Lesson 21: Margin of Error when Estimating a Population Mean
Date:
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NYS COMMON CORE MATHEMATICS CURRICULUM
M4
Lesson 21
ALGEBRA II
Lesson 21: Margin of Error when Estimating a Population
Mean
Student Outcomes
Students use data from a random sample to estimate a population mean.
Students calculate and interpret margin of error in context.
Students know the relationship between sample size and margin of error in the context of estimating a
population mean.
Lesson Notes
In the previous lesson, students estimated the population mean using the sample mean based on a random sample of
size . To determine how accurate their estimate was, they had to create a sampling distribution of the sample mean
based on computing sample means for a large number of random samples. Finally, they computed the margin of error
as twice the standard deviation of the sample means. Although the process was a lot of work, students developed a
conceptual understanding of margin of error.
In this lesson, students use a formula for the standard deviation of the sample mean,
, where is the standard
deviation of the sample and is the size of the sample. The margin of error, 2
, is based on a single random sample,
thus making the work much easier.
The formula
is used to calculate the standard deviation of the sample mean when the mean and the standard
deviation of the population are stated. Previously, the formula
()
was used to calculate the standard deviation of a
sample proportion when the number of successes was known. In both formulas, as gets larger, the standard deviation
gets smaller. Both methods are applications of the central limit theorem, which says that regardless of the shape of the
population from which samples are taken, the distributions of both the sample means and the sample proportions are
approximately normal.
Classwork
This lesson continues to discuss using the sample mean as an estimate of the population mean and judging its accuracy
based on the concept of margin of error. In the last lesson, the margin of error was defined as twice the standard
deviation of the sampling distribution of the sample mean. In this lesson, a formula will be given for the margin of error
that allows you to calculate the margin of error from a single random sample rather than having to create a sampling
distribution of sample means.
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M4
Lesson 21
ALGEBRA II
Example 1 (5 minutes): Estimating a Population Mean Using a Random Sample
Give students a few minutes to read the introductory material of this example, and remind them of the process they
used in the previous lesson to get an estimate of margin of error. Then, write the formula for margin of error on the
board, making sure that students understand that this will allow them to calculate an estimate of the margin of error
using data from a single random sample.
Example 1: Estimating a Population Mean Using a Random Sample
Provide a one-sentence summary of our findings from the previous lesson.
Sample response: We took lots of random samples of computer game ratings, computed their means, displayed the
distribution of their means, and, finally, computed a margin of error.
What were drawbacks of the calculation method?
Sample response: Many samples are required. If we had increased the sample size or the number of samples, the time
required to take all those samples, calculate their means, and analyze the distribution would have increased significantly.
In practice, you do not have to use that process to find the margin of error. Fortunately, just as was
the case with estimating a population proportion, there are some general results that lead to a
formula that allows you to estimate the margin of error using a single sample. You can then gauge
the accuracy of your estimate of the population mean by calculating the margin of error using the
sample standard deviation.
Exercises 1–5 (10 minutes)
Have students work independently on the calculations required to answer Exercises 1–3.
Then, work through Exercises 4 and 5 as a class.
Exercises 1–5
1. Suppose a random sample of size ten produced the following ratings in the computer games
rating example in the last lesson: , , , , , , , , , . Estimate the population mean
rating based on these ten sampled ratings.
The sample mean estimate for the population mean rating is
= . rating points.
2. Calculate the sample standard deviation. Round your answer to three decimal places.
The sample standard deviation is . rating points.
The standard deviation of the distribution of sample means is approximated by
,
where
is the standard deviation of the sample, and
is the size of the sample.
Scaffolding:
For struggling students –
the bigger the value of ,
the smaller the standard
deviation. From a
population where = 2, if
= 36, the standard
deviation is
; however, if
the sample is larger, say
81, the standard deviation
would be
.
For advanced students –
when a sample is taken
from a population, the
mean of the sample is the
same as the mean of the
population, but the
variance (square of the
standard deviation) is only
as large. Regardless of
the shape of the
population, the
distribution of the sample
means will approach
normal (central limit
theorem). The variance is
; therefore, by applying a
square root, the standard
deviation is
.
MP.2
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Lesson 21
ALGEBRA II
3. Use the formula given above to calculate the approximate standard deviation of the distribution of sample means.
Round your answer to three decimal places.
