The Normal Distribution
Date:
Name:
The Normal Distribution
YOU WILL NEED • calculator • ruler
Keep in Mind
A set of data that is normally distributed has a symmetric bell shape when graphed as a frequency polygon. Generally, measurements of living things (such as mass, height, and lifespan) have this type of distribution. The properties of a normal distribution are • The mean, median, and mode are close to equal and fall at the vertical line of symmetry • The amounts of data within one, two, and three standard deviations of the mean are as shown in the graph.
:235%i i.—2
2.35% ..
+1,
+3y
• ‘The area under the curve can be considered as equal to 1, since it represents 100% of the data.
Example The ages of the grandparents of Elaine’s classmates were collected. 68 60 56 70 74 71 65 75 61 83 62 82 74 85 72 75 81 71 64 71 90 78 72 73 76 80 71 70 69 74 69 65 a) Does the data approximate a normal curve? Sketch a graph to help you explain.
—__________
b) Hal is a new student in Elaine’s class. What is the likelihood that his grandparent is at least 80 years old? Solution a) I checked the “fit” with a normal curve:
Step 1. I calculated the mean and standard deviation for the ages. mean:
=
72.1 years
standard deviation:
122
5.q
a-
=
7.4 years
The Nomia! Distribution
Copyright © 2012 by Nelson Education Ltd.
( ;
Name:
Datci
Step 2. I created a frequency distribution table so that I could draw a frequency polygon. tervaIyears).
Ii?lidpo (yeq)
1 F recncy
55—60
57.5 62.5
1
60—65 65—70 70—75 75—80 80—85 85—90 90—95
F
TIP For age data, include (e.g.) 55 through 59 in the interval 55—60, since “59” could mean 59 years and 364 days. This gives a midpoint of 57.5.
4
67.5 72.5 77.5 82.5 87.5 92.5
5 12 4 4 1 1
Step 3. I plotted the midpoint of each interval and joined the points for my frequency polygon. The curve looks like a normal distribution. Step 4. I checked the percents of data within one and two standard deviations of the mean. Since the percents are close to 68% and 95% for the two intervals
around the mean, the data has a normal distribution. Iange
Aang
p—lcrtop.+lcr
64.7to79.5
—2otoi.+2o
57.3to86.9
Ages of Grandparents
12 ‘1o
z cr6 1
L14
2/-
Percent of Data
60
80
Age (years)
h) 80 years old or more corresponds exactly to being at least one standard deviation above the mean. Since the data is normal, the probability that Hal’s grandparent is at least 80 years old is about
(1 00%
—
68%), or 16%.
Practice
1. Ihe ages of members of a seniors’ lawn bowling club are normally distributed. -The mean is 65 years and the standard deviation is 3 years. What percent of the bowlers is in each of the following age groups? a) between 59 and 65 years old, inclusive
b) between 68 and 74 years old, inclusive
Copyright © 2012 by Nelson Education Ltd.
5)1 The Normal Distribution
123
Date:
Name:
c) older than 74 years
( 2. A teacher is analyzing these class results for three history tests. The results of all three tests were normally distributed.
‘
Standard
‘1 Test
a) Describe and compare the normal curves for tests 1 and 2.
i)
1
85
2 3
85 75
i 2.5 4.9 23
b) Compare the normal curves for tests 1 and 3.
c) Determine Jasmine’s marks on each test, given the information. Round to the nearest percent.
a Test
Jasmine’s Mark
Jasmine’s Mark (%)
1 2
—1.5a’
3
p.+3.5r
(
3. Is the data in each set normally distributed? Use a frequency polygon to help you explain. 11—15
16—20
21—25
26—30
31—35
36—40
4
6
15
18
9
4
b)
(. } 124
5.q
The Normal Distribution
Copyright © 2012 by Nelson Education Ltd.
Name: c)
Date: Interval Frequency
1—6
7—12
13—18
19—24
25—30
31—36
1
8
5
8
4
4
dpçHnt -j
NUMERICAL RESPONSE
4. Reggie recorded the points he scored playing basketball during the year. 22 32
24 12
23 11
10 10
0 31
0 10
3 32
36 29
28 14
0 21
15 22
28 30
a) Determine the mean and standard deviation of the set of data.
b) Complete the frequency table.
c) Create a frequency polygon, using the grid provided. d) Are Reggie’s basketball scores normally distributed? Explain your answer.
5. A manufacturer offers a warranty on its toasters. The toaster has a mean lifespan of 6.5 years, with a standard deviation of 0.5 years. How long should the toasters be covered by the warrantç if the manufacturer wants to repair no more than 2.5% of the toasters sold? Length of warranty:
years
Copyright @ 2012 by Ne’son Education Ltd.
