MATHEMATICS 20-2
Statistical Reasoning
High School collaborative venture with
Edmonton Christian, Institutional Services, Jasper Place,
Millwoods Christian, Queen Elizabeth and Victoria Schools
Edmonton Christian: Jenn Johnson
Institutional Services: Eric Hanson
Jasper Place: Jessica Noselski
Millwoods Christian: Ken Scharf
Queen Elizabeth: David Hernandez-Rivera
Victoria: Gina MacKechnie
Facilitator: John Scammell (Consulting Services)
Editor: Jim Reed (Contracted)
2010 - 2011
Mathematics 20-2
Statistical Reasoning
Page 2 of 47
TABLE OF CONTENTS
STAGE 1
DESIRED RESULTS
PAGE
Big Idea
4
Enduring Understandings
4
Essential Questions
4
Knowledge
5
Skills
6
STAGE 2
ASSESSMENT EVIDENCE
Transfer Task
Talking Feet Project
Teacher Notes for Transfer Task
Transfer Task
Rubric
Possible Solution
7
9
16
17
STAGE 3 LEARNING PLANS
Lesson #1
Standard Deviation
23
Lesson #2 The Normal Curve
30
Lesson #3
35
Z-scores
Lesson #4 Confidence Interval
43
Lesson #5 Confidence Intervals in Print and Media
47
Mathematics 20-2
Statistical Reasoning
Page 3 of 47
Mathematics 20-2
Statistical Reasoning
STAGE 1
Desired Results
Big Idea:
Statistics summarizes data and predict future outcomes in areas such as
advertisement, sales, sports and academics. We can make logical assumptions and
back them up with numerical data.
Implementation note:
Post the BIG IDEA in a prominent
place in your classroom and refer to
it often.
Enduring Understandings:
Students will understand that…




There are different measures of central tendency, which may be appropriate in
different situations.
Data can be presented in a misleading fashion.
Certain data is normally distributed (bell curve).
Predictions based on statistics will contain error.
Essential Questions:





Which measure of central tendency is better in certain situations?
o When is the mean the best measure of central tendency to use?
o When is the median the best measure of central tendency to use?
o When is the mode the best measure of central tendency to use?
When is it appropriate to use a sample set instead of an entire population?
Why would people misrepresent data using statistics?
When is the size of the standard deviation important?
Why are some sets of data normally distributed and others are not?
Implementation note:
Ask students to consider one of the
essential questions every lesson or two.
Has their thinking changed or evolved?
Mathematics 20-2
Statistical Reasoning
Page 4 of 47
Knowledge:
Enduring
Understanding
Specific
Outcomes
Students will know …
Students will understand…

*S1.3
There are different
measures of central
tendency, which may
be appropriate in
different situations.
Data can be presented
in a misleading
fashion.
S1.6
Certain data is
normally distributed
(bell curve).
Predictions based on
statistics will contain
error.

S1.1, 1.4, 1.5, 1.7,
1.8, 1.9



S2.1, 2.2

Students will understand…

how standard deviation effects the curve and
area under the curve
what standard deviation is
the characteristics of a normal distribution
how standard deviation effects the curve and
area under the curve
 what a z-score is and how it applies to the
normal distribution
Students will know …

8888
I*S =
how to identify measures of central tendency
Students will know …
Students will understand…


Students will know …
Students will understand…

Description of
Knowledge
how confidence levels, margin of error and
confidence intervals may vary depending on
the size of the random sample
the significance of a confidence interval,
margin of error or confidence level
Statistics
Mathematics 20-2
Statistical Reasoning
Page 5 of 47
Skills:
Enduring
Understanding
Specific
Outcomes
Students will be able to…
Students will understand…

There are different
measures of central
tendency, which may
be appropriate in
different situations.
*S1.5

S2.6

Data can be presented
in a misleading
fashion.
S2.5
S2.3
S2.4
Students will understand…

S1.7, S1.4
Certain data is
normally distributed
(bell curve).
*S = Statistics
Mathematics 20-2
interpret and explain confidence intervals and
margin of error, using examples found in print
of electronic media
 make inferences about a population from
sample data using given confidence intervals
 provide examples from print or electronic
media in which confidence intervals are used to
support a particular position
Students will be able to…




determine if a set approximates a normal curve
determine the z-score for a given value in a
normally distributed data set
solve a contextual problem
calculate the population standard deviation of a
data set
Students will be able to…
Students will understand…
Predictions based on
statistics will contain
error.

S1.8
S1.5, S1.9
S1.2

compare the properties of two or more normally
distributed data sets
support a position by analyzing statistical data
presented in the media
Students will be able to…
Students will understand…

