PSYCH 248 Chapter 7: PROBABILITY & SAMPLES: THE DISTRIBUTION OF THE SAMPLE MEANS

What is the goal of all research?

The goal of all research is to MAKE INFERENCES about the population based on our sample (inferential statistic).

There are TWO TYPES of sampling design:

1. Theoretical sampling:
- order matters (A, B) or (B,A)
- sampling with replacement

2. Experimental sampling:
- order does not matter (A, B) = (B, A)
- sampling without replacement

How does experimental sampling demonstrate that even from a small population (e.g., N = 6), there can be a large number of possible samples generated?

There are a large number of possible samples that can be generated even from a small population (e.g., N = 6; n = 2 → 15 samples)

Can samples produce different statistics even though they are from the same population?

Yes, samples produce different statistics even though they are from the same population

Why is sampling error important?

Important because it tells you how close your sample mean is to your population mean and samples differ with regard to the amount of sampling error they have

How is sampling error related to the distribution of sample means?

The smaller your sampling error the closer it is to your population mean

What tells us which sample's mean is the MOST OR LEAST representative?

PROBABILITY tells us which sample's mean is the MOST OR LEAST representative.

What is sampling distribution?

Distribution of ANY statistic obtained by selecting ALL of the possible samples of a specific size (n) from a population of a specific size (N).

What is the Distribution of Sample Means?

COLLECTION OF SAMPLE MEANS for all of the possible random samples of a particular size (n) that can be obtained for a population of a specific size N. (ONE TYPE OF SAMPLING DISTRIBUTION)

CHARACTERISTICS OF SAMPLING DISTRIBUTION OF M

1. SHAPE: Normal distribution (bell-shaped)
2. MEAN: Mean of sampling distribution = population mean
3. VARIABILITY: The larger the sample size, the closer the means are to the population mean
4. THE LARGER THE SAMPLE SIZE THE NARROWER THE DISTRIBUTION

What can we ask if we create a (normal) distribution of the sampling means from all of the possible samples?

We can ask PROBABILITY questions based on the proportion of area in the graph

Central limit theorem (CLT):

For any population with a mean of µ, the distribution of sample means for sample size n will:

Two propositions:
1. Central tendency: have a mean which is equal to the population mean: µ (EM: Expected value of M)
2. Shape: will approach a normal distribution as n increases

According to the Central Limit Theorem, what is the center, variability, and shape of the distribution of sample means?

Center is the population mean
Variability ≥ 30
Shape is normally distributed

What is the mean of the distribution of sample means called? What is it equivalent to? What does it mean to say it is unbiased?

Expected value of M (EM). It is equal to the mean of the population of scores, µ. To say it is unbiased means the mean sample statistic is equal to the population parameter

What is an unbiased statistic?

When you take the average of all of the statistics (pick one- M, SD) from a sample of certain size n, and it is equal to the Population Parameter.

What is the standard error of the mean? (σM; SEM)

The average distance between sample means and the population mean.

What is the formula for Standard Error of the Mean (SEM)?

σ /√n for z-test
√s2 /n for one sample t-test

As indicated in the formula, the MAGNITUDE of the standard error is determined by TWO THINGS:

1. Standard deviation of population: σ
2. Sample size: n

What is the law of large numbers?

As the SAMPLE SIZE increases, the standard error of the mean (average of SAMPLING ERROR between the sampling means and the population mean) should DECREASE.

What does standard error tell us about how representative a mean is

If SEM is small (as with larger sample sizes), the mean from a large sample should be MORE ACCURATE (more representative of/ closer to the population mean) than the mean from a small sample

How is sample size related to the spread of the distribution of sample means?

If the sample size is equal to or greater than 30 then it is normally distributed

What is the primary use of the distribution of sample means?

The primary use of the distribution of sample means is to determine the PROBABILITY associated with a specific sample's mean

Z-score formula for the Unit Normal Table

z = (X - µ) / σ
Using the distribution of z-scores (and unit normal table) we determined the proportion or probability associated with that score.

STEPS FOR CALCULATING THE PROBABILITY OF SAMPLE MEANS

1. ASSUME distribution is normal (otherwise empirical rule doesn't apply)
2. Calculate the standard error: σ /√n (σM)
3. Calculate z score: M - µ / σM
4. Find the proportion associated with z-score (using UNIT NORMAL TABLE)
If µ is not provided do: µ = z(σ) + M
If M is not provided do M = z(σ) + µ

The standard error of the mean (SEM) is the typical or average deviation between what two things? What aspect of the sampling distribution of the means does it measure?

Average deviation between sample means and the population mean. Measures variability of the sampling distribution of the means

How does the SEM change as σ increases/decreases?

σ /√n
SEM increases if σ increases
SEM decreases if σ decreases

How does the SEM change as n increases/decreases?

σ /√n
SEM increases if n decreases
SEM decreases if n increases

How does the Z-score change as mean difference increases/decreases?

M - µ / σM
Z-score increases if M - µ increases
Z-score decreases if M - µ decreases

​How does the Z-score change as standard error of the mean (SEM) increases/decreases?

M - µ / σM
Z-score increases if σM decreases
Z-score decreases if σM increases