Hypothesis Tests: Two Independent Samples Cal State Northridge Ψ320 Andrew Ainsworth PhD 2 Psy 320 - Cal State Major Points Northridge •What are independent samples? •Distribution of differences between means •An example •Heterogeneity of Variance •Effect size •Confidence limits 3 Psy 320 - Cal State Northridge Independent Samples •Samples are independent if the participants in each sample are not related in any way •Samples can be considered independent for 2 basic reasons ▫ First, samples randomly selected from 2 separate populations 1 4 Psy 320 - Cal State Independent Samples Northridge • Getting Independent Samples ▫ Method #1 Random Random Selection Selection 5 Psy 320 - Cal State Northridge Independent Samples •Samples can be considered independent for 2 basic reasons ▫ Second, participants are randomly selected from a single population ▫ Subjects are then randomly assigned to one of 2 samples 6 Psy 320 - Cal State Northridge Independent Samples • Getting Independent Samples ▫ Method #2 Population Random Selection and Random Assignment Sample #1 Sample #2 2 7 Psy 320 - Cal State Northridge Sampling Distributions Again • If we draw all possible pairs of independent samples of size n1 and n2, respectively, from two populations. • Calculate a mean X 1 and X 2 in each one. • Record the difference between these values. • What have we created? • The Sampling Distribution of the Difference Between Means 8 Psy 320 - Cal State Northridge Sampling Distribution of the Difference Between Means Population 1 Population 2 • Mean of sampling Calculate distribution X is µ Statistic 1 1 1 Sample 1 Sample 2 • Mean of sampling X11− X 21 distribution X 2 is µ2 2 Sample 1 Sample 2 X12− X 22 • So, the mean of a sampling 3 Sample 1 Sample 2 X13− X 23 distribution X 1 − X 2 ... ... is µ1- µ2 ∞ Sample 1 Sample 2 X1∞− X 2 ∞ 9 Psy 320 - Cal State Northridge Sampling Distribution of the Difference Between Means • Shape ▫ approximately normal if both populations are approximately normal ▫ or N1 and N 2 are reasonably large. 3 10 Psy 320 - Cal State Northridge Sampling Distribution of the Difference Between Means • Variability (variance sum rule) ▫ The variance of the sum or difference of 2 independent samples is equal to the sum of their variances σ2 σ 2 σ2= σ 2 + σ 2 =X1 + X 2 XX12− X 1 X 2 n1 n 2 σ2 σ 2 σ σ=XX1 + 2 ; Remembering that σ = X 1 XX1− 2 X 1 n1 n 2 n1 Think Pathagorasc 2= a 2 + b 2 11 Psy 320 - Cal State Northridge Sampling Distribution of the Difference Between Means • Sampling distribution of X1− X 2 σ2 σ 2 1+ 2 n1 n 2 µ1 - µ2 12 Psy 320 - Cal State Northridge Converting Mean Differences into Z- Scores • In any normal distribution with a known mean and standard deviation, we can calculate z-scores to estimate probabilities. (X− X )( −µ − µ ) z = 1 2 12 σ X1− X 2 • What if we don’t know the σ’s? • You got it, we guess with s! 4 13 Psy 320 - Cal State Northridge Converting Mean Differences into t-scores s σ • We can use X 1 − X 2 to estimate X1− X 2 (X− X )( −µ − µ ) t = 1 2 12 s • If, X1− X 2 ▫ The variances of the samples are roughly equal (homogeneity of variance) ▫ We estimate the common variance by pooling (averaging) 14 Psy 320 - Cal State Northridge Homogeneity of Variance • We assume that the variances are equal because: ▫ If they’re from the same population they should be equal (approximately) ▫ If they’re from different populations they need to be similar enough for us to justify comparing them (“apples to oranges” and all that) 15 Psy 320 - Cal State Northridge Pooling Variances • So far, we have taken 1 sample estimate (e.g. mean, SD, variance) and used it as our best estimate of the corresponding population parameter • Now we have 2 samples, the average of the 2 sample statistics is a better estimate of the parameter than either alone 5 16 Psy 320 - Cal State Northridge Pooling Variances • Also, if there are 2 groups and one groups is larger than it should “give” more to the average estimate (because it is a better estimate itself) • Assuming