The standard deviation of the distribution of sample means is
=
.
= . rating points.
4. Recall that the margin of error is twice the standard deviation of the distribution of sample means. What is the
value of the margin of error based on this sample? Write a sentence interpreting the value of the margin of error in
the context of this problem on computer game ratings.
Margin of error is (. ) = . rating points. The population mean rating for the computer games is
likely to be within . rating points of the sample mean estimate . .
5. Based on the sample mean and the value of the margin of error, what is an interval of plausible values for the
population mean?
Plausible values for the population mean rating are from . . = . to . + . = . rating
points.
Exercises 6–13 (20 minutes): The Gettysburg Address
Distribute a copy of the Gettysburg Address to each student in the class. (A copy is provided at the end of this lesson.)
Have students work individually or in pairs to answer the questions in this set of exercises. Then, discuss the answers to
the last question as a class. Consider challenging students to find the length of a typical word in the Gettysburg Address.
After students do this exercise “by hand” (using a calculator), you may want to show them an applet that displays three
different estimates regarding the Gettysburg Address. One is the mean word length. The other two are estimating
population proportions; one is the proportion of “long” words defined as words with more than four letters, and the
other is the proportion of nouns. The applet can be found at the following
site: http://www.rossmanchance.com/applets/GettysburgSample/GettysburgSample.html
.
This applet may require an updated version of an operating system to work correctly. If the applet does not work for all
students due to a computer’s operating system or network settings, attempt to demonstrate it for the whole class, as it
is an effective way to complement how students obtained their answers in the exercises. The applet allows the user to
specify a sample size (ten in this exercise) and the number of samples desired. Note that only one sample is to be used
to answer the questions in this exercise set.
To generate a sampling distribution for the sample mean (or proportion), enter a large number in the Num samples box,
such as 500. The Animate box shows the observations for each sample taken and the resulting values of the statistics
(mean or proportion) plotted on a histogram. (You may unclick the Animate box at any time to see the total results
immediately.)
Students should begin work on Exercise 6. Exercises 7–13 are provided as scaffolding if necessary. Students should be
able to clearly describe and fully implement a plan on their own. Sample responses are provided but will vary.
Exercises 6-13
The Gettysburg Address is considered one of history’s greatest speeches. Some students noticed that the speech was
very short (about words, depending on the version) and wondered if the words were also relatively short. To
estimate the mean length of words in the population of words in the Gettysburg Address, work with a partner on the
following steps. Your teacher will give you a copy of the Gettysburg Address with words numbered from to .
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Lesson 21
ALGEBRA II
6. Develop and describe a plan for collecting data from the Gettysburg Address and determining the typical length of a
word. Then, implement your plan, and report your findings.
Many answers are possible. Every answer should include the following:
A description of how a word sample is chosen, making sure to describe how randomization occurs
The actual sample chosen
Calculations of the sample mean, standard deviation, and margin of error
Interpretations in context of the sample mean, standard deviation, and margin of error
7. Use a random-number table or a calculator with a random-number generator to obtain ten different random
numbers from to .
8. Use the random numbers found in Exercise 7 as identification numbers for the words that will make up your random
sample of words from the Gettysburg Address. Make a list of the ten words in your sample.
the
resolve
score
consecrate
final
The
nobly
a
The
endure
9. Count the number of letters in each of the ten words in your sample.
10. Calculate the sample mean number of letters for the ten words in your sample.
The mean of the ten word lengths from Exercise 9 is
= . letters.
11. Calculate the sample standard deviation of the number of letters for the ten words in your sample. Round your
answer to three decimal places.
The standard deviation of the ten word lengths from Exercise 9 is . letters.
12. Use the sample standard deviation from Exercise 11 to calculate the margin of error associated with using your
sample mean as an estimate of the population mean. Round your answer to three decimal places.
The margin of error of this estimate is
.
= . letters.
13. Write a few sentences describing what you have learned about the mean length of the population of words in
the Gettysburg Address. Be sure to include an interpretation of the margin of error.
We estimate the mean word length of words in the Gettysburg Address to be . letters. The margin of error of this
estimate is . letters. So, plausible values for the population mean word length are from . to . letters.
