Post on 24-Dec-2015
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
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
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