Presentation on theme: "Hypothesis Tests: Two Independent Samples"— Presentation transcript:
1 Hypothesis Tests: Two Independent Samples Cal State Northridge320Andrew Ainsworth PhD
2 What are independent samples? Major PointsPsy Cal State NorthridgeWhat are independent samples?Distribution of differences between meansAn exampleHeterogeneity of VarianceEffect sizeConfidence limits
3 Samples can be considered independent for 2 basic reasons Psy Cal State NorthridgeIndependent SamplesSamples are independent if the participants in each sample are not related in any waySamples can be considered independent for 2 basic reasonsFirst, samples randomly selected from 2 separate populations
4 Independent Samples Getting Independent Samples Method #1 Psy Cal State NorthridgeGetting Independent SamplesMethod #1
5 Samples can be considered independent for 2 basic reasons Psy Cal State NorthridgeIndependent SamplesSamples can be considered independent for 2 basic reasonsSecond, participants are randomly selected from a single populationSubjects are then randomly assigned to one of 2 samples
6 Independent Samples Getting Independent Samples Method #2 Psy Cal State NorthridgeGetting Independent SamplesMethod #2
7 Sampling Distributions Again Psy Cal State NorthridgeIf we draw all possible pairs of independent samples of size n1 and n2, 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
8 Sampling Distribution of the Difference Between Means Psy Cal State NorthridgeSampling Distribution of the Difference Between MeansMean of sampling distribution is 1Mean of sampling distribution is 2So, the mean of a sampling distribution is 1- 2
9 Sampling Distribution of the Difference Between Means Psy Cal State NorthridgeSampling Distribution of the Difference Between MeansShapeapproximately normal ifboth populations are approximately normalorN1 and N2 are reasonably large.
10 Sampling Distribution of the Difference Between Means Psy Cal State NorthridgeSampling Distribution of the Difference Between MeansVariability (variance sum rule)The variance of the sum or difference of 2 independent samples is equal to the sum of their variances
11 Sampling Distribution of the Difference Between Means Psy Cal State NorthridgeSampling Distribution of the Difference Between MeansSampling distribution of1 - 2
12 Converting Mean Differences into Z-Scores Psy Cal State NorthridgeConverting Mean Differences into Z-ScoresIn 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!
13 Converting Mean Differences into t-scores Psy Cal State NorthridgeConverting Mean Differences into t-scoresWe can use to estimateIf,The variances of the samples are roughly equal (homogeneity of variance)We estimate the common variance by pooling (averaging)
14 Homogeneity of Variance Psy Cal State NorthridgeHomogeneity of VarianceWe 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 VariancesSo far, we have taken 1 sample estimate (e.g. mean, SD, variance) and used it as our best estimate of the corresponding population parameterNow we have 2 samples, the average of the 2 sample statistics is a better estimate of the parameter than either alone
16 Psy 320 - Cal State Northridge Pooling VariancesAlso, 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 Calculating Pooled Variance Psy Cal State NorthridgeCalculating Pooled VarianceRemembering thatsp2 is the pooled estimate: we pool SS and df to get
18 Calculating Pooled Variance Psy Cal State NorthridgeCalculating Pooled Variance
19 Calculating Pooled Variance Psy Cal State NorthridgeCalculating Pooled VarianceWhen sample sizes are equal (n1 = n2) it is simply the average of the 2When unequal n we need a weighted average
20 Calculating SE for the Differences Between Means Psy Cal State NorthridgeCalculating SE for the Differences Between MeansOnce we calculate the pooled variance we just substitute it in for the sigmas earlierIf n1 = n2 then
21 Example: Violent Videos Games Psy Cal State NorthridgeExample: Violent Videos GamesDoes playing violent video games increase aggressive behaviorTwo independent randomly selected/assigned groupsGrand 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
22 Hypotheses (Steps 1 and 2) Psy Cal State NorthridgeHypotheses (Steps 1 and 2)2-tailedh0: 1 - 2 = 0 or Type of video game has no impact on aggressive behaviorsh1: 1 - 2 0 or Type of video game has an impact on aggressive behaviors1-tailedh0: 1 - 2 ≤ 0 or GTA leads to the same or less aggressive behaviors as NBAh1: 1 - 2 > 0 or GTA leads to more aggressive behaviors than NBA
23 Distribution, Test and Alpha (Steps 3 and 4) Psy Cal State NorthridgeDistribution, Test and Alpha (Steps 3 and 4)We don’t know for either group, so we cannot perform a Z-testWe have to estimate with s so it is a t-test (t distribution)There are 2 independent groupsSo, an independent samples t-testAssume alpha = .05
26 Psy 320 - Cal State Northridge Calculate to (Step 6)Data
27 Psy 320 - Cal State Northridge Calculate to (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
28 Calculate to (Step 6) The t for 2 groups is: Psy Cal State NorthridgeCalculate to (Step 6)The t for 2 groups is:But under the null 1 - 2 = 0, so:
29 Psy 320 - Cal State Northridge Calculate to (Step 6)We need to calculate and to do that we need to first calculate
30 Calculate to (Step 6)Psy Cal State NorthridgeSince is _____ we can insert this into the equation
31 Make your decision (Step 7) Psy Cal State NorthridgeMake your decision (Step 7)Since ____ > ____ we would reject the null hypothesis under a 2-tailed testSince ____ > ____ we would reject the null hypothesis under a 1-tailed testThere is evidence that violent video games increase aggressive behaviors
32 Heterogeneous Variances Psy Cal State NorthridgeHeterogeneous VariancesRefers to case of unequal population variancesWe don’t pool the sample variances just add themWe adjust df and look t up in tables for adjusted dfFor adjustment use Minimumdf = smaller n - 1
33 Effect Size for Two Groups Psy Cal State NorthridgeEffect Size for Two GroupsExtension 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?
34 Effect Size Use either standard deviation or their pooled average. Psy Cal State NorthridgeEffect SizeUse either standard deviation or their pooled average.We will pool because neither has any claim to priority.Pooled st. dev. =
35 Effect Size, cont.Psy Cal State NorthridgeThis difference is approximately standard deviationsThis is a very large effect
36 Psy 320 - Cal State Northridge Confidence Limits
37 Psy 320 - Cal State Northridge Confidence Limitsp = .95 that based on the sample values the interval formed in this way includes the true value of 1 - 2The probability that interval includes 1 - 2 given the value ofDoes 0 fall in the interval? What does that mean?
38 Psy 320 - Cal State Northridge Computer ExampleSPSS 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 heterogeneityand modify degrees of freedom
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