5.1
The Normal D!stnbution
125
Name:
The Normal Distribution
YOU WILL NEED • calculator • ruler
Keep in Mind
A set of data that is normally distributed has a symmetric bell shape when graphed as a frequency polygon. Generally, measurements of living things (such as mass, height, and lifespan) have this type of distribution. The properties of a normal distribution are • The mean, median, and mode are close to equal and fall at the vertical line of symmetry • The amounts of data within one, two, and three standard deviations of the mean are as shown in the graph.
:235%i i.—2
2.35% ..
+1,
+3y
• ‘The area under the curve can be considered as equal to 1, since it represents 100% of the data.
Example The ages of the grandparents of Elaine’s classmates were collected. 68 60 56 70 74 71 65 75 61 83 62 82 74 85 72 75 81 71 64 71 90 78 72 73 76 80 71 70 69 74 69 65 a) Does the data approximate a normal curve? Sketch a graph to help you explain.
—__________
b) Hal is a new student in Elaine’s class. What is the likelihood that his grandparent is at least 80 years old? Solution a) I checked the “fit” with a normal curve:
Step 1. I calculated the mean and standard deviation for the ages. mean:
=
72.1 years
standard deviation:
122
5.q
a-
=
7.4 years
The Nomia! Distribution
Copyright © 2012 by Nelson Education Ltd.
( ;
Name:
Datci
Step 2. I created a frequency distribution table so that I could draw a frequency polygon. tervaIyears).
Ii?lidpo (yeq)
1 F recncy
55—60
57.5 62.5
1
60—65 65—70 70—75 75—80 80—85 85—90 90—95
F
TIP For age data, include (e.g.) 55 through 59 in the interval 55—60, since “59” could mean 59 years and 364 days. This gives a midpoint of 57.5.
4
67.5 72.5 77.5 82.5 87.5 92.5
5 12 4 4 1 1
Step 3. I plotted the midpoint of each interval and joined the points for my frequency polygon. The curve looks like a normal distribution. Step 4. I checked the percents of data within one and two standard deviations of the mean. Since the percents are close to 68% and 95% for the two intervals
around the mean, the data has a normal distribution. Iange
Aang
p—lcrtop.+lcr
64.7to79.5
—2otoi.+2o
57.3to86.9
Ages of Grandparents
12 ‘1o
z cr6 1
L14
2/-
Percent of Data
60
80
Age (years)
h) 80 years old or more corresponds exactly to being at least one standard deviation above the mean. Since the data is normal, the probability that Hal’s grandparent is at least 80 years old is about
(1 00%
—
68%), or 16%.
Practice
1. Ihe ages of members of a seniors’ lawn bowling club are normally distributed. -The mean is 65 years and the standard deviation is 3 years. What percent of the bowlers is in each of the following age groups? a) between 59 and 65 years old, inclusive
b) between 68 and 74 years old, inclusive
Copyright © 2012 by Nelson Education Ltd.
5)1 The Normal Distribution
123
Date:
Name:
c) older than 74 years
( 2. A teacher is analyzing these class results for three history tests. The results of all three tests were normally distributed.
‘
Standard
‘1 Test
a) Describe and compare the normal curves for tests 1 and 2.
i)
1
85
2 3
85 75
i 2.5 4.9 23
b) Compare the normal curves for tests 1 and 3.
c) Determine Jasmine’s marks on each test, given the information. Round to the nearest percent.
a Test
Jasmine’s Mark
Jasmine’s Mark (%)
1 2
—1.5a’
3
p.+3.5r
(
3. Is the data in each set normally distributed? Use a frequency polygon to help you explain. 11—15
16—20
21—25
26—30
31—35
36—40
4
6
15
18
9
4
b)
(. } 124
5.q
The Normal Distribution
Copyright © 2012 by Nelson Education Ltd.
Name: c)
Date: Interval Frequency
1—6
7—12
13—18
19—24
25—30
31—36
1
8
5
8
4
4
dpçHnt -j
NUMERICAL RESPONSE
4. Reggie recorded the points he scored playing basketball during the year. 22 32
24 12
23 11
10 10
0 31
0 10
3 32
36 29
28 14
0 21
15 22
28 30
a) Determine the mean and standard deviation of the set of data.
b) Complete the frequency table.
c) Create a frequency polygon, using the grid provided. d) Are Reggie’s basketball scores normally distributed? Explain your answer.
5. A manufacturer offers a warranty on its toasters. The toaster has a mean lifespan of 6.5 years, with a standard deviation of 0.5 years. How long should the toasters be covered by the warrantç if the manufacturer wants to repair no more than 2.5% of the toasters sold? Length of warranty:
years
Copyright @ 2012 by Ne’son Education Ltd.
5.1
The Normal D!stnbution
125
