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Chapter 13

Chapter 13. Using Inferential Statistics. 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

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Chapter 13

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  1. Chapter 13 Using Inferential Statistics

  2. 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

  3. 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

  4. Relationship Between Population and Samples When a Treatment Had No Effect

  5. Relationship Between Population and Samples When a Treatment Had An Effect

  6. Statistical Errors True State of Affairs Decision

  7. 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)

  8. 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)

  9. 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

  10. 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

  11. The Wilcoxon Signed Ranks Test • Used for a single-factor design with correlated samples

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

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