MP.5
MP.2
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M4
Lesson 21
ALGEBRA II
Closing (3 minutes)
Ask students to summarize the main ideas of the lesson in writing or with a neighbor. Use this as an
opportunity to informally assess comprehension of the lesson. The Lesson Summary below offers some
important ideas that should be included.
Exit Ticket (7 minutes)
Lesson Summary
When using the sample mean to estimate a population mean, it is important to know something about
how accurate that estimate might be.
Accuracy can be described by the margin of error.
The margin of error can be estimated using data from a single random sample (without the need to
create a simulated sampling distribution) by using the formula
,
where
is the standard
deviation of a single sample, and
is the sample size.
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Lesson 21
ALGEBRA II
Name Date
Lesson 21: Margin of Error when Estimating a Population Mean
Exit Ticket
A Health Group study recommends that the total weight of a male student’s backpack should not be more that 15% of
his body weight. For example, if a student weighs 170 pounds, his backpack should not weigh more than 25.5 pounds.
Suppose that ten randomly selected eleventh grade boys produced the following data:
Body weight
155
136
197
174
165
165
150
142
176
157
Backpack weight
29.8
27.2
32.5
34.8
31.8
28.8
31.1
26.0
28.3
31.4
a. For each student, calculate backpack weight as a percentage of body weight (round to one decimal place).
b. Based on the data in part (a), estimate the mean percentage of body weight that eleventh grade boys carry in
their backpacks.
c. Find the margin of error for your estimate of part (b). Round your answer to three decimal places. Explain
how you determined your answer.
d. Comment on the amount of weight eleventh grade boys at this school are carrying in their backpacks
compared to the recommendation by the Health Group.
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Lesson 21
ALGEBRA II
Exit Ticket Sample Solutions
A Health Group study recommends that the total weight of a male student’s backpack should not be more that % of
his body weight. For example, if a student weighs pounds, his backpack should not weigh more than . pounds.
Suppose that ten randomly selected eleventh grade boys produced the following data:
Body weight
Backpack weight
.
.
.
.
.
.
.
.
.
.
a. For each student, calculate backpack weight as a percentage of body weight (round to one decimal place).
Body weight
Backpack weight
.
.
.
.
.
.
.
.
.
.
Percentage
.
.
.
.
.
.
.
.
.
.
b. Based on the data in part (a), estimate the mean percentage of body weight that eleventh grade boys carry in
their backpacks.
The sample mean percentage is . percentage points.
c. Find the margin of error for your estimate of part (b). Explain how you determined your answer.
The standard deviation of the percentages is . percent. So, the margin of error is
.
=
. percentage points.
d. Comment on the amount of weight eleventh grade boys at this school are carrying in their backpacks
compared to the recommendation by the Health Group.
Based on the data in this study, plausible percentages of mean body weight percentage that eleventh graders
are carrying in their backpacks are . . = . to . + . = . percentage
points. The interval (. %, . %) is above the recommended % maximum. On average,
eleventh grade boys at this school are carrying too much weight in their backpacks.
Problem Set Sample Solutions
1. A new brand of hot dog claims to have a lower sodium content than the leading brand.
a. A random sample of ten of these new hot dogs results in the following sodium measurements (mg).
Estimate the population mean sodium content of this new brand of hot dog based on the ten sampled
measurements.
Based on the data, an estimate for the population mean sodium content of this new brand of hot dog is
. mg of sodium.
b. Calculate the margin of error associated with your estimate of the population mean from part (a). Round
your answer to three decimal places.
The margin of error is
.
= . mg.
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ALGEBRA II
c. The mean sodium content of the leading brand of hot dogs is known to be mg. Based on the sample
mean and the value of the margin of error for the new brand, is a mean sodium content of mg a
plausible value for the mean sodium content of the new brand? Comment on whether you think the new
brand of hot dog has a lower sodium content on average than the leading brand.
Plausible values for population mean sodium content are between . . = . mg and
. + . = . mg. This interval is well below the mg which is the mg content for the
leading brand. So, the new hot dog brand has lower mean sodium content.
d. Another random sample of new brand hot dogs is taken. Should this larger sample of hot dogs produce a
more accurate estimate of the population mean sodium content than the sample of size ? Explain your
answer by appealing to the formula for margin of error.
