1. Hypothesis Tests: Two Independent Samples
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320
Andrew Ainsworth PhD

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

3. Samples can be considered independent for 2 basic reasons
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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

4. Independent Samples Getting Independent Samples Method #1
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Getting Independent Samples
Method #1

5. Samples can be considered independent for 2 basic reasons
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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

6. Independent Samples Getting Independent Samples Method #2
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Getting Independent Samples
Method #2

7. Sampling Distributions Again
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If 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
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Sampling Distribution of the Difference Between Means
Mean of sampling distribution is 1
Mean of sampling distribution is 2
So, the mean of a sampling distribution is 1- 2

9. Sampling Distribution of the Difference Between Means
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Sampling Distribution of the Difference Between Means
Shape
approximately normal if
both populations are approximately normal or
N1 and N2 are reasonably large.

10. Sampling Distribution of the Difference Between Means
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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

11. Sampling Distribution of the Difference Between Means
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Sampling Distribution of the Difference Between Means
Sampling distribution of
1 - 2

12. Converting Mean Differences into Z-Scores
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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!

13. Converting Mean Differences into t-scores
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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)

14. Homogeneity of Variance
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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)

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

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. Calculating Pooled Variance
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Calculating Pooled Variance
Remembering that
sp2 is the pooled estimate: we pool SS and df to get

18. Calculating Pooled Variance
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Calculating Pooled Variance

19. Calculating Pooled Variance
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Calculating Pooled Variance
When sample sizes are equal (n1 = n2) it is simply the average of the 2
When unequal n we need a weighted average

20. Calculating SE for the Differences Between Means
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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 n1 = n2 then

21. Example: Violent Videos Games
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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

22. Hypotheses (Steps 1 and 2)
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Hypotheses (Steps 1 and 2)
2-tailed
h0: 1 - 2 = 0 or Type of video game has no impact on aggressive behaviors
h1: 1 - 2  0 or Type of video game has an impact on aggressive behaviors
1-tailed
h0: 1 - 2 ≤ 0 or GTA leads to the same or less aggressive behaviors as NBA
h1: 1 - 2 > 0 or GTA leads to more aggressive behaviors than NBA

23. Distribution, Test and Alpha (Steps 3 and 4)
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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 = .05

24. Decision Rule (Step 5) dftotal = (n1 – 1) + (n2 – 1) = n1 + n2 – 2
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Decision Rule (Step 5)
dftotal = (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
dftotal = (n1 – 1) + (n2 – 1) = = 16 OR dftotal = n1 + n2 – 2 = – 2 = 16
1-tailed t.05(16) = ____; If to > ____ reject 1-tailed
2-tailed t.05(16) = ____; If to > ____ reject
2-tailed

25. Psy 320 - Cal State Northridge
t Distribution

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:
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Calculate 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)
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Since is _____ we can insert this into the equation

31. Make your decision (Step 7)
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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

32. Heterogeneous Variances
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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 - 1

33. Effect Size for Two Groups
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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?

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

35. Effect Size, cont.
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This difference is approximately standard deviations
This is a very large effect

36. Psy 320 - Cal State Northridge
Confidence Limits

37. Psy 320 - Cal State Northridge
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?

38. Psy 320 - Cal State Northridge
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

39. SPSS output
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