concepts Sampling Distribution The distribution of every possible sample taken from a population The critical values of a statistic are the sampling distribution for that statistic Sampling Error The difference between a sample mean and the population mean The standard error of the mean is a measure of sampling error

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Degrees of Freedom The number of scores in sample with a known mean that are free to vary and is defined as n-1 Used to find the appropriate tabled critical value of a statistic Parametric vs. Nonparametric Statistics Parametric statistics make assumptions about the nature of an underlying population Nonparametric statistics make no assumptions about the nature of an underlying population

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Relationship Between Population and Samples When a Treatment Had No Effect

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Relationship Between Population and Samples When a Treatment Had An Effect

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Statistical Errors H o TrueH o False Reject H o Type I error Correct decision Do not reject H o Correct decision Type II error True State of Affairs Decision

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Parametric Statistics Assumptions Scores are sampled randomly from the population The sampling distribution of the mean is normal Within-groups variances are homogeneous Serious violation of one or more assumption(s) may bias a statistical analysis Two-Sample Tests t test for independent samples used when subjects were randomly assigned to your two groups t test for independent samples used when samples are not independent (e.g., repeated measure)

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. z test for the difference between two proportions is used to determine if two proportions differ significantly Beyond Two Samples The Analysis of Variance is used when you have more than two groups in an experiment The F-ratio is the statistic computed in an Analysis of Variance and is compared to critical values of F A significant overall F may require further planned or unplanned (post hoc) follow-up analyses The analysis of variance may be used with unequal sample size (weighted or unweighted means analysis)

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Factorial Designs The Analysis of Variance is also used to analyze data from multifactor designs Main effects and interactions can be evaluated If an interaction is significant, main effects are not normally interpreted Versions of the Analysis of Variance are available for mixed designs and other specialized designs

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Nonparametric Statistics Used if data violate assumptions of parametric statistics or if you have ordinal or nominal data Chi-square Used when your dependent variable is a dichotomous decision (e.g., yes or no) Chi-square for contingency tables is used when you have more than one variable to analyze Mann-Whitney U-Test Used when data scaled on at least an ordinal scale Good nonparametric alternative to the t-test when assumptions are violated

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. The Wilcoxon Signed Ranks Test Used for a single-factor design with correlated samples

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Special Topics Power of a statistical test Power refers to a statistic’s ability to detect differences between groups Power is affected by The alpha level chosen Sample size Whether a one-tailed or two-tailed test is used Effect size Power can be determined statistically

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Statistical vs. Practical Significance A statistically significant effect is not likely due to chance Statistical significance does not mean that a difference is important A finding may have practical significance if the finding has practical applications

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. The meaning of statistical significance The alpha level adopted (e.g., p <.05) tells you the likelihood of making a type I error A finding found to be significant at p <.01 is NOT more significant than one at p <.05

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Data Transformations Data may need to be transformed with a data transformation Adding or subtracting a constant from each datum does not change the shape of the original frequency distribution The mean and standard deviation do not change Multiplying by a constant does change the distribution and the mean and standard deviation This is a linear transformation You may need to transform data if the data do not meet assumptions of statistical tests Data must be rechecked for other problems

© 2005 The McGraw-Hill Companies, Inc., All Rights Reserved. Alternatives to Inferential Statistics Some research designs preclude using inferential statistics (e.g., single-subject design) Reliability of data may be checked using replication You should be able to repeat (replicate) a reliable finding Replication need not be limited to single-subject designs