The margin of error will be smaller. Sample size is in the denominator of formula for margin of error.
2. It is well known that astronauts increase their height in space missions because of the lack of gravity. A question is
whether or not we increase height here on Earth when we are put into a situation where the effect of gravity is
minimized. In particular, do people grow taller when confined to a bed? A study was done in which the heights of
six men were taken before and after they were confined to bed for three full days.
a. The before-after differences in height measurements (mm) for the six men were:
. . . . . . .
Assuming that the men in this study are representative of the population of all men, what is an estimate of
the population mean increase in height after three full days in bed?
Based on the given data, an estimate of the population mean increase in height after three full days in bed is
. mm.
b. Calculate the margin of error associated with your estimate of the population mean from part (a). Round
your answer to three decimal places.
The margin of error is approximated by
.
= . mm.
c. Based on your sample mean and the margin of error from parts (a) and (b), what are plausible values for the
population mean height increase for all men who stay in bed for three full days?
Plausible values for the population mean height increase for all men who stay in bed for three full days are
those between . . = . and . + . = . mm.
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M4
Lesson 21
ALGEBRA II
Exercises 6-13: Gettysburg Address
001 Four
045 any
089 nation
133 our
177 they
221 full
265 perish
002 score
046 nation,
090 might
134 poor
178 who
222 measure
266 from
003 and
047 so
091 live.
135 power
179 fought
223 of
267 the
004 seven
048 conceived
092 It
136 to
180 here
224 devotion,
268 earth.
005 years
049 and
093 is
137 add
181 have
225 that
006 ago,
050 so
094 altogether
138 or
182 thus
226 we
007 our
051 dedicated,
095 fitting
139 detract.
183 far
227 here
008 fathers
052 can
096 and
140 The
184 so
228 highly
009 brought
053 long
097 proper
141 world
185 nobly
229 resolve
010 forth
054 endure.
098 that
142 will
186 advanced.
230 that
011 upon
055 We
099 we
143 little
187 It
231 these
012 this
056 are
100 should
144 note,
188 is
232 dead
013 continent
057 met
101 do
145 nor
189 rather
233 shall
014 a
058 on
102 this.
146 long
190 for
234 not
015 new
059 a
103 But,
147 remember,
191 us
235 have
016 nation;
060 great
104 in
148 what
192 to
236 died
017 conceived
061 battlefield
105 a
149 we
193 be
237 in
018 in
062 of
106 larger
150 say
194 here
238 vain,
019 liberty,
063 that
107 sense,
151 here,
195 dedicated
239 that
020 and
064 war.
108 we
152 but
196 to
240 this
021 dedicated
065 We
109 cannot
153 it
197 the
241 nation,
022 to
066 have
110 dedicate,
154 can
198 great
242 under
023 the
067 come
111 we
155 never
199 task
243 God,
024 proposition
068 to
112 cannot
156 forget
200 remaining
244 shall
025 that
069 dedicate
113 consecrate,
157 what
201 before
245 have
026 all
070 a
114 we
158 they
202 us,
246 a
027 men
071 portion
115 cannot
159 did
203 that
247 new
028 are
072 of
116 hallow
160 here.
204 from
248 birth
029 created
073 that
117 this
161 It
205 these
249 of
030 equal.
074 field
118 ground.
162 is
206 honored
250 freedom,
031 Now
075 as
119 The
163 for
207 dead
251 and
032 we
076 a
120 brave
164 us
208 we
252 that
033 are
077 final
121 men,
165 the
209 take
253 government
034 engaged
078 resting
122 living
166 living,
210 increased
254 of
035 in
079 place
123 and
167 rather,
211 devotion
255 the
036 a
080 for
124 dead,
168 to
212 to
256 people,
037 great
081 those
125 who
169 be
213 that
257 by
038 civil
082 who
126 struggled
170 dedicated
214 cause
258 the
039 war,
083 here
127 here
171 here
215 for
259 people,
040 testing
084 gave
128 have
172 to
216 which
260 for
041 whether
085 their
129 consecrated
173 the
217 they
261 the
042 that
086 lives
130 it,
174 unfinished
218 gave
262 people,
043 nation,
087 that
131 far
175 work
219 the
263 shall
044 or
088 that
132 above
176 which
220 last
264 not
