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Accelerated Mathematics III
Unit 1
1st Edition
GETTYSBURG ADDRESS LEARNING TASK
1. Sampling Distribution of a Sample Mean from a Normal Population
The scores of individual students on the ACT entrance exam have a normal distribution with
mean 18.6 and standard deviation 5.9.
a. Use your calculator to simulate the scores of 25 randomly selected students who took the
ACT. Record the mean and standard deviations of these 25 people in the table below.
Repeat, simulating the scores of 100 people. (To do this, use the following command:
randNorm(,, n)L1.)
Population
25 people
100 people
Mean
18.6
Standard Deviation
5.9
b. As a class, compile the means for the sample of 25 people. Determine the mean and
standard deviation of this set of means. That is, calculate  x and  x . How does the mean
of the sample means compare with the population mean? How does the standard
deviation of the sample means compare with the population standard deviation?
c. Describe the plot of this set of means. How does the plot compare with the normal
distribution?
d. As a class, compile the means for the sample of 100 people. Determine the mean and
standard deviation of this set of means. That is, calculate  x and  x . How does the mean
of the sample means compare with the population mean? How does the standard
deviation of the sample means compare with the population standard deviation?
e. Describe the plot of this set of means. How does the plot compare with the normal
distribution?
f. Determine formulas for the mean of the sample means,  x , and the standard deviation of
the sample means,  x . Compare with a neighbor.
g. Just as we saw with proportions, the sample mean is an unbiased estimator of the
population mean. What does that mean?
The Sampling Distribution of a Sample Mean:
Choose a simple random sample of size n from a large population with mean  and
standard deviation . Then:
o The mean of the sampling distribution of x is ____.
o The standard deviation of the sampling distribution of x is ___________.
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Kathy Cox, State Superintendent of Schools
Copyright 2010 © All Rights Reserved
Unit 1: Page 1 of 8
Accelerated Mathematics III
Unit 1
1st Edition
h. Again, we must be cautious about when we use the formula for the standard deviation of
x . What was the rule when we looked at proportions? It is the same here.
i. Put these latter facts together with your response to part b to complete the following
statement:
Choose a simple random sample of size n from a population that has a normal
distribution with mean  and standard deviation . Then the sample mean x has a
____________distribution with mean ________ and standard deviation ____________.
This problem centered on a population that was known to be normally distributed. What about
populations that are not normally distributed? Can we still use the facts above? Let’s investigate!
2. How Long are the Words in the Gettysburg Address?
(Note: The Gettysburg Address and Word List are at the end of this task.)
a. To answer the question of how long the words in the Gettysburg address are, we want to
take a sample of the words. Propose different ways you might take a random sample of 5
words from the Gettysburg address.
b. The last page of this activity lists the words from the Gettysburg address. There are 268
words, and each word is assigned a number from 1 (001) to 268. To select a simple
random sample of 5 words, we need to generate 5 distinct random integers between 1 and
268. (Use the following command: randInt(1, 268, 5). If any of the numbers repeat,
repeat the command until you have 5 distinct integers.)
Find the words on the list that correspond to these integers. List the word lengths. Find
the average length of the 5 words in your sample, and record the mean length as x  5 .
Generate a new set of 5 distinct integers, and repeat. Also record each mean length on a
separate post-it note. Make sure the post-it is labeled, e.g. x  5 !
Random Integers
Word Lengths
Average Length
Sample One
Sample Two
x1  5 =
x2  5 =
c. Repeat the process two more times, this time choosing 10 words at a time. Calculate the
average length of the sample of 10 words and record as x 10  . Also record each mean
length on a separate post-it note. Make sure the post-it is labeled!
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Copyright 2010 © All Rights Reserved
Unit 1: Page 2 of 8
Accelerated Mathematics III
Unit 1
1st Edition
Sample One
Sample Two
x1 10 =
x2 10  =
Random Integers
Word Lengths
Average Length
d. Repeat the process one more time, this time choosing 25 words. Calculate the average
length of sample of 25 words and record as x  25 . Also record the mean length on a
post-it note. Make sure the post-it is labeled!
Random Integers
Word Lengths
Average Length
x  25 =
Clear a space on the floor, the wall, or the board. Use masking tape to make a number line
(horizontal axis) with average lengths marked from 2 to about 6 on the axis, with tick marks
every 0.1. Each interval should be a little more than the width of a post-it note.
e. Place the post-its with the average length of the samples of 5 words on the axis according
to the average length, making a post-it dotplot on the floor/wall/board.
Look at the shape of the final dotplot. Describe the distribution of the words’ lengths.
