Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Chapter 8 Introduction to the t Test Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall t Test for a Single Sample • Used to compare the mean of a sample with a population for which the mean is known but the variance is unknown • Unlike previous methods, one must now estimate the population variance from the scores in the sample Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall t Test for a Single Sample • Estimating population variance from sample scores – Variance in sample generally slightly smaller than population • Sample is a biased estimate of population • So, divide by N-1 rather than N to correct for bias – N-1 is known as the “degrees of freedom,” the number of scores that are free to vary S 2 (X M ) 2 N 1 S 2 SS N 1 2 ( X M ) df Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall SS df t Test for a Single Sample • Because variance is estimated, comparison distribution is not a normal curve – t distribution – Like a normal curve • Bell-shaped • Unimodal • Symmetrical – But has more scores at the extremes (i.e., heavier tails) and varies somewhat according to degrees of freedom • Sample mean thus has to be slightly more extreme to be significant with a t distribution than with a normal curve Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall t Test for a Single Sample • Comparison distribution is the distribution of means – Figuring variance of the distribution of means – Figuring standard deviation of the distribution of means S 2 SM N 2 S M S M2 Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall t Test for a Single Sample • Determine cutoff sample score for rejecting the null hypothesis (using t table) • Figure sample mean’s score on the t distribution (t score) M t SM Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall t Test for Dependent Means • Used to compare two sets of scores where there are two scores for each person – Repeated-measures – Within-subjects – Paired • Compares mean difference score across pairs of scores against a difference of 0 under the null hypothesis. • In other respects, t test for dependent means is just like a single sample t test with a population mean of 0 Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Assumptions • Assume that the population of individuals from which the sample was taken is normally distributed • In practice, one seldom knows if a population is normally distributed – OK because many distributions in nature do approximate a normal curve – The t test is often still fairly accurate even when this assumption is violated Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall Effect Size • Effect size for the t test for dependent means is – the mean of the difference scores – divided by the estimated SD of the population of individual difference scores • Studies using the t test for dependent means typically have larger effect sizes and more power than do studies with participants divided into two groups 1 2 d • Effect size conventions – Small = .20 – Medium = .50 – Large = .80 Aron, Aron, & Coups, Statistics for the Behavioral and Social Sciences: A Brief Course (3e), © 2005 Prentice Hall