Probability and the Sampling Distribution

Quantitative Methods in HPELS

440:210

Agenda

Introduction Distribution of Sample Means Probability and the Distribution of Sample

Means Inferential Statistics

Introduction Recall:

Any raw score can be converted to a Z-score Provides location relative to µ and

Assuming NORMAL distribution: Proportion relative to Z-score can be determined Z-score relative to proportion can be determined

Previous examples have looked at single data points Reality most research collects SAMPLES of multiple data

points Next step convert sample mean into a Z-score

Why? Answer probability questions

Introduction

Two potential problems with samples:1. Sampling error

Difference between sample and parameter

2. Variation between samples Difference between samples from same taken

from same population

How do these two problems relate?

Agenda

Introduction Distribution of Sample Means Probability and the Distribution of

Sample Means Inferential Statistics

Distribution of Sample Means Distribution of sample means = sampling

distribution is the distribution that would occur if:

Infinite samples were taken from same population The µ of each sample were plotted on a FDG

Properties: Normally distributed µM = the “mean of the means”

M = the “SD of the means”

Figure 7.1, p 202

Distribution of Sample Means

Sampling error and Variation of Samples Assume you took an infinite number of

samples from a populationWhat would you expect to happen?Example 7.1, p 203

Assume a population consists of 4 scores (2, 4, 6, 8)

Collect an infinite number of samples (n=2)

Total possible outcomes: 16

p(2) = 1/16 = 6.25% p(3) = 2/16 = 12.5%

p(4) = 3/16 = 18.75% p(5) = 4/16 = 25%

p(6) = 3/16 = 18.75% p(7) = 2/16 = 12.5%

p(8) = 1/16 = 6.25%

Central Limit Theorem

For any population with µ and , the sampling distribution for any sample size (n) will have a mean of µM and a standard deviation of M, and will approach a normal distribution as the sample size (n) approaches infinity

If it is NORMAL, it is PREDICTABLE!

Central Limit Theorem

The CLT describes ANY sampling distribution in regards to:

1. Shape

2. Central Tendency

3. Variability

Central Limit Theorem: Shape

All sampling distributions tend to be normal

Sampling distributions are normal when:The population is normal or,Sample size (n) is large (>30)

Central Limit Theorem: Central Tendency

The average value of all possible sample means is EXACTLY EQUAL to the true population meanµM = µ

If all possible samples cannot be collected?µM approaches µ as the number of

samples approaches infinity

µ = 2+4+6+8 / 4

µ = 5

µM = 2+3+3+4+4+4+5+5+5+5+6+6+6+7+7+8 / 16

µM = 80 / 16 = 5

Central Limit Theorem: Variability

The standard deviation of all sample means is denoted as M

M = /√n

Also known as the STANDARD ERROR of the MEAN (SEM)

SEMMeasures how well statistic estimates

the parameterThe amount of sampling error between

M and µ that is reasonable to expect by

chance

Central Limit Theorem: Variability

Central Limit Theorem: Variability

SEM decreases when:Population decreasesSample size increases

Other properties:When n=1, M = (Table 7.2, p 209)

As SEM decreases the sampling distribution “tightens” (Figure 7.7, p 215)

M = /√n

Agenda

Introduction Distribution of Sample Means Probability and the Distribution of Sample

Means Inferential Statistics

Probability Sampling Distribution

Recall:A sampling distribution is NORMAL and

represents ALL POSSIBLE sampling outcomes

Therefore PROBABILITY QUESTIONS can be answered about the sample relative to the population

Probability Sampling Distribution Example 7.2, p 209 Assume the following about SAT scores:

µ = 500 = 100 n = 25 Population normal

What is the probability that the sample mean will be greater than 540?

Process:1. Draw a sketch2. Calculate SEM3. Calculate Z-score4. Locate probability in normal table

Step 1: Draw a sketch

Step 2: Calculate SEM

SEM = M = /√n

SEM = 100/√25

SEM = 20

Step 3: Calculate Z-score

Z = 540 – 500 / 20

Z = 40 / 20

Z = 2.0

Step 4: Probability

Column C

p(Z = 2.0) = 0.0228

Agenda

Introduction Distribution of Sample Means Probability and the Distribution of Sample

Means Inferential Statistics

Looking Ahead to Inferential Statistics

Review:Single raw score Z-score probability

Body or tailSample mean Z-score probability

Body or tail

What’s next?Comparison of means experimental

method

Textbook Assignment

Problems: 13, 17, 25 In your words, explain the concept of a

sampling distribution In your words, explain the concept of the

Central Limit Theorum