CABT SHS Statistics & Probability - The Standard Normal Distribution

CABT Statistics & Probability – Grade 11 Lecture Presentation

The Normal
Distribution
A Grade 11
Statistics & Probability
Lecture
3

The Normal Distribution
The NORMAL DISTRIBUTION is a
continuous, symmetric, bell-shaped
distribution of a random variable. The
graph of this distribution is called a
NORMAL CURVE.
a normal curve

The equation of the theoretical normal
distribution is given by the formula
 
 2
2
2
1
2


 



x
f x e
where  is the mean of the distribution,  is
the standard deviation, and e and  are
irrational constants (e = 2.718… and  =
3.1415….)

Properties of the Normal
Distribution
1. The distribution curve is
bell-shaped.
2. The curve is symmetric
about its center, the mean.
3. The mean, the median,
and the mode coincide at
the center.
4. The width of the curve is
determined by the
standard deviation of the
distribution.CABT Statistics & Probability – Grade 11 Lecture Presentation

Properties of the Normal
Distribution
5. The tails of the curve flatten
out indefinitely along the
horizontal axis, always
approaching the axis but
never touching it. That is,
the curve is asymptotic to
the base line.
6. The area under the curve is
1. Thus, it represents the
probability or proportion or
the percentage associated
with specific sets of
asymptotic to the x-axis

The Distribution of Area
Under the Normal Curve
- a.k.a. the empirical rule or the
“68% - 95% - 99.7%” rule
The area under the part of a normal curve that
lies:
• within 1 standard deviation of the mean is
approximately 0.68, or 68%;
• within 2 standard deviations, about 0.95, or
95%
within 3 standard deviations, about 0.997,

The Distribution of Area
Under the Normal Curve

The Standard Normal
Distribution
The STANDARD NORMAL
DISTRIBUTION of a random variable
is a normal distribution with mean  =
0 and standard deviation  = 1.
The letter Z is used to denote the
standard normal random variable. The
specific value z of the r.v. Z is called
the z-score.

The Standard Normal
Distribution
The probability function of a random
variable Z with a standard normal
distribution by is given by
 
2
2
1
2

 
z
y p z e


The Standard Normal
Distribution
The graph of the standard normal
distribution
is shown below:

Areas Under the Standard
Normal Curve
The Table of Areas under the Normal Curve
is also known as the z-Table.
The z-score is a measure of relative standing.
It is calculated by subtracting the mean from
the measurement X and then dividing the
result by the standard deviation.
The final result, the z-score, represents the
distance between a given measurement X and
the mean, expressed in standard deviations.

The standard normal distribution table to be
used in this course gives areas under the
standard normal curve for the variable Z
ranging from 0 to a positive number z.
Normal Curve

In the table, the area A
gives the
PROBABILITY that
the value of Z lies
between 0 and a
constant z0; i.e.
 00A P Z z  
Normal Curve

Four-Step Process in Finding the Areas Under
the Normal Curve Given a z-Value
Step 1. Express the given z-value into a three-digit
form.
Step 2. Using the z-Table, find the first two digits on
the left column.
Step 3. Match the third digit with the appropriate
column on the right.
Step 4. Read the area (or probability) at the
intersection of the row and the column. This is
Normal Curve

Find the area that corresponds to
z = 1.
The area is A = 0.3413
Normal Curve

z = 2.58.
Normal Curve

FACT: The area between 0 and a positive
value z is the same as the area between z
and 0.
To find the area between z and 0, use the
value in the table corresponding to positive z.
Both regions have the same
area.CABT Statistics & Probability – Grade 11 Lecture Presentation
Normal Curve

The area that corresponds to
z = 2.58 is the same as the area
that corresponds to z = 2.58,
which is
A = 0.4951.
Normal Curve

Probabilities and Areas Under
the Standard Normal Curve
z = 1.15.

What is the probability that the value
of a standard normal random variable
Z lies between
a. 0 and 1.28?
b. 2.07 and 0?
0 1.28
A = 0.3997
02.0
7A = 0.4808
Normal Curve

Finding areas of other regions
tails - right of a positive z
or left of a negative z
between two z values
with the same sign
between two z values
with opposite signs
cumulative - left of a positive z
or right of a negative z
Normal Curve

Finding areas of other regions
CASE REGION INVOLVED ILLUSTRATION
1
tails - right of a positive z
or left of a negative z
2
between two z values with
the same sign
3
between two z values with
opposite signs
4
cumulative - left of a
positive z or right of a
negative z
Normal Curve

Probabilities and Areas Under the
Standard Normal Curve
To find the area at any
tail:
• Look up the z score to
get the area.
• Subtract the area from
0.5.
CASE 1: Finding areas of region in the TAILS

CASE 1: Finding areas of region in the TAILS
To the RIGHT of or GREATER THAN +z:
  00.5A P Z z A   
To the LEFT of or LESS THAN z:
   0.5     A P Z z P Z z
  00.5A P Z z A    
Let A0 be the area between 0 and +z (value in
table)

Find the area under the standard
normal curve for z greater than 2.
2
Look for the value in the
table corresponding to z =
2:
0 0.4772A 
Subtract the table value from
0.5 to find the area.
0.5 0.4772 0.0228A   
Normal Curve

