15 Inferential Statistics

Inferential Statistics using sample data to make inferences about populations Statistic a numerical index based on sample data Parameter a numerical characteristic of a population Two major divisions estimation hypothesis testing

Sampling Distributions the theoretical probability distribution of the values of a statistic that would result if you selected all possible samples of a particular size from a population Sampling distribution of the mean the theoretical probability distribution of the means of all possible samples of a particular size selected from a population how its done draw a random sample of a certain size from the population, calculate and write down the value of the sample mean draw another random sample (of the same size), calculate and write down the value of this sample mean continue this process an infinite number of times or until all possible samples of a particular size (e.g., 30 people per sample) have been recorded then display all of the sample means obtained. If you construct a line graph of all of these sample means, you will have a depiction of your sampling distribution of the mean

Sampling Distributions Standard error the standard deviation of a sampling distribution A sampling distribution can be made for any sample statistic

Hypothesis Testing The branch of inferential statistics focused on determining when the null hypothesis can or cannot be rejected in favor of the alternative hypothesis The process of testing a predicted relationship or hypothesis by making observations and then comparing the observed facts with the hypothesis or predicted relationship Null hypothesis a statement regarding the population parameter no relationship exists between the independent and dependent variables researcher hopes to “nullify” Alternative hypothesis states that there is a relationship between independent and dependent variables

Hypothesis Testing Steps of hypothesis testing state the null and alternative hypotheses both are stated in population parameters e.g., null hypothesis: H0: μM = μF e.g., alternative hypothesis: H1: μM ≠ μF begin by assuming that the null hypothesis is true (that the independent variable has no effect) the researcher wants to reject H0 and accepts H1 determine the standard for rejecting the null hypothesis i.e., identify the level of significance typically set at .05 if you set alpha at .05, you will incorrectly reject the null hypothesis only 5% of the time or less i.e., you will only conclude 5% of the time that there is a relationship in the population, when there really is not a relationship

Hypothesis Testing Steps of hypothesis testing input the data into a statistical program, such as SPSS or SAS and run the appropriate statistical test critical region the area on a null hypothesis sampling distribution where the observed value of the statistic, if it fell in this area, would be considered a rare event

Hypothesis Testing Steps of hypothesis testing 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 p value (probability value) the likelihood of the observed value (or a more extreme value) of a statistic, if the null hypothesis were true

Hypothesis Testing Steps of hypothesis testing compute effect size, interpret findings, and make judgment of practical significance of results many different effect size indicators Cohen’s d, eta squared (η2), omega squared (ω2), and the amount of variance explained by one or more independent variables practical significance claim made when a statistically significant finding seems large enough to be important i.e., whether the difference between the means or the observed relationship is “big enough to matter” for practical decisions e.g., to continue the line of research, to make policy decisions, or to make clinical recommendations statistically significant results are not always practically significant

Hypothesis Testing Directional alternative hypotheses an alternative hypothesis that includes a “less than sign” (<) or a “greater than sign” (>) i.e., the researcher wants to test the hypothesis that one population mean is greater than (or less than) another example null hypothesis: H0: μTraining ≤ μNo Training alternative hypothesis: H1: μTraining > μNo Training advantage increases statistical power disadvantage cannot reject null if effect is opposite of prediction typically not used in practice even when researcher believes there is direction to their hypothesis