Transcript Probability and the Sampling Distribution

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
 Provides location relative to µ and 
a Z-score
 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 Zscore
 Why? Answer
probability questions
Introduction

Two potential problems with samples:
1.
Sampling error

2.
Variation between samples


Difference between sample and parameter
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:



Properties:




Infinite samples were taken from same population
The µ of each sample were plotted on a FDG
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 population

 What
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
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!
 For
Central Limit Theorem

The CLT describes ANY sampling
distribution in regards to:
Shape
2. Central Tendency
3. Variability
1.
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 mean
 µM = 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)
Central Limit Theorem: Variability

SEM
Measures how well statistic estimates
the parameter
The amount of sampling error between
M and µ that is reasonable to expect by
chance
Central Limit Theorem: Variability

SEM decreases when:
 decreases
 Sample size increases
 Population

M = /√n
Other properties:
 When
 As
n=1, M
=  (Table 7.2, p 209)
SEM decreases the sampling distribution
“tightens” (Figure 7.7, p 215)
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
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


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.
2.
3.
4.
Draw a sketch
Calculate SEM
Calculate Z-score
Locate probability in normal table
Step 1: Draw a sketch
Step 2: Calculate SEM
Step 3: Calculate Z-score
Step 4: Probability
SEM = M = /√n
Z = 540 – 500 / 20
Column C
SEM = 100/√25
Z = 40 / 20
p(Z = 2.0) = 0.0228
SEM = 20
Z = 2.0
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 tail
 Sample


mean  Z-score  probability
Body or tail
What’s next?
 Comparison
method
of means  experimental
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
