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Probability and Samples
• Sampling Distributions
• Central Limit Theorem
• Standard Error
• Probability of Sample Means
Inferential Statistics
tomorrow and beyond
Population
Sample
last week and today
Probability
When we take a sample from a population we can talk
about the probability of
- getting a certain type of individual when we
sample once
last Thursday
- getting a certain type of sample mean when n>1
today
Distribution of Individuals in a Population
6
5
4
3
2
1
10 20 30 40 50 60 70
p(X > 50) = ?
Distribution of Individuals in a Population
6
5
4
3
2
1
10 20 30 40 50 60 70
p(X > 50) =
1 = 0.11
9
Distribution of Individuals in a Population
6
5
4
3
2
1
10 20 30 40 50 60 70
p(X > 30) = ?
Distribution of Individuals in a Population
6
5
4
3
2
1
10 20 30 40 50 60 70
p(X > 30) =
6 = 0.66
9
Distribution of Individuals in a Population
6
normally distributed
 = 40,  = 10
5
4
3
2
1
10 20 30 40 50 60 70
p(40 < X < 60) = ?
Distribution of Individuals in a Population
6
5
normally distributed
 = 40,  = 10
4
3
2
1
10 20 30 40 50 60 70
p(40 < X < 60) = p(0 < Z < 2) = 47.7%
Distribution of Individuals in a Population
6
normally distributed
 = 40,  = 10
5
4
3
2
1
10 20 30 40 50 60 70
p(X > 60) = ?
Distribution of Individuals in a Population
6
5
normally distributed
 = 40,  = 10
4
3
2
1
10 20 30 40 50 60 70
p(X > 60) = p(Z > 2) = 2.3%
For the preceding calculations to be accurate, it is
necessary that the sampling process be random.
A random sample must satisfy two requirements:
1. Each individual in the population has an equal
chance of being selected.
2. If more than one individual is to be selected, there
must be constant probability for each and every
selection (i.e. sampling with replacement).
Distribution of Sample Means
A distribution of sample means is:
the collection of sample means for all the possible
random samples of a particular size (n) that can be
obtained from a population.
Population
6
5
4
3
2
1
1
2
3
4
5
6
7
8
9
Distribution of Sample Means
from Samples of Size n = 2
Sample #
Scores
Mean ( X )
1
2, 2
2
2
2,4
3
3
2,6
4
4
2,8
5
5
4,2
3
6
4,4
4
7
4,6
5
8
4,8
6
9
6,2
4
10
6,4
5
11
6,6
6
12
6,8
7
13
8,2
5
14
8,4
6
15
8.6
7
16
8.8
8
Distribution of Sample Means
from Samples of Size n = 2
6
5
4
3
2
1
1
2
3
4
5
6
7
8
9
sample mean
We can use the distribution of sample means to answer
probability questions about sample means
Distribution of Sample Means
from Samples of Size n = 2
6
5
4
3
2
1
1
2
3
4
5
6
7
sample mean
p( X > 7) = ?
8
9
Distribution of Sample Means
from Samples of Size n = 2
6
5
4
3
2
1
1
2
3
4
5
6
7
8
sample mean
p( X > 7) =
1 =6%
16
9
Distribution of Individuals in Population
6
5
4
3
2
1
 = 5,  = 2.24
Distribution of Sample Means
6
5
1 2 3 4 5 6 7 8 9
X = 5, X = 1.58
4
3
2
1
1 2 3 4 5 6 7 8 9
sample mean
Distribution of Individuals
6
5
4
3
2
1
 = 5,  = 2.24
Distribution of Sample Means
6
5
1 2 3 4 5 6 7 8 9
p(X > 7) = 25%
X = 5, X = 1.58
4
3
2
1
1 2 3 4 5 6 7 8 9
sample mean
p(X> 7) = 6% , for n=2
A key distinction
Population Distribution – distribution of all individual scores in
the population
Sample Distribution – distribution of all the scores in your
sample
Sampling Distribution – distribution of all the possible sample
means when taking samples of size n from the population. Also
called “the distribution of sample means”.
Distribution of Individuals in Population
6
5
4
3
2
1
 = 5,  = 2.24
Distribution of Sample Means
6
5
1 2 3 4 5 6 7 8 9
X = 5, X = 1.58
4
3
2
1
1 2 3 4 5 6 7 8 9
sample mean
Distribution of Sample Means
6
5
Things to Notice
4
3
2
1.
The sample means tend to pile up
around the population mean.
2.
The distribution of sample means is
approximately normal in shape, even
though the population distribution was
not.
3.
The distribution of sample means has
less variability than does the population
distribution.
1
1 2 3 4 5 6 7 8 9
sample mean
What if we took a larger sample?
Distribution of Sample Means
from Samples of Size n = 3
24
22
20
X = 5, X = 1.29
18
16
14
12
p( X > 7) =
10
8
6
4
2
1
2
3
4
5
6
7
sample mean
8
9
1 =2%
64
Distribution of Sample Means
As the sample gets bigger, the
sampling distribution…
1. stays centered at the population
mean.
2. becomes less variable.
3. becomes more normal.
Central Limit Theorem
For any population with mean  and standard deviation ,
the distribution of sample means for sample size n …
1. will have a mean of 
2. will have a standard deviation of

