### Chapter 1: Introduction to Statistics

```COURSE: JUST 3900
TIPS FOR APLIA
Chapter 7:
Distribution of Sample Means
Developed By:
Ethan Cooper (Lead Tutor)
John Lohman
Michael Mattocks
Aubrey Urwick
Key Terms: Don’t Forget
Notecards
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
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Sampling Error (p. 201)
Distribution of Sample Means (p. 201)
Sampling Distribution (p. 202)
Central Limit Theorem (p. 205)
Expected Value of M (p. 206)
Standard Error of M (p. 207)
Law of Large Numbers (p. 207)
Formulas

Standard Error of M:  =

−

z-Score Formula:  =

=
2

=
2

Central Limit Theorem

Question 1: A population has a mean of µ = 50 and a
standard deviation of σ = 12.
a)
b)
c)
d)
For samples of size n = 4, what is the mean (expected value)
and the standard deviation (standard error) for the distribution
of sample means?
If the population distribution is not normal, describe the shape
of the distribution of sample means based on n = 4.
For samples of size n = 36, what is the mean (expected value)
and the standard deviation (standard error) for the distribution
of sample means?
If the population distribution is not normal, describe the shape
of the distribution of sample means based on n = 36.
Central Limit Theorem

Question 1 Answers:
a)
Expected Value of M: µ = 50
Standard Error of M:  =
b)
c)
=
12
4
=
12
2
=6
The distribution of sample means does not satisfy either
criterion to be normal. It would not be a normal distribution.
Expected Value of M: µ = 50
Standard Error of M:  =
d)

=
12
36
=
12
6
=2
Because the sample size is greater than n = 30, the distribution
of sample means is a normal distribution.
Understanding the Sampling
Distribution of M

Question 2: As sample size increases, the value of
expected value of M also increases. (True or False?)
Understanding the Sampling
Distribution of M

Question 2 Answer:

False. The expected value of M does not depend on sample
size; it will always be equal to the population mean: µ.
Understanding the Sampling
Distribution of M

Question 3: As sample size increases, the value of the
standard error also increases. (True or False?)
Understanding the Sampling
Distribution of M

Question 3 Answer:

False. The standard error decreases as sample size increases.
In Question 1a, the standard error was  =

=
12
4
=
12
2
= 6.
However in Question 1c, in which the sample size was increased from
n = 4 to n = 36, the standard error decreased:  =

=
12
36
=
12
6
= 2.
Using z-Scores with the
Distribution of Sample Means

Question 4: For a population with a mean of µ = 40 and
a standard deviation of σ = 8, find the z-score
corresponding to a sample mean of M = 44 for each of
the following sample sizes.
a)
b)
n=4
n = 16
Using z-Scores with the
Distribution of Sample Means

Question 4 Answers;
a)
The standard error is  =
=
b)
−

=
44−40
4
=
4
4
−

=
44−40
2
=
4
2
=
8
4
=
8
16
8
2
= = 4, and
= 1.00
The standard error is  =
=

= 2.00
8
4
= = 2, and
Using z-Scores with the
Distribution of Sample Means

Question 5: What is the probability of obtaining a sample
mean greater than M = 60 for a random sample of n = 16
scores selected from a normal population with a mean of
µ = 65 and a standard deviation of σ = 20?
Using z-Scores with the
Distribution of Sample Means
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Question 5 Answer:


M corresponds to  =

p(M > 60) = p(z > -1.00) = 0.8413 (or 84.13%)
Z = -1.00
=
60−65
5
=
20
4
The standard error is  =
−

=
20
16

=
−5
5
= 5,
= −1.00,
Using z-Scores with the
Distribution of Sample Means

Question 6: A positively skewed distribution has µ = 60
and σ = 8.
a)
b)
What is the probability of obtaining a sample mean greater than
M = 62 for a sample of n = 4 scores?
What is the probability of obtaining a sample mean greater than
M = 62 for a sample of n = 64 scores?
Using z-Scores with the
Distribution of Sample Means
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Question 6 Answer:
a)
The distribution does not satisfy either of the criteria for being
normal. Therefore, you cannot use the unit normal table, and it
is impossible to find the probability.
Remember: A distribution of sample means is normal if at least one of the following
condition are met:
1) The population from which the samples are selected is normal,
2) The number of scores (n) in each sample is relatively large, around 30 or more.
b)
With n = 64, the distribution of sample means is nearly normal.

8
8
The standard error is  =
=
= = 1,

M corresponds to  =
−

=
64
62−60
1
=
8
2
1
= 2.00,
p(M > 62) = p(z > 2.00) = 0.0228 (or 2.28%)
Three Different Distributions

Question 7: A population has a mean of µ = 100 and a
standard deviation of σ = 15. A sample of n = 25 scores
is taken with a mean of M = 101.2 and a standard
deviation of s = 11.5.
a)
b)
c)
On average, how much difference should there be between the
population mean and a single score selected from this
population?
On average, how much difference should there be between the
sample mean and a single score selected from that sample?
On average, how much difference should there be between the
population mean and the sample mean of any sample
consisting of n = 25 scores?
Three Different Distributions

Question 7:
b)
σ = 15
s = 11.5
c)
=
a)

=
15
25
=
15
5
=3
Frequently Asked Questions
FAQs

What effect does sample size have on the standard
error?


As sample size increases, standard error decreases. This is
because large samples are more representative of the
population. Thus we can expect less difference, or error.
As sample size decreases, standard error increases. This is
because smaller samples are less representative of the
population. Thus we can expect a greater difference, or error.
Frequently Asked Questions
FAQs


What’s the difference between standard deviation and
standard error?
Go back and review the slides for Question 7.



The standard deviation deals with means and SCORES.
If Aplia asks a question about the average difference between
the population mean and a SCORE, we are looking for the
population standard deviation, not the standard error.
The sample standard deviation deals with the sample mean and
scores for a particular sample.
If you see the words “mean” and “score,” think standard deviation, not standard error.

Standard error, however, measures the average distance
between the population mean and any sample mean of size n.
If you see the words “sample mean” and “population mean,” without reference to
scores, think standard error.
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