Make a second axis and label it with increments of 0.1. Again, each interval should be a little
more than the width of a post-it note.
f. Plot the means for the sample size of 10. What is the shape of the dotplot for the
distribution of x 10  ? How does it compare with the previous distribution?
Make a third axis on which to create a dotplot of the means for the sample size 25.
g. Plot the means for the sample size of 25. What is the shape of the dotplot for the
distribution of x  25 ? How does it compare with the previous distribution?
h. Divide into groups for the next part of this task. There should be at least 4 groups. Each
group will take one of the dotplots above, as well as the entire list of words, and find the
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Copyright 2010 © All Rights Reserved
Unit 1: Page 3 of 8
Accelerated Mathematics III
Unit 1
1st Edition
mean and standard deviation of the sample means. (Because there are so many words,
two groups should separately determine the overall mean and standard deviation and
check each other.)
Post a table similar to the one below on the board so that groups can post their results.
Record all groups’ result here.
Mean
Standard Deviation
Shape of the Distribution
Population
Samples of 5
Samples of 10
Samples of 25
i. Previously, we stated that if samples were taken from a normal distribution, that the mean
and standard deviation of the sampling distribution of sample means was also normal
with  x   and  x 

. In this activity, we did not begin with a normal distribution.
n
However, compare the means for the samples of 5, 10, and 25 with the overall mean of
the word lengths. Then compare the standard deviations with the standard deviation of all
the word lengths. Do these formulas appear to hold despite the population of word
lengths being obviously non-normal? Explain.
j. This brings us to the Central Limit Theorem (CLT) for Sample Means
Choose a simple random sample of size n from any population, regardless of the original
shape of the distribution, with mean  and finite standard deviation . When n is large,
the sampling distribution of the sample mean x is approximately normal with mean
______ and standard deviation ________.
Note: The statement “when n is large” seems a bit ambiguous. A good rule of thumb is
that the sample size should be at least 30, as we can see in the dotplots above. When the
sample sizes were smaller, i.e. 5 and 10, the plots were still quite right skewed.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Copyright 2010 © All Rights Reserved
Unit 1: Page 4 of 8
Accelerated Mathematics III
Unit 1
1st Edition
3. Applying the CLT for Sample Means
Whenever we have a normal distribution, we are able to use normal tables and normal
calculations to find probabilities. Due to the CLT, we can use normal calculations to determine
probabilities about sample means drawn from large samples.
a. First, we need to recall how to standardize scores, that is, find z-scores. State the formula
for z-scores that you learned in Math 3.
Recall that whenever we standardize values, we take the estimate (or statistic) minus the
corresponding parameter and divide that difference by the corresponding standard
deviation. We will standardize sample means in the same way. Substitute the statistics
and parameters for sample means into the z-score formula to obtain our standardization
formula for sample means.
b. Consider an IQ test with scores that vary according to a normal distribution with a mean
of 100 and a standard deviation of 15. Find the probability that a single person scores
higher than 110.
c. Suppose you take a sample of 20 individuals who took this IQ test. What are the mean
and standard deviation of the sampling distribution of the average score of this size
sample?
d. Find the probability that the mean score of these 20 people is greater than 110.
e. Find the probability that the mean score of these 20 people is between 95 and 110.
f. How would you expect your answers to parts d and e would be different if the sample
size were 50 instead of 20?
g. Calculate these probabilities and comment on your conjecture.
h. Which of your answers, if any, to parts b, c, d, e, and g would be affected if the
distribution of the candy bar weights was not normally distributed? Explain.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Copyright 2010 © All Rights Reserved
Unit 1: Page 5 of 8
Accelerated Mathematics III
Unit 1
1st Edition
The Gettysburg Address
Four score and seven years ago our fathers brought forth on this continent, a
new nation, conceived in Liberty, and dedicated to the proposition that all
men are created equal.
Now we are engaged in a great civil war, testing whether that nation, or any
nation so conceived and so dedicated, can long endure. We are met on a great
battlefield of that war. We have come to dedicate a portion of that field, as a
final resting place for those who here gave their lives that that nation might
live. It is altogether fitting and proper that we should do this.
But, in a larger sense, we cannot dedicate -- we can not consecrate -- we can
not hallow -- this ground. The brave men, living and dead, who struggled here,
have consecrated it, far above our poor power to add or detract. The world
will little note, nor long remember what we say here, but it can never forget
what they did here. It is for us the living, rather, to be dedicated here to the
unfinished work which they who fought here have thus far so nobly advanced.
