11-3 Use Normal
Distributions
Algebra II
The Normal Distribution
(Bell Curve)
Average contents 50
Mean = μ = 50
Standard deviation = σ = 5
REMEMBER your box plot
Middle 50%
LQ
UQ
range
Normal Distribution

The normal distribution is a
theoretical probability
the area under the curve adds up to one
The normal distribution is a
A Normal distribution is a theoretical model of the
wholetheoretical
population. It isprobability
perfectly symmetrical about
the central value; the mean μ represented by zero.
the area under the curve adds up to one
The X axis is divided up into deviations from the
As well
as the
theshaded
meanarea
theisstandard
mean.
Below
one deviation
deviation
(σ) must also be known.
from
the mean.
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
Two standard deviations
from the mean
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
Three standard deviations
from the mean
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-5
-4
-3
-2
-1
0
1
2
3
4
5
Fill in % for each section

0.15%/2.35%/13.5%/34%/34%/13.5
%/2.35%/0.15%
A handy estimate – known as the
Imperial Rule for a set of normal data:
68% of data will fall within 1σ of the μ
P( -1
<
z
<
1
)
=
0.683
=
0
1
2
68.3%
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-5
-4
-3
-2
-1
3
4
5
95% of data fits within 2σ of the μ
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-5
-4
P( -2 <
-3
-2
z
<
-1
2
0
)
1
=
2
0.954
3
=
4
95.4%
5
99.7% of data fits within 3σ of the μ
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-5
-4
-3
-2
P( -3 < z < 3
-1
0
)
=
1
2
0.997
3
=
4
5
99.7%
Ex. 1 ) guided practice






A.) P(x ≤ x¯)
B.) P(x ≥ x¯)
C.) P(x¯ ≤ x ≤ x¯ + 2Ơ)
D.) P ( x¯-Ơ ≤ x ≤ x¯ )
E.) P ( x ≤ x¯ -3Ơ )
F.) P( x ≥ x¯ + Ơ)
Interpret normally distributed
data (when given mean and
standard deviation)



Ex. 2) The score on an exam for entrance to
a firefighter program are normally
distributed with a mean of 200 points and
standard deviation of 20 points.
A.) About what percent of the candidates
score lower than 160 points?
B.) Candidates with scores above 220 are
admitted into the training program. About
what percent of the candidates are accepted
into the program?
11-3B Z-score

Ex. 1 A P(x ≤ 90)

Scientists conducted aerial surveys of
a seal sanctuary & recorded the
number of x seals they observed
during each survey. The number of
seals observed were normally
distributed with a mean of 73 seals &
a standard deviation of 14.1 seals.
Find the probability that at most 90
seals were observed during a survey.
Ex. 1B

If Ex. 1 said at least 90 or P(x≥90),
then 1 - _____ (decimal)
Ex. 2 Given mean = 64
standard dev. = 7

Find P(50 ≤ x ≤ 70)
Simple problems solved using the imperial
rule - firstly, make a table out of the rule
<-3
0%
-3 to - -2 to 2
1
2%
14%
-1 to
0
0 to 1
1 to 2
2 to 3
>3
34%
34%
14%
2%
0%
The heights of students at a
college were found to follow
a bell-shaped distribution
with μ of 165cm and σ of 8
cm.
What proportion of students
are smaller than 157 cm
first standardis e
x

16%
z
157  165
first 157cm is
 1
8
or 1 below the 
Simple problems solved using the imperial
rule - firstly, make a table out of the rule
<-3
0%
-3 to - -2 to 2
1
2%
14%
-1 to
0
0 to 1
1 to 2
2 to 3
>3
34%
34%
14%
2%
0%
The heights of students at a
college were found to follow
a bell-shaped distribution
with μ of 165cm and σ of 8
cm.
Above roughly what height
are the tallest 2% of the
students?
The tallest 2% of students are beyond 2 of 
165 + 2 x 8 = 181 cm
Task – class 10 minutes
finish for homework

Exercise A
Page 76
Assignment

The Bell shape curve happens
so when recording continuous
random variables that an
equation is used to model the
shape exactly.
1  12 x2
y
e
2
Put it into your
calculator and use the
graph function.
Sometimes you will see
it using phi =.
1  12 z 2
 ( z) 
e
2
Luckily you don’t have to use the equation each time and you
don’t have to integrate it every time you need to work out the
area under the curve – the normal distribution probability
There are normal distribution tables
How to read the Normal distribution
table
Φ(z) means the area under the curve
on the left of z
How to read the Normal distribution
table
Φ(0.24) means the area under the
curve on the left of 0.24 and is this
value here:
Values of Φ(z)

Φ(-1.5)=1- Φ(1.5)
Values of Φ(z)


Φ(0.8)=0.78814 (this is for the left)
Area = 1-0.78814 = 0.21186
Values of Φ(z)


Φ(1.5)=0.93319
Φ(-1.00)
=1- Φ(1.00)
=1-0.84134
=0.15866

Shaded area =
Φ(1.5)- Φ(-1.00)
= 0.93319 - 0.15866
= 0.77453
Task

Exercise B page 79
Solving Problems using
the tables
NORMAL DISTRIBUTION
 The area under the curve is the probability of getting less than the z
score. The total area is 1.
 The tables give the probability for z-scores in the distribution
X~N(0,1), that is mean =0, s.d. = 1.
ALWAYS SKETCH A DIAGRAM
 Read the question carefully and shade the area you want to find. If
the shaded area is more than half then you can read the probability
directly from the table, if it is less than half, then you need to
subtract it from 1.
NB If your z-score is negative then you would look up the positive from
the table. The rule for the shaded area is the same as above: more
than half – read from the table, less than half subtract the reading
from 1.
You will have to standardise if the
mean is not zero and the standard
deviation is not one
Task

Exercise C page 168
Normal distribution
problems in reverse


Percentage points table on page 155
Work through examples on page 84
and do questions Exercise D on page
85
Key chapter points



The probability distribution of a continuous random
variable is represented by a curve. The area under
the curve in a given interval gives the probability of
the value lying in that interval.
If a variable X follows a normal probability
distribution, with mean μ and standard deviation σ,
we write X ̴ N (μ, σ2)
The variable Z=
is called the standard
normal variable corresponding to X
Key chapter points cont.


If Z is a continuous random variable
such that Z ̴ N (0, 1) then
Φ(z)=P(Z<z)
The percentage points table shows, for
probability p, the value of z such that
P(Z<z)=p