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Hypothesis Tests: Two Independent Samples Cal State Northridge 320 Andrew Ainsworth PhD

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Major Points What are independent samples? Distribution of differences between means An example Heterogeneity of Variance Effect size Confidence limits 2 Psy Cal State Northridge

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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 3 Psy Cal State Northridge

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Independent Samples Psy Cal State Northridge 4 Getting Independent Samples ▫Method #1

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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 5 Psy Cal State Northridge

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Independent Samples Psy Cal State Northridge 6 Getting Independent Samples ▫Method #2

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Sampling Distributions Again If we draw all possible pairs of independent samples of size n 1 and n 2, respectively, from two populations. Calculate a mean and in each one. Record the difference between these values. What have we created? The Sampling Distribution of the Difference Between Means 7 Psy Cal State Northridge

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Sampling Distribution of the Difference Between Means Psy Cal State Northridge 8 Mean of sampling distribution is 1 Mean of sampling distribution is 2 So, the mean of a sampling distribution is 1 - 2

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Sampling Distribution of the Difference Between Means Shape ▫approximately normal if both populations are approximately normal ▫or N 1 and N 2 are reasonably large. 9 Psy Cal State Northridge

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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 10 Psy Cal State Northridge

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Sampling Distribution of the Difference Between Means Psy Cal State Northridge 11 Sampling distribution of 1 - 2

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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. What if we don’t know the ’s? You got it, we guess with s! 12 Psy Cal State Northridge

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Converting Mean Differences into t-scores We can use to estimate If, ▫The variances of the samples are roughly equal (homogeneity of variance) ▫We estimate the common variance by pooling (averaging) 13 Psy Cal State Northridge

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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) 14 Psy Cal State Northridge

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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 15 Psy Cal State Northridge

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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 16 Psy Cal State Northridge

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Calculating Pooled Variance Remembering that s p 2 is the pooled estimate: we pool SS and df to get 17 Psy Cal State Northridge

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Calculating Pooled Variance 18 Psy Cal State Northridge

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Calculating Pooled Variance When sample sizes are equal (n 1 = n 2 ) it is simply the average of the 2 When unequal n we need a weighted average 19 Psy Cal State Northridge

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Calculating SE for the Differences Between Means Once we calculate the pooled variance we just substitute it in for the sigmas earlier If n 1 = n 2 then 20 Psy Cal State Northridge

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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 21 Psy Cal State Northridge

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Hypotheses (Steps 1 and 2) 2-tailed ▫h 0 : 1 - 2 = 0 or Type of video game has no impact on aggressive behaviors ▫h 1 : 1 - 2 0 or Type of video game has an impact on aggressive behaviors 1-tailed ▫h 0 : 1 - 2 ≤ 0 or GTA leads to the same or less aggressive behaviors as NBA ▫h 1 : 1 - 2 > 0 or GTA leads to more aggressive behaviors than NBA 22 Psy Cal State Northridge

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Distribution, Test and Alpha (Steps 3 and 4) We don’t know for either group, so we cannot perform a Z-test We have to estimate with s so it is a t-test (t distribution) There are 2 independent groups So, an independent samples t-test Assume alpha = Psy Cal State Northridge

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Decision Rule (Step 5) df total = (n 1 – 1) + (n 2 – 1) = n 1 + n 2 – 2 ▫Group 1 has 8 subjects – df = 8 – 1= 7 ▫Group 2 has 10 subjects – df = 10 – 1 = 9 df total = (n 1 – 1) + (n 2 – 1) = = 16 OR df total = n 1 + n 2 – 2 = – 2 = 16 1-tailed t.05 (16) = ____; If t o > ____ reject 1-tailed 2-tailed t.05 (16) = ____; If t o > ____ reject 2-tailed 24 Psy Cal State Northridge

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t Distribution 25 Psy Cal State Northridge

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Calculate t o (Step 6) 26 Data Psy Cal State Northridge

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Calculate t o (Step 6) We have means of (GTA) and 8.4 (NBA), but both of these are sample means. We want to test differences between sample means. ▫Not between a sample and a population mean 27 Psy Cal State Northridge

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Calculate t o (Step 6) The t for 2 groups is: But under the null 1 - 2 = 0, so: 28 Psy Cal State Northridge

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Calculate t o (Step 6) We need to calculate and to do that we need to first calculate 29 Psy Cal State Northridge

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Calculate t o (Step 6) Since is _____ we can insert this into the equation 30 Psy Cal State Northridge

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Make your decision (Step 7) Since ____ > ____ we would reject the null hypothesis under a 2-tailed test Since ____ > ____ we would reject the null hypothesis under a 1-tailed test There is evidence that violent video games increase aggressive behaviors 31 Psy Cal State Northridge

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Heterogeneous Variances Refers to case of unequal population variances We don’t pool the sample variances just add them We adjust df and look t up in tables for adjusted df For adjustment use Minimum df = smaller n Psy Cal State Northridge

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Effect Size for Two Groups Extension of what we already know. We can often simply express the effect as the difference between means (e.g. 1.85) We can scale the difference by the size of the standard deviation. ▫Gives d (aka Cohen’s d) ▫Which standard deviation? 33 Psy Cal State Northridge

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Effect Size Psy Cal State Northridge 34 Use either standard deviation or their pooled average. ▫We will pool because neither has any claim to priority. ▫Pooled st. dev. =

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Effect Size, cont. Psy Cal State Northridge 35 This difference is approximately 1.12 standard deviations This is a very large effect

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Confidence Limits 36 Psy Cal State Northridge

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Confidence Limits p =.95 that based on the sample values the interval formed in this way includes the true value of 1 - 2 The probability that interval includes 1 - 2 given the value of Does 0 fall in the interval? What does that mean? 37 Psy Cal State Northridge

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Computer Example SPSS reproduces the t value and the confidence limits. All other statistics are the same as well. Note different df depending on homogeneity of variance. Remember: Don’t pool if heterogeneity and modify degrees of freedom 38 Psy Cal State Northridge

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39 SPSS output Psy Cal State Northridge

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