Description of
Skills
S2.3

S2.4

make inferences about a population from
sample data using given confidence intervals
provide examples from print or electronic
media in which confidence intervals are used to
support a particular position
Implementation note:
Teachers need to continually ask
themselves, if their students are
acquiring the knowledge and skills
needed for the unit.
Statistical Reasoning
Page 6 of 47
STAGE 2
1
Assessment Evidence
Desired Results Desired Results
Talking Feet Project
Teacher Notes
There is one transfer task to evaluate student understanding of the concepts relating to
statistical reasoning. A photocopy-ready version of the transfer task is included in this section.
Implementation note:
Students must be given the transfer task & rubric at
the beginning of the unit. They need to know how
they will be assessed and what they are working
toward.
Note: This task will take several days to complete.
 The task is best used as a final cumulative activity.
 Question 8: Students should have the same scale for male and female foot
length, as well as for male and female height. This will allow for comparison of
the distribution by inspection.
Students must know how to:




calculate z-scores using tables or calculator
use the confidence interval formula
calculate standard deviation and mean
calculate margin of error
Each student will:



demonstrate their understanding of normal distribution including standard
deviation and z-scores
interpret statistical data using confidence intervals and margin of error
solve problems involving interpretation of standard deviation
Implementation note:
Teachers need to consider what performances and
products will reveal evidence of understanding?
What other evidence will be collected to reflect
the desired results?
Mathematics 20-2
Statistical Reasoning
Page 7 of 47
Talking Feet Project - Student Transfer Task
What does your foot length say about your height?
Can you predict people's height by how long their feet
are?
If a Grade 11 student's foot is 27 cm long, how tall is that student
likely to be? In this activity, you will use data collected from your
class to determine whether a relationship exists between foot length
and height.
Example:
Students from South Africa, the United Kingdom, Australia and New
Zealand measured and recorded their height and foot length in
centimetres and entered this information into the Census at School
database. A random sample was selected from the combined data to
create the scatter plot below, where each dot shows an ordered pair
of height relative to foot length.
Scatter plot of height and foot length for 15-year-old students
Source: http://www.censusatschool.ca/02/pdf/02_023-eng.pdf
Notes: http://www.censusatschool.ca/02/pdf/02_023a-eng.pdf
Mathematics 20-2
Statistical Reasoning
Page 8 of 47
Talking Feet Project
Part 1: Data Collection
1. Collect data on foot length in centimetres and height in centimetres
from at least two grade 11 students not in your class (at least one
male, one female) using the Talking Feet Survey Form provided.
2. Fill in the Predicting Height from Foot Length Class Data table
provided for all data collected by your class.
3. Create a scatter plot similar to the one shown on the previous page.
4. Manually draw a line of best fit on the scatter plot.
5. From the graph:
a. Is there a relationship that exists between foot length and
height?
b. Is your graph similar to the above example from other
countries?
6. Determine the mean and standard deviation for the following:
a. Male foot length
b. Male height
c. Female shoe length
d. Female height
7. Create a normal distribution curve for:
a. Male foot length
b. Male height
c. Female foot length
d. Female height
8. Label the following distribution curve with:
a. Mean
b. 3 standard deviations above and below the mean
9. Using your normal distribution curve answer the following:
a. What percent of students have a height less than 150 cm?
i. a male
ii. a female
b. What percent of students have a foot size larger than 26 cm?
i. a male
ii. a female
c. In two or three sentences explain your findings regarding the percentages
found in questions (a) and (b).
Talking Feet Project
10. Calculate the z-score for your foot size and height.
11. What standard deviation for your foot length would be
necessary so that you would have the same z-score for your
height (prove your answer algebraically)?
12. Of the four sets of data collected which ones, if any, falls within
the 68-95-99 Rule for a normal distribution curve? Provide an
explanation if necessary.
13. Calculate the 95% confidence interval for each set of data and
explain what it means.
14. What is the margin of error for each set of data? Explain what
factors could influence this.
Conclusion:
What does your foot length say about your height?
Can you predict people's height by how long their feet are?
If a Grade 11 student's foot is 27 cm long, how tall is that student likely
to be?
Talking Feet Survey Form
Gender
(M / F)
Participants Name
Foot Length
(cm)
Grade
Height (cm)
11
11
11
11