2
NOTE: The area under the standard
normal curve for z greater than 2 (or to the
right of 2) is the same as the area for z
less than 2 (or to the left of 2).
The table value at z = 2 is
0 0.4772A 
The area corresponding to z < 2 is
0.5 0.4772 0.0228A   
Normal Curve

Find the area to the left of z = 1.5.
1.5
Look for the value in the table
corresponding to z = 1.5:
0 0.4332A 
Subtract the table value from
0.5 to find the area.
0.5 0.4332A   0.0668

Between two z scores on
the same side of the mean
(or with the SAME SIGN):
• Look up both z scores to
get the areas.
• Subtract the smaller
area from the larger
area.
CASE 2: Finding areas between two values
of z with the SAME SIGN

of z with the SAME SIGN
Let z1 = number nearer zero
z2 = number farther from zero
If A1 = area corresponding to z1
A2 = area corresponding to z2
2 1A A A 
The area of the region between z1 and
z2 is

Find the area of the region
between
z = 1.23 and z = 2.57Let z1 = 1.23 and z2 =
2.57.
1.23 2.57
From the table:
For z1 : A1 = 0.3907
For z2 : A2 = 0.4911
The area of the region is
2 1A A A  0.4911 0.3907 
0.1104
Normal Curve

Note: The area of the region between
z = 1.23 and z = 2.57 is EQUAL to the
area between z = 2.57 and z =  1.23.
Let z1 = 1.23 and z2 =  2.57. From the table:
-1.23-2.57
For z1 : A1 = 0.3907
For z2 : A2 = 0.4911
2 1A A A  0.1104

between
z = 0.96 and z = 0.36.Let z1 = 0.36 and z2 =  0.96. From the table:
1.23 2.57
For z1 (use z = 0.36): A1 =
0.1406
For z2 (use z = 0.96): A2 =
0.3395
2 1A A A  0.3395 0.1406 
0.1909

Between two z scores on
DIFFERENT sideS of the
mean (or with DIFFERENT
SIGNS):
• Look up both z scores to
get the areas.
• Add the two areas.
of z with DIFFERENT SIGNS

of z with DIFFERENT SIGNS
Let z1 = negative z value
z2 = positive z value
If A1 = area corresponding to z1
A2 = area corresponding to z2
1 2A A A 
The area of the region between z1 and
z2 is
Normal Curve

between
z = 2.46 and z = 1.55.Let z1 = 2.46 and z2 = 1.55. From the table:
-2.46 1.55
For z1 (use z = 2.46): A1 =
0.4931
For z2 (use z = 1.55): A2 =
0.4394
1 2A A A  0.4931 0.4394 
0.9325CABT Statistics & Probability – Grade 11 Lecture Presentation

To find the area to the left of
any positive z score or to
the right of a negative z
score:
CASE 4: Finding areas of regions to the
LEFT of a positive z or to the RIGHT of a
negative z
• Look up the z score to get
the area.
• Add 0.5 to the area.

Find the area of the region to the
left of z = 2.37.
From the table, the area corresponding to z =
2.37 is
00.5A A 
0.5 0.4911  0.9911
0 0.4911A 
2.37

Find the area of the region to the
right of z = 2.37.
From the table, the area corresponding to z =
2.37 is the same for z = 2.37:
00.5A A 
0.5 0.4911 
0.9911
0 0.4911A 
2.37
Normal Curve

Check your
understanding
Determine the area of the indicated
region under the standard normal
curve.1. to the left of z = 1.31
2. to the right of z = 1.92
3. to the left of z = 2
4. between z = 1.23 and z = 1.9
5. between z = 1.98 and z = 1.46
6. between z = 3 and z = 1.5
0.9049
0.9726
0.0228
0.0806
0.0482
0.9319

Recall that the area under the graph of a
continuous probability function corresponds to
the value of a probability in an interval.
PROBABILITY CORRESPONDING AREA
P(Z > a) to the right of a
P(Z < a) to the left of a
P( a < Z < b) between a and b
NOTE: The area won’t change even if “>” and
“<” are replaced by “” and “”, respectively.

If Z is a standard normal random
variable, what is the probability
that
a. 0 < Z < 0.33?
b. Z > 2?
c. Z < 1.67?
d. 1.03 < Z < 0.99?
Direct table value
Case 1
Case 4
Case 2
Case 3

If Z is a standard normal random
variable, what is the probability
that
a. 0 < Z < 0.33?
b. Z > 2?
c. Z < 1.67?
d. 1.03 < Z < 0.99?
A = 0.1293
A = 0. 5 –
0.4772
=
0.0228A = 0. 5 +
0.4525
=
0.9525A = 0. 3485 – 0.3389=
0.0096A = 0. 4986 + 0.0793=
0.5779

Check your
understanding
Given the random variable Z with a
standard normal distribution,
determine the following probabilities:
1. P(– 0.75 < Z < 0)
2. P(Z > 1.92)
3. P(Z < 1.11)
4. P(0.33 < Z < 0.99)
5. P(Z > 0.2)
0.2734
0.5 + 0.4726 = 0.9726
0.5 + 0.3665 = 0.8665
0.3389  0.1293 = 0.2096
0.5  0.0793 = 0.4207

CABT SHS Statistics & Probability - The Standard Normal Distribution

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CABT SHS Statistics & Probability - The Standard Normal Distribution