statistics for the social sciences psychology 340 spring 2005 sampling distribution

Statistics for the Social Sciences

Psychology 340Spring 2005

Sampling distribution


Outline

• Review 138 stuff: – What are sample distributions– Central limit theorem– Standard error (and estimates of)– Test statistic distributions as transformations


Flipping a coin example

HHH

HHT

HTH

HTT

THH

THT

TTH

TTT

Number of heads3

2

1

0

2

2

1

1

2n= 23 = 8 total outcomes


Flipping a coin example

Number of heads3

2

1

0

2

2

1

1

X f p

3 1 .125

2 3 .375

1 3 .375

0 1 .125

Number of heads0 1 2 3

.1

.2

.3

.4

probability

.125 .125.375.375

Distribution of possible outcomes(n = 3 flips)


Hypothesis testing

Can make predictions about likelihood of outcomes based on this distribution.

Distribution of possible outcomes(of a particular sample size, n)

• 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


Distribution of sample means

• Comparison distributions considered so far were distributions of individual scores

• Mean of a group of scores– Comparison distribution is distribution of means



• A simple case– Population:

– All possible samples of size n = 2

2 4 6 8

Assumption: sampling with replacement



• A simpler case– Population:

– All possible samples of size n = 2

2 4 6 8

2

4

62

2

82

2

4 4

4

6

8

28

8

8

8

84

64

6

6

6

6

4

6

8

2

4 2

mean mean mean2

3

4

5

3

4

5

6

4

5

6

7

5

6

7

8

There are 16 of them



2

4

6

8

2

4

6

8

2

4 6

2

6

2

6

4 6

4

6

8

28

8

8

8

4

4

4

6

8

2

2

mean mean mean2

3

4

5

3

4

5

6

4

5

6

7

5

6

7

8

means2 3 4 5 6 7 8

5

234

1

In long run, the random selection of tiles leads to a predictable pattern



means2 3 4 5 6 7 8

5

234

1

X f p

8 1 0.0625

7 2 0.1250

6 3 0.1875

5 4 0.2500

4 3 0.1875

3 2 0.1250

2 1 0.0625

• Sample problem:– What’s the probability of getting a sample with a mean of 6 or more?

P(X > 6) =

.1875 + .1250 + .0625 = 0.375

• Same as before, except now we’re asking about sample means rather than single scores



• Distribution of sample means is a “virtual” distribution between the sample and population

PopulationDistribution of sample meansSample


Properties of the distribution of sample means

• Shape– If population is Normal, then the dist of sample means will be Normal

Population Distribution of sample means

N > 30

– If the sample size is large (n > 30), regardless of shape of the population



– The mean of the dist of sample means is equal to the mean of the population

Population Distribution of sample means

€

μsame numeric value

different conceptual values

• Center



• 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

μ =2 + 4 + 6 + 84= 5


means2 3 4 5 6 7 8

5

234

1

2+3+4+5+3+4+5+6+4+5+6+7+5+6+7+816

=

= 5



• Spread– The standard deviation of the distribution of sample mean depends on two things• Standard deviation of the population• Sample size



• Spread• Standard deviation of the population

μX1X2

X3 μX1X2

X3

• The smaller the population variability, the closer the sample means are to the population mean



• Spread• Sample size

μ

n = 1

X




μ

n = 10

X




μ

n = 100

X

The larger the sample size the smaller the spread



• Spread• Standard deviation of the population• Sample size

– Putting them together we get the standard deviation of the distribution of sample means

€

σX

=σ

n– Commonly called the standard error


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)



• Keep your distributions straight by taking care with your notation

Sample

s

X

Population

σ

μ


€

σX



• 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 μ and a standard deviation of as n approaches infinity

€

σn

(good approximation if n > 30).


Performing your statistical test

• What are we doing when we test the hypotheses?– Computing a test statistic: Generic test

€

test statistic =observed difference

difference expected by chance

Could be difference between a sample and a population, or between different samples

Based on standard error or an estimate

of the standard error


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

Z =(X−μX )

σ X


“Generic” statistical test

An example: One sample z-test

Memory example experiment:• We give a n = 16 memory patients a memory improvement treatment.

• 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?

• After the treatment they have an average score of = 55 memory errors.

€

X

• Step 1: State your hypotheses

H0

:the memory treatment sample are the same (or worse) as the population of memory patients.HA: Their memory is better than the population of memory patients

μTreatment > μpop > 60

μTreatment < μpop < 60







€

X

• Step 2: Set your decision criteria

H0: μTreatment > μpop > 60 HA: μTreatment < μpop < 60

= 0.05One -tailed







€

X

= 0.05One -tailed

• Step 3: Collect your data








€

X

= 0.05One -tailed• Step 4: Compute your

test statistics

€

zX

=X − μ

X

σX

€

=55 − 60

816

⎛ ⎝ ⎜

⎞ ⎠ ⎟

= -2.5








€

X

= 0.05One -tailed

€

zX

= −2.5

• Step 5: Make a decision about your null hypothesis

μ-1-2 1 2

5%

Reject H0








€

X

= 0.05One -tailed

€

zX

= −2.5

• Step 5: Make a decision about your null hypothesis

- Reject H0- Support for our HA, the evidence suggests that the treatment decreases the number of memory errors


statistics for the social sciences psychology 340 spring 2005 sampling distribution

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