11
11
11
11
11
11
11
References:
US &
Canada
Shoe
Size
Foot
Length
(cm)
M
3½
4
4½
5
5½
6
6½
7
7½
8
8½
9
10½
11½
12½
14
F
5
5½
6
6½
7
7½
8
8½
9
9½
10
10½
12
13
14
15½
--- 22.8 23.1 23.5 23.8 24.1 24.5 24.8 25.1 25.4 25.7
26
26.7
27.3
27.9
28.8
29.2
Height in feet and
inches
4 feet 0 inches
4 feet 1 inches
4 feet 2 inches
4 feet 3 inches
4 feet 4 inches
4 feet 5 inches
4 feet 6 inches
4 feet 7 inches
4 feet 8 inches
4 feet 9 inches
4 feet 10 inches
4 feet 11 inches
Height
(cm)
121.92
124.46
127.00
129.54
132.08
134.62
137.16
139.70
142.24
144.78
147.32
149.86
Height in feet and
inches
5 feet 0 inches
5 feet 1 inches
5 feet 2 inches
5 feet 3 inches
5 feet 4 inches
5 feet 5 inches
5 feet 6 inches
5 feet 7 inches
5 feet 8 inches
5 feet 9 inches
5 feet 10 inches
5 feet 11 inches
Height
(cm)
152.40
154.94
157.48
160.02
162.56
165.10
167.64
170.18
172.72
175.26
177.80
180.34
Height in feet and
inches
6 feet 0 inches
6 feet 1 inches
6 feet 2 inches
6 feet 3 inches
6 feet 4 inches
6 feet 5 inches
6 feet 6 inches
6 feet 7 inches
Height
(cm)
182.88
185.42
187.96
190.50
193.04
195.58
198.12
200.66
Predicting Height from Foot Length Class Data
Grade 11 Male Data:
Foot Length
(cm)
Height (cm)
Grade 11 Male Data Continued:
Foot Length
(cm)
Height (cm)
Predicting Height from Foot Length Class Data
Grade 11 Female Data:
Foot Length
(cm)
Height (cm)
Grade 11 Female Data Continued:
Foot Length
(cm)
Height (cm)
Glossary
central tendency – Measures of central tendency are numbers that indicate the
center of a set of ordered numerical data. The three common measures of central
tendency are the mean, median and the mode.
confidence interval – The interval within which the value of a random variable is
estimated to lie with a stated degree of probability
histogram - A graph consisting of bars used to visually represent a frequency table
where at least one of the scales represents continuous data
line of best fit - A line on a scatter plot that best defines or expresses the trend
shown in the plotted points
margin of error - The proportion added to and subtracted from the result to construct
the confidence interval
mean – The mean (or "arithmetic mean") is a measure of central tendency of a set of
data represented by numbers. Adding all of the values and dividing by the number of
values calculate the mean. The symbol for the arithmetic mean is a letter with a
segment above it.
median - The median is a measure of central tendency of a set of data represented by
numbers. The median is the "middle" of a set of numbers in ascending or descending
order. The symbol for the median is usually the letter "M".
mode – The mode is a measure of central tendency of a set of data represented by
numbers. The mode is the most frequently occurring number. There is no standard
symbol associated with the mode.
normal distribution curve - A bell-shaped curve showing a particular distribution of
probability over the values of a random variable.
scatter plot - A graph consisting of individual points whose coordinates represent
values of an independent and a dependent variable
standard deviation - A measure of the dispersion of a frequency distribution
z-score - A standard score that measures how many standard deviation units away
from the mean a particular value lays.
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Assessment
Mathematics 20-2
Statistical Reasoning
Rubric
Level
Criteria
Content
Excellent
5
Procedures and
calculations are
efficient and
effective
May contain
minor errors that
does not affect
understanding
Proficient
4
Procedures
and
calculations
are reasonable
and may
contain minor
errors
May contain
error(s) that
hinder a
complete
solution
Adequate
3
Procedures
and
calculations
are basic
May contain a
major
mathematical
error or
omission
Limited
2
Insufficient
1
Procedures and
calculations are
basic
Develops an
initial start that
may be partially
correct or could
led to a correct
solution
May contain
several major
mathematical
errors or
omissions
Reasoning
Makes significant
comparisons and
connections with
data
Makes
reasonable
comparisons
and
connections
with data
Makes some
comparisons
and
connections
with data
Makes minimal
comparisons
and
connections
with data
Makes minimal
or no
comparisons
and
connections
with data
Communication
Uses significant
mathematical
language to
explain
understanding
Uses
mathematical
language to
explain
understanding
Uses common
language to
explain
understanding
Communication
is weak
Communication
is weak or
absent
When work is judged to be limited or insufficient, the teacher makes decisions
about appropriate intervention to help the student improve.
Possible Solution to Talking Feet Project
1.
Talking Feet Survey Form
Participants
Name
Pam
Jill
Bob
Jim
Amale
Kevin
Kelly
Amara
Ralph
Tarek
Sean
Gender (M / F)
Grade
F
F
M
M
F
M
F
F
M
M
M
11
11
11
11
11
11
11
11
11
11
11
Foot Length
(cm)
27.5
23.0
29.5
26.5
25.0
27.0
21.0
23.0
26.0
26.0
19.0
Height (cm)
180
163
185
166
172
178
162
148
167
179
147
2.
Predicting Height from Foot Length Class Data
Grade 11 Male Data:
Foot Length (cm)
