1. Statistics for the Social Sciences
Psychology 340
Spring 2010
Sampling distributions

2. Reminders
I posted Quiz 3 this morning – it covers 3 chapters (5, 6, & 7), Due Thurs Feb 4
Homework 3 is due Tues Feb 2
Don’t forget that Exam 1 is coming up (Thurs Feb 11)

3. What are sample distributions
Outline
What are sample distributions
Central limit theorem
Standard error (and estimates of)
Test statistic distributions as transformations

4. Based on standard error or an estimate of the standard error
Testing Hypotheses
Looking ahead:
Core logic of hypothesis testing
Considers the probability that the result of a study could have come about if the experimental procedure had no effect
If this probability is low, scenario of no effect is rejected and the theory behind the experimental procedure is supported
How do we determine this?
Based on standard error or an estimate of the standard error

5. Flipping a coin example
Number of heads
HHH 3
HHT 2
HTH 2
HTT 1 2
THH
THT 1
TTH 1
TTT 2n
= 23 = 8 total outcomes

6. Flipping a coin example
Number of heads 3
Distribution of possible outcomes
(n = 3 flips) 2 X f p 3 1
.125 2
.375
Number of heads 1 2 3 .1 .2 .3 .4
probability 2
.375
.375 1 2
.125
.125 1 1

7. Hypothesis testing Distribution of Sample Means
Distribution of possible outcomes
(of a particular sample size, n)
Can make predictions about likelihood of outcomes based on this distribution.
In hypothesis testing, we compare our observed samples with the distribution of possible samples (transformed into standardized distributions)
This distribution of possible outcomes is often Normally Distributed

8. Hypothesis testing Distribution of Sample Means
Distribution of possible outcomes
(of a particular sample size, n)
Mean of a group of scores
Comparison distribution is distribution of means

9. Distribution of sample means
Distribution of sample means is a “theoretical” distribution between the sample and population
Mean of a group of scores
Comparison distribution is distribution of means
Population
Distribution of sample means
Sample

10. Distribution of sample means
A simple case
Population: 2 4 6 8
All possible samples of size n = 2
Assumption: sampling with replacement

11. Distribution of sample means
# of possible samples =Nn
N = Population size
n = Sample size
A simple case
Population: 2 4 6 8
All possible samples of size n = 2
There are 16 of them
mean 2 2 2 6 4 5 6 4 7 8 4 6 8 2 2 4 3 8 4 6 2 4 6 4 8 2 8 2 5 4 2 3 4 4

12. Distribution of sample means2 3 4 5 6 7 8 1
In long run, the random selection of tiles leads to a predictable pattern
mean
mean
mean 2 2 2 4 6 5 8 2 5 2 4 3 4 8 6 8 4 6 2 6 4 6 2 4 8 6 7 2 8 5 6 4 5 8 8 8 4 2 3 6 6 6 4 4 4 6 8 7

13. Distribution of sample means2 3 4 5 6 7 8 1
Sample problem:
What’s the probability of getting a sample with a mean of 6 or more? X f p 8 1
0.0625 7 2
0.1250 6 3
0.1875 5 4
0.2500
P(X > 6) =
= 0.375
Same as before, except now we’re asking about sample means rather than single scores

14. Distribution of sample means
Distribution of sample means is a “theoretical” distribution between the sample and population
# of possible samples =Nn
N = Population size
n = Sample size
Nn =10010
=100,000,000,000,000,000,000
possible samples
Rather than computing all of these means and looking at their distribution, we use something called:
The Central Limit Theorem
Population σ μ
Distribution of sample means
Sample s X
N=100
n=10

15. Properties of the Distribution of Sample Means
Shape
If population is Normal, then the dist of sample means will be Normal
If the sample size is large (n > 30), regardless of shape of the population
Distribution of sample means
Population
N > 30

16. Properties of the Distribution of Sample Means
Center
The mean of the dist of sample means is equal to the mean of the population
Population
Distribution of sample means
same numeric value
different conceptual values

17. Properties of the Distribution of Sample Means
Center
The mean of the dist of sample means is equal to the mean of the population
Consider our earlier example 2 4 6 8
Population
Distribution of sample means
means 2 3 4 5 6 7 8 1 4
μ =
= 5 16 =
= 5

18. Properties of the Distribution of Sample Means
Spread
The standard deviation of the distribution of sample mean depends on two things
Standard deviation of the population
Sample size

19. Properties of the Distribution of Sample Means
Spread
Standard deviation of the population
The smaller the population variability, the closer the sample means are to the population mean μ X 1 2 3 μ X 1 2 3

