From Last week
Q22
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A multiple choice test has 48
questions, each with four response
choices. If a student is simply
guessing at the answers
What is the probability of guessing
correctly for any question?
Q22
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
A multiple choice test has 48
questions, each with four response
choices. If a student is simply
guessing at the answers
On average, how many questions
would a student get correct for the
entire test?
Q22


A multiple choice test has 48
questions, each with four response
choices. If a student is simply
guessing at the answers
What is the probability that a
student would get more than 15
answers correct simply by
guessing?
Q22


A multiple choice test has 48
questions, each with four response
choices. If a student is simply
guessing at the answers
What is the probability that a
student would get 15 or more
answers correct simply by
guessing?
The Distribution of Sample
Means
Review
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z-scores close to zero indicate that
the sample mean is relatively close to
the population mean
z-scores beyond 2 represent extreme
values that are quite different from
the population mean
The Distribution of Sample Means
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
Definition: the set of means from all
the possible random samples of a
specific size (n) selected from a
specific population
Theoretical distribution because
normally you wouldn’t have all the
information about a population
available, and that’s why we have to
use samples to make inferences
about the population
Characteristics of the distribution
of sample mean
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The sample means tend to cluster
around the population mean
The distribution of sample means
can be used to answer probability
questions about sample means
The distribution of sample means is
approximately normal in shape
Central Limit Theorem
1.
The mean of the distribution of sample means is
called the Expected Value of M
2. The standard deviation of the distribution of sample
means is called the Standard Error of M
M 
3.

n
The shape of the distribution of sample means tends
to be normal.
Example


Imagine a population that is normally
distributed with µ=110 and σ=24
If we take a sample from this
population, how accurately would the
sample mean represent the population
mean?
q6


For a population with a mean of
µ=70 and a σ=20, how much error,
on average, would you expect
between the sample mean and the
population mean for each of the
following sample sizes
N=4 scores
q6


For a population with a mean of
µ=70 and a σ=20, how much error,
on average, would you expect
between the sample mean and the
population mean for each of the
following sample sizes
N=16 scores
q6


For a population with a mean of
µ=70 and a σ=20, how much error,
on average, would you expect
between the sample mean and the
population mean for each of the
following sample sizes
N=25 scores
q8
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
If the population standard deviation
is 8, how large a sample is
necessary to have a standard error
that is?
Less than 4 points?
q8


If the population standard deviation
is 8, how large a sample is
necessary to have a standard error
that is?
Less than 2 points?
q8


If the population standard deviation
is 8, how large a sample is
necessary to have a standard error
that is?
Less than 1 point?
Probability and Sample Means

Because the distribution of sample
means tends to be normal, the zscore value obtained for a sample
mean can be used with the unit
normal table to obtain probabilities.
z-Scores and Location within the
Distribution of Sample Means

Within the distribution of sample
means, the location of each sample
mean can be specified by a z-score,
M–μ
z = ─────
σM
Example
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GRE quantitative scores are considered
to be normally distributed with a  =
500 and  = 100.
An exceptional group of 16 graduate
school applicants had a mean GRE
quantitative score of 710.
What is the probability of randomly
selecting 16 graduate school applicants
with an even greater mean GRE
quantitative score?
p (> 710 ) = ?
q11
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
A sample of n=4 scores has a mean
of m=75. Find the z-score for this
sample
If it was obtained from a population
with  = 80 and  = 10
q11


A sample of n=4 scores has a mean
of m=75. Find the z-score for this
sample
If it was obtained from a population
with  = 80 and  =40
q14

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The population IQ scores forms a
normal distribution, with a mean of
100 and standard deviation of 15.
What is the probability of obtaining
a sample mean greater than M=97
For a random sample of n=9
people?
q14


The population IQ scores forms a
normal distribution, with a mean of
100 and standard deviation of 15.
What is the probability of obtaining
a sample mean greater than M=97
For a random sample of n=25
people?
Example 2


Scores on a test form a normal
distribution with µ=70 and σ=12.
With a sample size of n=16.
What is the probability of obtaining
a sample of at least 75?
70
0
75

What proportion of the sample
means will be lower than 73?

What is the probability of obtaining
a sample with a mean less than 64?

What proportion of the sample
means will be within 2 points of the
population mean?
Example 3
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A normal distribution with µ=80 and
σ=15, sample size of n=36.
What sample means would mark off
the most extreme 5% of the
distribution?
What should you do?
q17
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
A population of scores forms a
normal distribution with a mean of
µ=80 and σ=10
What proportion of the scores have
values between 75 and 85?
q17


A population of scores forms a
normal distribution with a mean of
µ=80 and σ=10
For samples of n=4, what
proportion of the samples will have
means between 75 and 85?
q17


A population of scores forms a
normal distribution with a mean of
µ=80 and σ=10
For samples of n=16, what
proportion of the samples will have
means between 75 and 85?