29.5
Grade 11 Female Data:
Height (cm)
185
Foot Length (cm)
27.5
Height (cm)
180
26.5
166
23.0
163
27.0
178
25.0
172
26.0
167
21.0
162
26.0
179
23.0
148
19.0
147
23.0
145
24.0
155
25.0
171
28.0
172
23.0
157
23.0
141
23.0
157
25.0
150
25.5
166
27.0
159
23.0
163
25.5
172
26.0
165
Mathematics 20-2
Statistical Reasoning
Page 16 of 47
3.
Height vs. Foot Length
190
185
180
Height (cm)
175
170
165
160
155
150
145
140
135
130
15
17
19
21
23
25
27
Foot Length (cm)
4. Students will manually draw in line of best fit
5. Answers will vary depending on student graph
Questions 6, 7, 8
a. Male foot length
Mathematics 20-2
Statistical Reasoning
Page 17 of 47
29
b. Male height
c. Female foot length
d. Female height
162.4
9.4
Mathematics 20-2
Statistical Reasoning
Page 18 of 47
9. Using your normal distribution curve answer the following;
a. What percent of students have a height less than150cm?
i. a male
p(x £ 150)
z=
x-m
s
150 - 164.3
= -1.08
13.3
z = -1.08 (from table)
Percent = 14%
ii. a female
p(x £ 150)
z=
x-m
s
150 - 162.4
= -1.32
9.4
Percent = 9%
b. What percent of students have a foot size larger than 26 cm?
iii. a male
p ( x ³ 26)
x -m
z=
s
26 - 25.5
z=
= 0.19
2.57
1- 0.58 = 42%
Percent = 42%
iv. a female
p ( x ³ 26)
x -m
z=
s
26 - 24
z=
= 1.16
1.72
1- 0.8770 = 12%
Percent = 12%
c. In two or three sentences explain your findings regarding the per cents found in
questions (a) and (b).
Answers will vary.
z=
Mathematics 20-2
Statistical Reasoning
Page 19 of 47
10. Calculate the z-score for your foot size and height.
(Female sample)
Height: 160 cm
Shoe Size: 24.5 cm
160 - 162.4
Height : z =
= -0.26
9.4
24.5 - 24
ShoeSize : z =
1.72
11. What standard deviation for your foot length would be necessary so that you
would have the same z-score for your height (prove your answer algebraically)?
Adjusting Standard Deviation
x -m
s
24.5 - 24
-0.26 =
s
0.26s
0.5
=
-0.26 -0.26
s = -1.9 = 1.9
z=
Mathematics 20-2
Statistical Reasoning
Page 20 of 47
12. Of the four set of data collected which ones, if any, falls within the 68-95-99 Rule
for a normal distribution curve? Provide an explanation if necessary.
68-95-99 Rule:
Male Height
6
= 50%
12
12
±2s ® 137 - 190.9 =
= 100%
12
12
±3s ® 124.4 - 204.5 =
= 100%
12
±1s ® 151- 177.6 =
Female Height
10
 83%
12
12
2  143.6  181.2 
 100%
12
12
3  134.2  190.6 
 100%
12
1  153  171.8 
Male Foot Length
10
 83%
12
12
2  20.36  30.64 
 100%
12
12
3  17.76  33.21 
 100%
12
1  22.93  28.08 
Female Foot Length
10
= 83%
12
12
±2s ® 20.56 - 27.44 =
= 100%
12
12
±3s ® 18.84 - 29.16 =
= 100%
12
±1s ® 22.28 - 25.72 =
** None of the 4 distributions fit the 68-95-99 Rule
Mathematics 20-2
Statistical Reasoning
Page 21 of 47
13. Calculate the 95% confidence interval for each set of data and explain what it
means.
95% Confidence Interval
m ± 1.96s
Male Height :164.3 ± 1.96(13.3) Þ 138.0 to 190.4
Female Height :162.4 ± 1.96(9.4) Þ 144.0 to 180.8
Male Foot Length : 25.5 ± 1.96(2.57) Þ 20.5 to 30.5
Female Foot Length : 24 ± 1.96(1.72) Þ 20.6 to 27.4
14. What is the margin of error for each set of data? Explain what factors could
influence this.
Margin of Error
upper confidence level - mean = error
lower confidence limit - mean = error
Male Height:
190.4 - 164.3 = 26.1
Female Height:
180.8 - 162.4 = 18.4
138.2 - 164.3 = -26.1
144.0 - 162.4 = -18.4
Margin of Error: 164 ± 26 cm
Margin of Error: 162 ±18 cm
Male Foot Length:
30.5 - 25.5 = 5
Female Foot Length:
20.6 - 24 = 3.4
20.5 - 25.5 = -5
27.4 - 24 = -3.4
Margin of Error: 26 ± 5 cm
Margin of Error: 24 ± 3 cm
Mathematics 20-2
Statistical Reasoning
Page 22 of 47
STAGE 3
Learning Plans
Lesson 1
Standard Deviation
STAGE 1
BIG IDEA: Statistics summarizes data and predict future outcomes in areas such as advertisement,
sales, sports and academics. We can make logical assumptions and back them up with numerical data.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …
 There are different measures of central
tendency, which may be appropriate in
different situations.
 Data can be presented in a misleading fashion.
 Which measure of central tendency is better in
certain situations?
 When is the mean the best measure of
central tendency to use?
 When is the median the best measure of
central tendency to use?
 When is the mode the best measure of
central tendency to use?
 When is it better to use mean versus median?
 When is it appropriate to use a sample set
instead of an entire population?
 Why would people misrepresent data using
statistics?
 When is the size of the standard deviation
important?
KNOWLEDGE:
SKILLS:
Students will know …
 how to identify measures of central tendency
 what standard deviation is
 the characteristics of a normal distribution
Students will be able to …
 calculate the population standard deviation of a
data set.
Implementation note:
Each lesson is a conceptual unit and is not intended to
be taught on a one lesson per block basis. Each
represents a concept to be covered and can take
anywhere from part of a class to several classes to
complete.
Mathematics 20-2
Statistical Reasoning
Page 23 of 47
Lesson Summary