equal variances and pooling allows us to use an estimate of the population that is smaller than if we did not assume they were equal 17 Psy 320 - Cal State Northridge Calculating Pooled Variance SS SS • Remembering that s2 = = df n −1 2 sp is the pooled estimate: we pool SS and df to get SS 2 pooled SS12+ SS SS 12 + SS SS 12 + SS sp == = = dfpooled dfdf121+( n −+− 1) ( n 2 1) nn 12 +− 2 18 Psy 320 - Cal State Northridge Calculating Pooled Variance 2 SS1+ SS 2 sp = n1+ n 2 − 2 Method 1 Method 2 SS X X 2 2 1=∑(i 1 − 1 ) SS1=( n 1 − 1) s 1 2 2 SS2=∑( Xi 2 − X 2 ) SS2=( n 2 − 1) s 2 6 19 Psy 320 - Cal State Northridge Calculating Pooled Variance • When sample sizes are equal ( n1 = n2) it is simply the average of the 2 s2+ s 2 s2 = X1 X 2 p 2 • When unequal n we need a weighted average 2 2 2 (n1− 1) s 1 + ( n 2 − 1) s 2 sp = n1+ n 2 − 2 20 Psy 320 - Cal State Northridge Calculating SE for the Differences Between Means s2 s 2 s =p + p X1− X 2 n1 n 2 • Once we calculate the pooled variance we just substitute it in for the sigmas earlier • If n1 = n2 then s2 s 2 s2 s 2 s =p += p 1 + 2 X1− X 2 nn1 2 nn 12 21 Psy 320 - Cal State Northridge Example: Violent Videos Games • Does playing violent video games increase aggressive behavior • Two independent randomly selected/assigned groups ▫ Grand Theft Auto: San Andreas (violent: 8 subjects) VS. NBA 2K7 (non-violent: 10 subjects) ▫ We want to compare mean number of aggressive behaviors following game play 7 22 Psy 320 - Cal State Northridge Hypotheses (Steps 1 and 2) • 2-tailed ▫ h0: µ1 - µ2 = 0 or Type of video game has no impact on aggressive behaviors ▫ h1: µ1 - µ2 ≠ 0 or Type of video game has an impact on aggressive behaviors • 1-tailed ▫ h0: µ1 - µ2 ≤ 0 or GTA leads to the same or less aggressive behaviors as NBA ▫ h1: µ1 - µ2 > 0 or GTA leads to more aggressive behaviors than NBA 23 Psy 320 - Cal State Northridge Distribution, Test and Alpha (Steps 3 and 4) • We don’t know σ for either group, so we cannot perform a Z-test • We have to estimate σ with s so it is a t-test (t distribution) • There are 2 independent groups • So, an independent samples t-test • Assume alpha = .05 24 Psy 320 - Cal State Northridge Decision Rule (Step 5) • df total = ( n1 – 1) + ( n2 – 1) = n1 + n2 – 2 ▫ Group 1 has 8 subjects – df = 8 – 1= 7 ▫ Group 2 has 10 subjects – df = 10 – 1 = 9 • df total = ( n1 – 1) + ( n2 – 1) = 7 + 9 = 16 OR df total = n1 + n2 – 2 = 8 + 10 – 2 = 16 • 1-tailed t.05 (16) = ____; If t o > ____ reject 1-tailed • 2-tailed t.05 (16) = ____; If t o > ____ reject 2-tailed 8 df Critical Values of Student's t 25 1-tailed 0.1 0.05 0.025 0.01 0.005 0.0005 2-tailed 0.2 0.1 0.05 0.02 0.01 0.001 Psy 320 - Cal State Northridge 1 3.078 6.314 12.706 31.821 63.657 636.619 2 1.886 2.92 4.303 6.965 9.925 31.598 3 1.683 2.353 3.182 4.5415 5.841 12.941 4 1.533 2.132 2.776 3.747 4.604 8.61 5 1.476 2.015 2.571 3.365 4.032 6.859 6 1.44 1.943 2.447 3.143 3.707 5.959 7 1.415 1.895 2.365 2.998 3.499 5.405 8 1.397 1.86 2.306 2.896 3.355 5.041 9 1.383 1.833 2.262 2.821 3.25 4.781 10 1.372 1.812 2.228 2.764 3.169 4.587 t 11 1.363 1.796 2.201 2.718 3.106 4.437 12 1.356 1.782 2.179 2.681 3.055 4.318 Distribution 13 1.35 1.771 2.16 2.65 3.012 4.221 14 1.345 1.761 2.145 2.624 2.977 4.14 15 1.341 1.753 2.131 2.602 2.947 4.073 16 1.337 1.746 2.12 2.583 2.921 4.015 17 1.333 1.74 2.11 2.567 2.898 3.965 18 1.33 1.734 2.101 2.552 2.878 3.922 19 1.328 1.729 2.093 2.539 2.861 4.883 20 1.325 1.725 2.086 2.528 2.845 3.85 21 1.323 1.721 2.08 2.518 2.831 3.819 22 1.321 1.717 2.074 2.508 2.819 3.792 23 1.319 1.714 2.069 2.5 2.807 3.767 24 1.318 1.711 2.064 2.492 2.797 3.745 25 1.316 1.708 2.06 2.485 2.787 3.725 26 Psy 320 - Cal State Northridge Subject GTA NBA Calculate t (Step 6) 1 12 9 o 2 9 9 3 11 8 4 13 6 5 8 11 6 10 7 Data 7 9 8 8 10 11 9 8 10 7 Mean 10.25 8.4 s 1.669 1.647 s2 2.786 2.713 27 Psy 320 - Cal State Northridge t Calculate o (Step 6) • We have means of 10.25 (GTA) and 8.4 (NBA), but both of these are sample means.