(189) It is rather for us to be here dedicated to the great task remaining
before us -- that from these honored dead we take increased devotion to that
cause for which they gave the last full measure of devotion -- that we here
highly resolve that these dead shall not have died in vain -- that this nation,
under God, shall have a new birth of freedom -- and that government of the
people, by the people, for the people, shall not perish from the earth.
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Copyright 2010 © All Rights Reserved
Unit 1: Page 6 of 8
Accelerated Mathematics III
Unit 1
1st Edition
Gettysburg Address Word List
(page 1)
Number
001
002
003
004
005
006
007
008
009
010
011
012
013
014
015
016
017
018
019
020
021
022
023
024
025
026
027
028
029
030
031
032
033
034
035
036
037
038
039
040
041
042
043
044
045
Word
Four
score
and
seven
years
ago.
our
fathers
brought
forth
upon
this
continent
a
new
nation:
conceived
in
Liberty,
and
dedicated
to
the
proposition
that
all
men
are
created
equal.
Now
we
are
engaged
in
A
great
civil
war,
testing
whether
that
nation,
or
any
Length
4
5
3
5
5
3
3
7
7
5
4
4
9
1
3
6
9
2
7
3
9
2
3
11
4
3
3
3
7
5
3
2
3
7
2
1
5
5
3
7
7
4
6
2
3
Number
046
047
048
049
050
051
052
053
054
055
056
057
058
059
060
061
062
063
064
065
066
067
068
069
070
071
072
073
074
075
076
077
078
079
080
081
082
083
084
085
086
087
088
089
090
Word
nation
so
conceived
and
So
dedicated,
Can
Long
endure.
We
Are
met
on
A
great
battlefield
of
That
war.
We
Have
Come
To
dedicate
a
portion
Of
That
field
as
A
final
resting
place
For
those
Who
Here
Gave
their
lives
That
That
nation
might
Length
6
2
9
3
2
9
3
4
5
2
3
3
2
1
5
11
2
4
3
2
4
4
2
8
1
7
2
4
5
2
1
5
7
5
3
5
3
4
4
5
5
4
4
6
5
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Copyright 2010 © All Rights Reserved
Unit 1: Page 7 of 8
Number
091
092
093
094
095
096
097
098
099
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
Word
live.
It
is
altogether
fitting
and
proper
that
we
should
do
this.
But
in
a
larger
sense,
we
cannot
dedicate,
we
cannot
consecrate,
we
cannot
hallow
this
ground.
The
brave
men,
living
and
dead,
who
struggled
here
have
consecrated
it,
far
above
our
poor
power
Length
4
2
2
10
7
3
6
4
2
6
2
4
3
2
1
6
5
2
6
8
2
6
10
2
6
6
4
6
3
5
3
6
3
4
3
9
4
4
11
2
3
5
3
4
5
Accelerated Mathematics III
Unit 1
1st Edition
Gettysburg Address Word List
(page 2)
Number
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
Word
to
add
or
detract.
The
world
will
little
note,
nor
long
remember
what
we
say
here,
but
it
can
never
forget
what
they
did
here.
It
is
for
us
the
living,
rather,
to
be
dedicated
here
to
the
unfinished
work
which
they
who
fought
here
Length
2
3
2
7
3
5
4
6
4
3
4
8
4
2
3
4
3
2
3
5
6
4
4
3
4
2
2
3
2
3
6
6
2
2
9
4
2
3
10
4
5
4
3
6
4
Number
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
Word
Have
Thus
Far
So
Nobly
advanced.
It
Is
rather
For
Us
Here
to
Be
dedicated
To
The
Great
Task
remaining
before
us,
That
From
These
honored
Dead
We
Take
increased
devotion
to
That
Cause
To
Which
They
Gave
The
Last
Full
measure
Of
devotion,
That
Length
4
4
3
2
5
8
2
2
6
3
2
4
2
2
9
2
3
5
4
9
6
2
4
4
5
7
4
2
4
9
8
2
4
5
2
5
4
4
3
4
4
7
2
8
4
Georgia Department of Education
Kathy Cox, State Superintendent of Schools
Copyright 2010 © All Rights Reserved
Unit 1: Page 8 of 8
Number
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
Word
we
here
highly
resolve
that
these
dead
shall
not
have
died
in
vain,
that
this
nation,
under
God,
shall
have
a
new
birth
of
freedom,
and
that
government
of
the
people,
by
the
people,
for
the
people,
shall
not
perish
from
the
earth.
Length
2
4
6
7
4
5
4
5
3
4
4
2
4
4
4
6
5
3
5
4
1
3
5
2
7
3
4
10
2
3
6
2
3
6
3
3
6
5
3
6
4
3
5