Students will understand what standard deviation means.
Students will be able to calculate the standard deviation of a given set of data.
Lesson Plan
Hook
Team A
Team B
Both of these teams have the same mean height.
 How are the two teams different?
 Which team would you rather have?
 Is there a way to mathematically describe the differences between these two
basketball teams?
Mathematics 20-2
Statistical Reasoning
Page 24 of 47
Lesson
We use a statistical measure called the standard deviation to help describe the
spread of data.
We want to calculate the standard deviation for the amount of gold coins each pirate
on a ship has. Source: http://standard-deviation.appspot.com/
There are 100 pirates on the ship. In statistical terms, this means we have a
population of 100. If we know the amount of gold coins each of the 100 pirates have,
we use the standard deviation equation for an entire population:
s=
å(x - x )
2
N
where,
𝜎 = the standard deviation
x = each value in the population
x = the mean of the values
N = the number of values (the population)
What if we don't know the amount of gold coins each of the 100 pirates have? For
example, we only had enough time to ask 5 pirates how many gold coins they have.
In statistical terms this means we have a sample size of 5 and in this case we use
the standard deviation equation for a sample of a population:
s=
å(x - x )
2
N -1
where,
s = the standard deviation
x = each value in the sample
x = the mean of the values
N = the number of values (the sample size)
The rest of this example will be done in the case where we have a sample size of 5
pirates; therefore we will be using the standard deviation equation for a sample of a
population.
Here are the amounts of gold coins the 5 pirates have:
4, 2, 5, 8, 6.
Mathematics 20-2
Statistical Reasoning
Page 25 of 47
Now, let's calculate the standard deviation:
1. Calculate the mean:
x=
åx
N
=
x1 + x 2 + ××× + xN
N
=
4+2+5+8+6
5
=5
2. Calculate x - x for each value in the sample:
x 1 -x = 4 – 5 = -1
x 2 -x = 2 – 5 = -3
x 3 -x = 5 – 5 = 0
x 4 -x = 8 – 5 = 3
x 5 -x = 6 – 5 = 1
𝑥1 − 𝑥̅ = 4 – 5 = -1
3. Calculate
å(x - x )
å(x - x ) = (x
2
1
-x
2
:
) + (x
2
2
-x
)
2
(
+ ××× + xN - x
)
2
= (-1)2 + (-3) 2 + 02 + 32 + 12
= 20
4. Calculate the standard deviation:
s=
s=
å(x - x )
2
N -1
20
5 -1
= 2.24
Mathematics 20-2
Statistical Reasoning
Page 26 of 47
The standard deviation for the amounts of gold coins the pirates have is 2.24 gold
coins.
Note to teachers: Show how to calculate standard deviation using the graphing
calculator if desired.
Practice problems:
1. Consider the following sets of data.
A = {9, 10, 11, 7, 13}
B = {10, 10, 10, 10, 10}
a.
b.
c.
d.
Calculate the mean of each data set.
Calculate the standard deviation of each data set.
Which set has the largest standard deviation?
Is it possible to answer question c. without calculations of the standard
deviation?
2. The frequency table of the monthly salaries of 20 people is shown below.
salary ($)
frequency
3500
5
4000
8
4200
5
4300
2
a. Calculate the mean of the salaries of the 20 people.
b. Calculate the standard deviation of the salaries of the 20 people.
Going Beyond
A given data set has a mean μ and a standard deviation σ.
1. What are the new values of the mean and the standard deviation if the same
constant k is added to each data value in the given set? Explain.
2. What are the new values of the mean and the standard deviation if each data
value of the set is multiplied by the same constant k? Explain.
Mathematics 20-2
Statistical Reasoning
Page 27 of 47
Resources
Principle of Mathematics - Nelson
Section 5.3 (pages 254 - 265)
Supporting
*Online practice questions:
http://www.regentsprep.org/Regents/math/algtrig/ATS2/NormalPrac.htm*
http://www.analyzemath.com/statistics/mean.html
Glossary
mean – The mean (or "arithmetic mean") is a measure of central tendency of a set of
data represented by numbers. Adding all the values and dividing by the number of
values calculates the mean. The symbol for the arithmetic mean is a letter with a
segment above it.
median - The median is a measure of central tendency of a set of data represented
by numbers. The median is the "middle" of a set of numbers in ascending or
descending order. The symbol for the median is usually the letter "M".
mode – The mode is a measure of central tendency of a set of data represented by
numbers. The mode is the most frequently occurring number. There is no standard
symbol associated with the mode.
standard deviation - A measure of the dispersion of a frequency distribution
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Mathematics 20-2
Statistical Reasoning
Page 28 of 47
Lesson 2
The Normal Curve
STAGE 1
BIG IDEA: Statistics summarizes data and predict future outcomes in areas such as advertisement,
sales, sports and academics. We can make logical assumptions and back them up with numerical data.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …
 Certain data is normally distributed (bell curve).
 When is it appropriate to use a sample set
instead of an entire population?
 When is the size of the standard deviation
important?
 Why are some sets of data normally distributed
and others are not?
KNOWLEDGE:
SKILLS:
Students will know …
 the characteristics of a normal distribution
 how standard deviation effects the curve and
area under the curve
Students will be able to …
 determine if a set approximates a normal curve
Lesson Summary


