1. 15 Inferential Statistics

2. Inferential Statistics • Inferential statistics involve using sample data to make inferences about populations • a statistic is a numerical index based on sample data • a parameter is a numerical characteristic of a population

3. Sampling distributions • A sampling distribution is a theoretical distribution of values of a statistic consisting of every possible sample of a given size from a population • standard error – the standard deviation of a sampling distribution • test statistic – statistic that follows a known sampling distribution and is used in significance testing

4. Estimation • A branch of inferential statistics involved in estimating population parameters • point estimation – use value of sample statistic as estimate of the value of population parameter (e.g., sample mean to estimate population mean)

5. Estimation (cont'd) • A branch of inferential statistics involved in estimating population parameters • interval estimation • confidence interval – includes a range of numbers that will contain the population parameter with a certain degree of certainty. e.g., 95% confidence intervals include a range of values that will contain the population parameter 95% of the time

6. Hypothesis Testing • Branch of inferential statistics used when testing the predicted relationship between variables • null hypothesis - a statement regarding the population parameter – typically that no relationship exists between the independent and dependent variables

7. Hypothesis Testing (cont'd) • Branch of inferential statistics used when testing the predicted relationship between variables • alternative hypothesis – states that there is a relationship between independent and dependent variables

8. Hypothesis Testing (cont'd) • Steps of hypothesis testing • state the null and alternative hypotheses • begin by assuming that the null hypothesis is true (that the independent variable has no effect) • determine the standard for rejecting the null hypothesis (i.e., identify the level of significance)

9. Hypothesis Testing (cont'd) • Steps of hypothesis testing • calculate the test statistic (e.g., t-test) • make a decision – if result of test statistic is unlikely to occur by chance (that is, if the p value is less than the alpha level), reject the null hypothesis • calculate effect size indicators to determine practical significance

10. Hypothesis Testing (cont'd) • Directional alternative hypotheses • predicts the direction of an effect • increases statistical power • cannot reject null if effect is opposite of prediction

12. Hypothesis Testing (cont'd) • Hypothesis testing errors • Type I error occurs when the researcher incorrectly rejects the null hypothesis • Type II error occurs when the researcher fails to reject a false null hypothesis

13. Hypothesis Testing (cont'd) • Hypothesis testing errors • reducing the alpha level reduces the risk of a Type I error but increases the risk of a Type II error • researchers are usually more concerned about Type I errors

15. Hypothesis Testing in Practice • The basic steps of hypothesis testing are used with a number of different research designs and statistical techniques • The t Test for Correlation Coefficients • used to determine whether an observed correlation coefficient is statistically significant • null hypothesis assumes that correlation = 0

16. Hypothesis Testing in Practice (cont'd) • One-way Analysis of Variance (ANOVA) • compares two or more group means • null assumes that all population means are equal; alternative is that at least two are different

17. Hypothesis Testing in Practice (cont'd) • One-way Analysis of Variance (ANOVA) • if null is rejected, post-hoc tests needed to determine which groups are different (if more than two groups are compared) • post-hoc tests allow multiple comparisons without inflating risk of a Type I error • common post-hoc tests include Tukey’s HSD, Neuman-Keuls, and Bonferroni

18. Hypothesis Testing in Practice (cont'd) • Analysis of Covariance (ANCOVA) • extension of ANOVA • includes a quantitative independent variable as a “covariate” • increased power over ANOVA

19. Hypothesis Testing in Practice (cont'd) • Two-way ANOVA • includes two categorical independent variables • tests three null hypotheses • main effects for each independent variable • interaction • a significant interaction generally takes precedence over main effects

20. Hypothesis Testing in Practice (cont'd) • One-Way Repeated Measures ANOVA • similar to one-way ANOVA but independent variable is within participants • The t test for Regression Coefficients • tests the significance of regression coefficients obtained in regression analysis • semi-partial correlation squared (sr2) – amount of variance in the dependent variable explained by a single independent variable

21. Hypothesis Testing in Practice (cont'd) • Chi-Square Test for Contingency Tables • tests the relationship observed in a contingency table • two categorical variables • null hypothesis states that there is no relationship between the two variables

22. Hypothesis Testing and Research Design • The following tables list the appropriate statistical analyses to be used with research designs discussed in the text