20. Properties of the Distribution of Sample Means
Spread
Sample size
n = 1 μ X

21. Properties of the Distribution of Sample Means
Spread
Sample size
n = 10 μ X

22. Properties of the Distribution of Sample Means
Spread
Sample size μ
n = 100
The larger the sample size the smaller the spread X

23. Properties of the Distribution of Sample Means
Spread
Standard deviation of the population
Sample size
Putting them together we get the standard deviation of the distribution of sample means
Commonly called the standard error

24. Standard error
The standard error is the average amount that you’d expect a sample (of size n) to deviate from the population mean
In other words, it is an estimate of the error that you’d expect by chance (or by sampling)

25. Distribution of sample means
Keep your distributions straight by taking care with your notation
Population σ μ
Distribution of sample means
Sample s X

26. Distribution of sample means: Properties
All three of these properties are combined to form the Central Limit Theorem
For any population with mean μ and standard deviation σ, the distribution of sample means for sample size n will approach a normal distribution with a mean of μX and a standard deviation of as n approaches infinity
(good approximation if n > 30).

27. Looking ahead: Statistical tests
What are we doing when we test hypotheses?
Computing a test statistic: Generic test
Could be difference between a sample and a population, or between different samples
Based on standard error or an estimate of the standard error

28. Hypothesis Testing With a Distribution of Means
It is the comparison distribution when a sample has more than one individual
Find a Z score of your sample’s mean on a distribution of means

29. Hypothesis Testing With a Distribution of Means
Logic is like what we were doing with single scores
Suppose that you got a 630 on the SAT (μ = 500, σ = 100, & Normal). What percent of the people who take the SAT get your score or worse?
From the table:
z(1.3) =.0968
That’s 9.68% above this score
So 90.32% got your score or worse
However, now we are looking at the likelihood of getting a sample mean rather than a single score.
An now the distribution of possibilities is the distribution of sample means

30. Looking ahead: Hypothesis testing
An example: One sample z-test
Memory example experiment:
Step 1: State your hypotheses
One -tailed
We give a n = 16 memory patients a memory improvement treatment.
H0:
the memory treatment sample are the same as those in the population of memory patients.
After the treatment they have an average score of = 55 memory errors.
μTreatment ≥ (μpop = 60)
How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60, σ = 8?
HA:
the memory treatment sample make fewer errors the the population
μTreatment < (μpop = 60)

31. Looking ahead: Hypothesis testing
An example: One sample z-test
Memory example experiment:
H0: μTreatment ≥ (μpop = 60)
HA: μTreatment < (μpop = 60)
We give a n = 16 memory patients a memory improvement treatment.
One -tailed
Step 2: Set your decision criteria
α = 0.05
After the treatment they have an average score of = 55 memory errors.
How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60, σ = 8?

32. Looking ahead: Hypothesis testing
An example: One sample z-test
Memory example experiment:
H0: μTreatment ≥ (μpop = 60)
HA: μTreatment < (μpop = 60)
We give a n = 16 memory patients a memory improvement treatment.
One -tailed
α = 0.05
Step 3: Collect your data
After the treatment they have an average score of = 55 memory errors.
How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60, σ = 8?

33. Looking ahead: Hypothesis testing
An example: One sample z-test
Memory example experiment:
H0: μTreatment ≥ (μpop = 60)
HA: μTreatment < (μpop = 60)
We give a n = 16 memory patients a memory improvement treatment.
One -tailed
α = 0.05
Step 4: Compute your test statistics
After the treatment they have an average score of = 55 memory errors.
How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60, σ = 8?
= -2.5

34. Looking ahead: Hypothesis testing
An example: One sample z-test
Memory example experiment:
H0: μTreatment ≥ (μpop = 60)
HA: μTreatment < (μpop = 60)
We give a n = 16 memory patients a memory improvement treatment.
One -tailed
α = 0.05
After the treatment they have an average score of = 55 memory errors.
Step 5: Make a decision about your null hypothesis
How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60, σ = 8? 5%
Reject H0

35. Looking ahead: Hypothesis testing
An example: One sample z-test
Memory example experiment:
H0: μTreatment ≥ (μpop = 60)
HA: μTreatment < (μpop = 60)
We give a n = 16 memory patients a memory improvement treatment.
One -tailed
α = 0.05
After the treatment they have an average score of = 55 memory errors.
Step 5: Make a decision about your null hypothesis
- Reject H0
How do they compare to the general population of memory patients who have a distribution of memory errors that is Normal, μ = 60, σ = 8?
- Support for our HA, the evidence suggests that the treatment decreases the number of memory errors