Anthony Greene1 Two Sample t-test: Hypothesis of Differences Between Two Groups 1.Is Group “A” Different Than Group “B”? 2.Does an Experimental Manipulation Have an Effect? Is an experimental group different than a control group? If so, then the experimental manipulation had an effect

Anthony Greene2 Use of the Two Independent Sample t-test This is the most universally used inferential statistic Why? Population parameters μ and σ are almost never known Most experiments require a comparison, and that requires at least two groups

Anthony Greene3 Significant Differences? M 1 = 40 M 2 =60

Anthony Greene4 Significant Differences? M 1 = 40 M 2 =60

Anthony Greene5 Where You’ve Been Thus Far 1.Computation of descriptives 2.Probability theory, especially standard normal distributions (z-scores) 3.Hypothesis Testing Using z-scores Single sample t-test Two sample t-test

Anthony Greene6 Overview of Procedure Two independent sample t-test a)  1 and  2 are hypothesized or predicted (not computed and generally not known): M 1 and M 2 are computed b)  M 1 and  M 2 are unknown (  is unknown) : s M 1 and s M 2 are computed c)Degrees freedom (df) is computed

Anthony Greene7 The Basic Idea A new distribution is used that is normally distributed This time the parent distribution is

Anthony Greene8 The Basic Idea So the sampling distribution is has the following mean and standard error:

Anthony Greene9 The Basic Idea So the basic t-test has the form:

Anthony Greene10 Alternate Forms

Anthony Greene11 The Basic Idea Since the usual H 0 is that μ 1 =μ 2 OR μ 1 - μ 2 = 0

Anthony Greene12 The Basic Idea What is s p ? It’s the pooled variance and its meant to allow you to make a comparison of means even if the σs aren’t equal

Anthony Greene13 Annual salaries ($1000s) for 30 faculty members in public institutions and 35 faculty members in private institutions

Anthony Greene14 Process for comparing two population means using independent samples Compare M 1 and M 2 Based on the Pooled Variance Make a Decision Compute M 1 Compute M 2

Anthony Greene15 Notation for parameters and statistics when considering two populations MM

Anthony Greene16 The t-test for two population means (Slide 1 of 3) Step 1The null hypothesis is H 0 :  1 =  2 or  1 -  2 = 0; the alternative hypothesis is one of the following: H a :  1   2 H a :  1  2 (Two Tailed)(Left Tailed)(Right Tailed) Step 2Decide on the significance level,  Step 3The critical values are ±t  /2 -t  +t  (Two Tailed)(Left Tailed)(Right Tailed) with df = n 1 +n

Anthony Greene17 The t-test for two population means (Slide 2 of 3)

Anthony Greene18 The t-test for two population means (Slide 3 of 3) Step 4Compute the value of the test statistic Where Step 5If the value of the test statistic falls in the rejection region, reject H 0, otherwise do not reject H 0.

Anthony Greene19 Summary statistics Public Institution Private Institution Sample Mean Sum of Squares Sample Size3035

Anthony Greene20 Computing the t-value

Anthony Greene21 Criterion for deciding whether or not to reject the null hypothesis

Anthony Greene22 Criterion for deciding whether or not to reject the null hypothesis

Anthony Greene23 Sample Problem: Minutes required to comprehend the self-study manual

X1X1 (X-μ) 2 X2X M 1 -M SpSp sM1-M2sM1-M t M SS Sample Problem

Remember the goal is always to see if the effect is large compared to random variation. 1.In the t-test this is done by the ratio: (diff. in means)/(randomness). For a reasonable sample this ratio must exceed ~ In the graph this is done by comparing the mean difference to the error bars which are each 1 s.e. If the difference is greater than 2 s.e. (~1.96) then the difference is significant. Sample Problem