Download Probability and Samples: The Distribution of Sample Means

Probability and Samples: The
Distribution of Sample Means
Chapter 7
Chapter Overview
• Samples and Sampling Error
• The Distribution of Sample Means
• Probability and the Distribution of Sample
Means
• Computations
Q? What is the purpose of obtaining a sample?
A. To provide a description of a population
What happens when the sample
mean differs from population mean?
• Sampling Error: The discrepancy, or amount
of error, between a sample statistic and its
corresponding population parameter.
• 2 separate samples from the same
population will probably differ.
– different individual
– different scores
– different sample means
Predicting the characteristics of a
sample
• Distribution of Sample Means: the
collection of sample means for all the
possible random samples of a particular
size (n) that can be obtained from a
population
• Distribution of sample means are statistics,
not single scores.
• Sampling distribution: a distribution of
statistics obtained by selecting all the
possible samples of a specific size from a
population.
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Example 7.1
Figure 7.1
Frequency distribution for a population of four scores: 2, 4, 6, 8
Let’s construct a distribution of
sample means
• What do we need to know
– Population parameters (scores)
• 2,4,6,8
– Specify an (n)
– Examine all possible samples
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Table 7.1
The possible samples of n = 2 scores from the population in Figure 7.1
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Figure 7.2
Figure 7.2
The distribution of sample means for n = 2
Characteristics of sample means
• Sample means tend to pile up around the
population mean
• The distribution of sample means is
approximately normal in shape.
• The distribution of sample means can be
used to answer probability questions about
sample means
What do we use when we have a large n
and do not want to calculate all of the
possible samples ?
Central Limit Theorem
• CLT: For any population with mean of 
and a standard deviation , the distribution
of sample means for sample size n will
approach a normal distribution with a mean
of  and a standard deviation of /n (square
root of n) as n approaches infinity.
n
n
CLT: Facts
• Describes the distribution of two sample of
sample means for any population, no matter
what shape, mean, or standard deviation.
• The distribution of sample means
“approaches” a normal distribution by the
time the size reaches n= 30.
Central Limit Theorem Cont’d
• Distribution of sample means tends to be a
normal distribution particularly if one of the
following is true:
– The population from which the sample is drawn
is normal.
– The number of scores (n) in each sample is
relatively large (n>30)
Expected value of X
• Sample means should be close to the
population mean aka the expected value of x
• Expected value of X: the mean of the
distribution of sample means will be equal
to  (the population mean)
X
Standard Error of X
• Notation: x = standard distance between x and 
• The standard deviation of the distribution of
sample means.
• Measures the standard amount of difference
one should expect between X and  simply
due to chance
Magnitude of the Standard error
is determined by
• The size of the sample
• The standard deviation of the population
from which the sample is selected
• Law of large numbers: the > n, the more
probable the sample mean will be close to
the population mean.
Learning Check pg 151
1) A population of scores is normal with
=80 and =20
a) Describe the distribution of sample means for
samples of size n=16 selected from this
population. (Describe shape, central
tendency, and variability, for the distribution)
b) How would the distribution of sample means
be changed if the sample size were n=100
instead of n=16.
• 2) As sample size increases, the value of the
standard error also increases? (True or
False)
• 3)Under what circumstances will the
distribution of sample means be a normal
shaped distribution?
Learning Check 7.2 pg 152
•
•
SAT scores with a normal distribution
with a =500 and =100
In a random sample of n=25 students,
what is the probability that the sample
mean would be greater than 540?
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Figure 7.3
A distribution of sample means
Z-scores for Sample Means
• Z-scores describe the position of any
specific sample w/in the distribution
• The z-score for each distribution can be
calculated using:
z=X-
x
General Concepts
• Standard error: samples will not provide
perfectly accurate representations of the
population
• Standard error provides a method for
defining and and measuring sampling error.
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Figure 7.6
The structure of research study
Copyright © 2002 Wadsworth Group. Wadsworth is an imprint of the
Wadsworth Group, a division of Thomson Learning
Figure 7.8
Showing standard error in a graph