Students will recognise the properties of a normal curve.
Students will be able to determine whether data fits a normal distribution.
Mathematics 20-2
Statistical Reasoning
Page 29 of 47
Lesson Plan
Hook: The Bell Curve Student
Source: http://www.youtube.com/watch?v=PXSBsgBNUgo
Lesson
The properties of a normal distribution (bell curve):
1. The total area under the curve is equal to 1.
2. The normal curve extends infinitely to the left and right (i.e. does not actually reach
the horizontal axis).
3. The normal curve is symmetrical about the mean (i.e. 50% of area under the curve
is to the left of the mean and 50% is to the right).
4. The area under the curve represents all the data.
5. The mean, median and mode are the same value.
Mathematics 20-2
Statistical Reasoning
Page 30 of 47
Task 1:
 Have students order themselves from shortest to tallest. The middle person
will represent the mean height.
 How many students represent 68% of the class?
o Example: in a class of 26, 17 students would represent 68%. Those
students would represent one standard deviation above and below the
mean.
 Two standard deviations above and below the mean include 95% of the data.
How many students represent 95% of the class?
 Three standard deviations above and below the mean include 98% of the data.
How many students represent 98% of the class?
**Be creative! Wrap the students in "caution tape" to represent the different
standard deviations.
Probing Question: are the people included in the 68% also included in the 95%?
68-95-99 Rule:
 From the mean to one standard deviation above and below, 68.26% (~68%) of
the data will fall under the curve.
 From the mean to two standard deviations above and below, 95.44% (~95%) of
the data will fall under the curve.
 From the mean to three standard deviations above and below, 99.74% (~99%)
of the data will fall under the curve.
Mathematics 20-2
Statistical Reasoning
Page 31 of 47
Task 2:
Energizer wants to examine the data of the life of an AA Battery. The company tested
44 batteries to determine the mean life of the batteries as well as the standard
deviation. The lifetime, in hours, of the batteries tested is shown below.
899
1049
901.7
768.2
952
932
903
849
830.5
903
904
845
872
922
905
897.9
874
953.7
908
837.1
875
952.5
910
840
880
962.9
915
849.9
881
975
919.8
851
882.3
987
920
851.4
885
997.2
922
854.8
899
949
898
862.1
Calculate the mean and standard deviation of the data.
Complete the table:
Interval
<750
750-800 800-850 850-900 900-950
950-1000
1000-1050
>1050
# of
batteries
% of
batteries
in the
interval
Using the table, create a histogram to display the data.
If we were to place a curve on the histogram, the following shape would be obtained:
Label the interval & data values and the standard deviation away from the mean.
What happens to our curve if we change the standard deviation? The following
reference may help: http://www.analyzemath.com/statistics/graph_normal.html.
*As our standard deviation increases, what happens to the curve?
*As our standard deviation decreases, what happens to the curve?
Mathematics 20-2
Statistical Reasoning
Page 32 of 47
Going Beyond
Have students come up with their own survey question and plot the data in a
histogram? Does it form a normal curve? Why or why not?
Resources
Principles of Mathematics - Nelson
Section 5.4 (pages 266-268)
Supporting
http://www.learnalberta.ca/content/t4tet/courses/senior/amath30/lessons.html
http://www.learnalberta.ca/content/meda/html/areasunderthenormaldistributions/index.
html
http://www.learnalberta.ca/content/mea30i/APM30/statistics/statistics_desc.html
http://www.intmath.com/counting-probability/14-normal-probability-distribution.php
http://stattrek.com/Lesson2/Normal.aspx
Glossary
central tendency – Measures of central tendency are numbers that indicate the
center of a set of ordered numerical data. The three common measures of central
tendency are the mean, median and the mode.
histogram - A graph consisting of bars used to visually represent a frequency table
where at least one of the scales represents continuous data
normal distribution curve - A bell-shaped curve showing a particular distribution of
probability over the values of a random variable
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Mathematics 20-2
Statistical Reasoning
Page 33 of 47
Lesson 3
Z-scores
STAGE 1
BIG IDEA: Statistics summarizes data and predict future outcomes in areas such as
advertisement, sales, sports and academics. We can make logical assumptions and back them
up with numerical data.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …
 Certain data is normally distributed (bell
curve).