n
3. will approach a normal distribution as
n approaches infinity
Notation
the mean of the sampling distribution
X  
the standard deviation of sampling distribution
(“standard error of the mean”)
X 

n
Standard Error
The “standard error” of the mean is:
The standard deviation of the distribution of sample
means.
The standard error measures the standard amount of
difference between x-bar and  that is reasonable to
expect simply by chance.
SE =

n
Standard Error
The Law of Large Numbers states:
The larger the sample size, the smaller the standard
error.
This makes sense from the formula for
standard error …
Distribution of Individuals in Population
6
5
4
3
2
1
 = 5,  = 2.24
Distribution of Sample Means
6
5
1 2 3 4 5 6 7 8 9
X = 5, X = 1.58
4
3
X 
2
1
1 2 3 4 5 6 7 8 9
sample mean
2.24
 1.58
2
Sampling Distribution (n = 3)
24
22
20
X = 5
X = 1.29
18
16
14
12
X 
10
8
6
4
2
1
2
3
4
5
6
7
sample mean
8
9
2.24
 1.29
3
Clarifying Formulas
Population
Sample
Distribution of
Sample Means
X


X

X 
X  
N
ss

N
n
X 
ss
s
n 1
notice
 
2
X
2
n

n
Central Limit Theorem
For any population with mean  and standard deviation ,
the distribution of sample means for sample size n …
1. will have a mean of 
2. will have a standard deviation of

n
3. will approach a normal distribution as
n approaches infinity
What does this mean in
practice?
Practical Rules Commonly Used:
1. For samples of size n larger than 30, the distribution of the sample
means can be approximated reasonably well by a normal distribution.
The approximation gets better as the sample size n becomes larger.
2. If the original population is itself normally distributed, then the sample
means will be normally distributed for any sample size.
normal population
non-normal population
small n
large n
X is normal
X is normal
X is nonnormal
X is normal
Probability and the Distribution of Sample Means
The primary use of the distribution of sample
means is to find the probability associated with any
specific sample.
Probability and the Distribution of Sample Means
Example:
Given the population of women has normally
distributed weights with a mean of 143 lbs and
a standard deviation of 29 lbs,
1. if one woman is randomly selected, find the probability that her
weight is greater than 150 lbs.
2. if 36 different women are randomly selected, find the probability
that their mean weight is greater than 150 lbs.
Given the population of women has normally distributed
weights with a mean of 143 lbs and a standard deviation of
29 lbs,
1. if one woman is randomly selected, find the probability that her
weight is greater than 150 lbs.
Population distribution
z = 150-143 = 0.24
29
0.4052
 = 143 150
= 29
0 0.24
Given the population of women has normally distributed
weights with a mean of 143 lbs and a standard deviation of
29 lbs,
2. if 36 different women are randomly selected, find the probability
that their mean weight is greater than 150 lbs.
X 
29
36
Sampling distribution
z = 150-143 = 1.45
4.33
0.0735
 = 143 150
= 4.33
0 1.45
Probability and the Distribution of Sample Means
Example:
Given the population of women has normally
distributed weights with a mean of 143 lbs and
a standard deviation of 29 lbs,
1. if one woman is randomly selected, find the probability that her
weight is greater than 150 lbs.
P( X  150)  .41
2. if 36 different women are randomly selected, find the probability
that their mean weight is greater than 150 lbs. P( X  150)  .07
Practice
Example:
Given a population of 400 automobile models,
with a mean horsepower = 105 HP, and a
standard deviation = 40 HP,
1. What is the standard error of the sample mean for a sample of
size 1? 40
2. What is the standard error of the sample mean for a sample of
size 4?
20
3. What is the standard error of the sample mean for a sample of
size 25?
8
Practice
Example:
Given a population of 400 automobile models,
with a mean horsepower = 105 HP, and a
standard deviation = 40 HP,
1. if one model is randomly selected from the population, find the
probability that its horsepower is greater than 120. .35
2. If 4 models are randomly selected from the population, find the
probability that their mean horsepower is greater than 120 .23
3. If 25 models are randomly selected from the population, find the
probability that their mean horsepower is greater than 120 .03
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