KNOWLEDGE:
SKILLS:
Students will know …
 the characteristics of a normal distribution
 how standard deviation effects the curve
and area under the curve
 what a z-score is and how it applies to the
normal distribution
Students will be able to …
 compare the properties of two or more
normally distributed data sets
 determine the z-score for a given value in
a normally distributed data set
 solve a contextual problem
When is the size of the standard deviation
important?
Lesson Summary




Students will calculate the mean and standard deviation of two given sets of
data.
Students will display their findings in a bell curve.
Students will calculate z-scores.
Students will solve a contextual problem.
Mathematics 20-2
Statistical Reasoning
Page 34 of 47
Lesson Plan
Hook
Are girls smarter then boys? Prove it. State your opinion and supporting evidence in
a few sentences.
Lesson Goal
Students will be able to determine and explain the z-score for a given value in a
normally distributed data set.
Students will be able to compare two sets of normally distributed data by converting to
z-scores.
Activate Prior Knowledge




bell curve properties
standard deviation calculation
mean of a sample set calculation
basic understanding of a normal distribution
Task:
The following marks are from a 20-2 math class. Calculate the mean and the
standard deviation.
Boys
72
70
60
55
55
53
54
64
75
92
Girls
68
80
60
62
65
60
50
81
71
53
Mean for boys = 65
Standard Deviation = 11.8
Mean for girls = 65
Standard Deviation = 9.8
Show your results on a bell curve.
Mathematics 20-2
Statistical Reasoning
Page 35 of 47
Solution:
Girls Data
Boys Data
Leading Questions:
 What is the standard deviation of both graphs?
 How do they compare?
 How does it affect the shape of the graph?
 Are there any extreme values (high vs. low)?
 Who is smarter? How can you tell?
Notes:
A z-score is a standard score that measures how many standard deviation units away
from the mean a particular value lays.
Converting values into z-scores allows for an easy comparison of two data sets.
1. Convert the boys graph into z-scores using the following z-score formula
z-score =
z=
raw value - mean
standard deviation
x-m
s
Solution:
76.8 - 65
=1
11.8
86.6 - 65
=2
z=
11.8
z=
z=
100.4 - 65
=3
11.8
Mathematics 20-2
53.2 - 65
= -1
11.8
41.4 - 65
= -2
z=
11.8
z=
z=
29.6 - 65
= -3
11.8
Statistical Reasoning
Page 36 of 47
Notes:
So, your bell curve for the boy’s turns into:
Using the mean and standard deviation from the boy’s graph, we can convert the girl’s
values in order to compare the two sets of data.
i.e. If the girls mark was 74.8% where would this value lie on the boys curve?
Girls Data
74.8 - 65
= 0.83
11.8
84.6 - 65
= 1.83
z=
11.8
z=
z=
94.4 - 65
= 2.49
11.8
Mathematics 20-2
Boys Data
55.2 - 65
= -0.83
11.8
45.4 - 65
= -1.83
z=
11.8
z=
z=
35.6 - 65
= -2.49
11.8
Statistical Reasoning
Page 37 of 47
When the two sets of z-scores are compared on one bell curve the following results:
2. Ray’s final exam marks are shown below, together with the class mean and
standard deviation for each subject. By calculating z-scores, determine in which
subject Ray performed best relative to the rest of his class. Display your findings
in a bell curve.
Subject
Math
English
Social
Ray’s Mark
74
79
68
Mean Mark Standard Deviation
68
12
73
14
66
11
Solution:
Convert to z scores
74 - 68
= 0.5
z(Math) =
12
79 - 73
= 0.43
z(English) =
14
z(Social) =
68 - 66
= 0.18
11
Mathematics 20-2
Statistical Reasoning
Page 38 of 47
3. Using the data from task 2, what percent of students in Ray’s class scored:
a. lower than him in Math (use tables)
b. lower than him in English (technology)
c. higher than him in Social (either)
Solution:
a. 69%
b. 67%
c. 43%
Going Beyond
Manipulate the z-score formula to find the value (x), the mean (µ) or the standard
deviation (σ).
Assessment
Practice Questions:
1. An average light bulb manufactured by the Acme Corporation lasts 300 days with
a standard deviation of 50 days. Assuming that bulb life is normally distributed,
what percent of Acme light bulbs will last at most 365 days?
2. Suppose scores on an IQ test are normally distributed. If the test has a mean of
100 and a standard deviation of 10, what percent of students will score between
90 and 110?
3. Find the area under the standard normal curve for the following, using the z-table.
Sketch each one.
a.
b.
c.
d.
e.
between z = 0 and z = 0.78
between z = -0.56 and z = 0
between z = -0.43 and z = 0.78
between z = 0.44 and z = 1.50
to the right of z = -1.33
4. It was found that the mean length of 100 parts produced by a lathe was 20.05 mm
with a standard deviation of 0.02 mm. What percent of parts selected will have a
length
a. between 20.03 mm and 20.08 mm
Mathematics 20-2
Statistical Reasoning
Page 39 of 47
b. between 20.06 mm and 20.07 mm
c. less than 20.01 mm
d. greater than 20.09 mm
5. A company pays its employees an average wage of $3.25 an hour with a standard
deviation of $0.60. If the wages are approximately normally distributed, determine
a. the proportion of the workers getting wages between $2.75 and $3.69 an hour
b. the minimum wage of the highest 5%
6. The average life of a certain type of motor is ten years, with a standard deviation of
two years. If the manufacturer is willing to replace only 3% of the motors that fail,
how long a guarantee should he offer? Assume that the lives of the motors follow a
normal distribution.
Resources
Principles of Mathematics - Nelson
Section 5.5 (pages 283 - 295)
Supporting
http://wise.cgu.edu/sdtmod/reviewz.asp
More Practice Questions:
http://www.webster.edu/~woolflm/zscores.html
Glossary
z-score - A standard score that measures how many standard deviation units away
from the mean a particular value lays
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Mathematics 20-2
Statistical Reasoning
Page 40 of 47
Lesson 4
Confidence Interval
STAGE 1
BIG IDEA: Statistics summarizes data and predict future outcomes in areas such as advertisement,
sales, sports and academics. We can make logical assumptions and back them up with numerical data.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …
 Data can be presented in a misleading fashion.
 Predictions based on statistics will contain
error.
 Why would people misrepresent data using
statistics?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 how confidence levels, margin of error and
confidence intervals may vary depending on
the size of the random sample
 the significance of a confidence interval,
margin of error or confidence level
 make inferences about a population from
sample data using given confidence intervals
Lesson Summary
Mathematics 20-2
Statistical Reasoning
Page 41 of 47



Students will review z-scores, median, standard deviation and 68-95-99 Rule
through warm up.
An example is provided for students, which demonstrates how a confidence
intervals is derived and used (using 95%C.I. =   1.96 )
Students will work on tasks that require them to use mean, standard deviation,
z-scores to determine confidence interval.
Lesson Plan
Lesson Goal
Students will determine the 95% confidence intervals for data that is given/collected.
Students will determine confidence intervals for data that is given/collected using the
formula: 95%C.I. = µ ± 1.96σ.
Activate Prior Knowledge


calculate standard deviation, mean and z-scores
68-95-99 Rule
Lesson
Warm up:
Recall the 68-95-99 Rule: (These values are approximate.)
68% of the data falls within ______ standard deviations of the mean.
95% of the data falls within ______ standard deviations of the mean.
99% of the data falls within ______ standard deviations of the mean.
Mathematics 20-2
Statistical Reasoning
Page 42 of 47
Investigation: The 68-95-99 Rule stated that, in a normal distribution, approximately
95% of the data lie within two standard deviations of the mean.
Now let’s find the precise z-scores that represent the SYMMETRIC INTERVAL that
contains 95% of the data.
1. Use your calculator to determine the area under the standard normal curve
between z = -2 and z = 2. What percent of data is in this range?
2. Now use your calculator to determine the area between z = -1.99 and z = 1.99.
What percent of data is in this range?
3. Repeat this process by deducting 0.01 from both the upper and lower bound.
Continue until you find a range of z-scores for which the area under the standard
normal curve is less than 0.95.
a. Which range of z-scores resulted in an area close to 0.95 or 95%?
b. Why is the area not exactly 95%? How could we find an area that is exactly
95%?
Example: A headline in a local newspaper reports that "70% of Residents Opposed to
Proposed Bylaw." The accompanying article states that the headline was based on an
opinion poll the city administration conducted. The article also states that the poll was
based on a random survey of 1000 residents and that the results are considered to be
"accurate to within three percentage points nineteen times out of twenty."
http://www.learnalberta.ca/content/t4tes/courses/senior/amath30/lessons/lesson015.html
For data that have a normal distribution with a mean µ and standard deviation σ,
a 95% confidence interval is:
µ ± 1.96σ
This is the range of values that lie within 1.96 standard deviations of the mean.
The percent of data that lies in that range is 0.95 or 95%.
Note: Round the lower bound down and round the upper bound up.
Task: A chocolate bar manufacturing company has found that the mean mass of its
chocolate bars is 56 grams and the standard deviation is 2 grams. Construct a 95%
confidence interval for the mass of chocolate bars, to the nearest whole number.
Mathematics 20-2
Statistical Reasoning
Page 43 of 47
Resources
Principles of Mathematics - Nelson
Section 5.6 (pages 295 - 301)
Assessment
1. Determine a 95% confidence interval for each set of information given below.
a) µ = 50, σ = 2
b) µ = 80, σ = 5
c) µ = 5.8, σ = 0.02
d) µ = 120, σ = 1
2. A sample of 250 trees in a logging area has a mean diameter of 52 cm, with a
standard deviation of 8.5 cm. Determine a 95% confidence interval for this data.
[35.34 – 68.66]
Glossary
confidence interval – The interval within which the value of a random variable is
estimated to lie with a stated degree of probability
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Mathematics 20-2
Statistical Reasoning
Page 44 of 47
Lesson 5
Confidence Intervals in Print and Media 4
STAGE 1
BIG IDEA: Statistics summarizes data and predict future outcomes in areas such as advertisement,
sales, sports and academics. We can make logical assumptions and back them up with numerical data.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …
 Data can be presented in a misleading fashion.
 Predictions based on statistics will contain
error.
 When is it appropriate to use a sample set
instead of an entire population?
 Why would people misrepresent data using
statistics?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …
 how confidence levels, margin of error and
confidence intervals may vary depending on
the size of the random sample
 the significance of a confidence interval,
margin of error or confidence level
 support a position by analyzing statistical data
presented in the media
 interpret and explain confidence intervals and
margin of error, using examples found in print
of electronic media
 make inferences about a population from
sample data using given confidence intervals
 provide examples from print or electronic
media in which confidence intervals are used
to support a particular position
Lesson Summary




Make inferences about a population from sample data, using given confidence
intervals, and explain the reasoning.
Provide examples from print or electronic media in which confidence intervals
and confidence levels are used to support a particular position.
Interpret and explain confidence intervals and margin of error, using examples
found in print or electronic media.
Support a position by analyzing statistical data presented in the media.
Mathematics 20-2
Statistical Reasoning
Page 45 of 47
Lesson Plan
Hook
I don't think he got the size of the circles right...
Lesson
Let's look at examples of statistics in the media: [Choose one or more to talk about as
a class]
http://sxxz.blogspot.com/2005/09/bush-poll-numbers-margin-of-error.html
http://www.cbc.ca/sports/football/story/2011/02/03/sp-nfl-schedule-fans.html
http://www.ctv.ca/CTVNews/Health/20101115/teens-drugs-101115/
http://www.edmontonjournal.com/business/Oilsands+companies+have+problem+surv
ey/1155444/story.html
http://www.edmontonjournal.com/Voters+worried+about+cost+higher+education+favo
ur+greater+government+investment+says+poll/4251335/story.html
http://www.globalnews.ca/story.html?id=4237642
http://news.bbc.co.uk/2/hi/uk_news/magazine/7605118.stm
Looking through the articles, as a class answer the following questions:
 How does the margin of error affect the data?
 What do the confidence intervals used in the data actually mean?
 Is this data skewed to favour a certain position? Were there enough people
sampled to represent the whole population?
 Overall, was this poll/survey done well?
 How could something be changed to make it better/worse?
Mathematics 20-2
Statistical Reasoning
Page 46 of 47
Going Beyond
Have students create their own misleading and skewed news article to "support" a
particular position. Their article should include raw data as well as confidence
intervals and/or margin of error.
Resources
Principles of Mathematics - Nelson
Section 5.6 (page 304)
Assessment
Have students find their own examples of confidence intervals and margin of error in
the media. Have them interpret and explain the data, then side with a certain position
and use the data they found to support their position.
Suggested resource(s) is/are:
 The Edmonton Journal Website
Glossary
margin of error - The proportion added to and subtracted from the result to construct
the confidence interval
Glossary hyperlinks redirect you to the Learn Alberta Mathematics Glossary
(http://www.learnalberta.ca/content/memg/index.html). Some terms can be found in more
than one division. Some terms have animations to illustrate meanings.
Mathematics 20-2
Statistical Reasoning
Page 47 of 47