Normal Curve Area Problems
involving only z (not x yet)
To accompany Hawkes lesson 6.2
A few slides are from Hawkes
Plus much original content by D.R.S.
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.2 Reading a Normal Curve Table
Probability of a Normal Curve:
The probability of a random variable having a value in
a given range is equal to the area under the curve in
that region.
The Key Idea behind
all of this is that
Probability IS Area !!!
Shaded area
Is 0.1587
(out of total
area 1.0000)
Probability is
0.1587, too!
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.2 Reading a Normal Curve Table
Probability of a Normal Curve:
The probability of a random variable having a value in
a given range is equal to the area under the curve in
that region.
This picture shows a normal
distribution with mean μ=75
and std deviation σ=5.
The probability that X > 80 is
the same as the area under the
curve to the right of x = 80.
This first simple basic batch of
problems about area and probability
• These are all z problems, not involving x
• They are all worked with the Standard Normal
Distribution curve
– Where mean μ is the standard normal’s mean = 0
– And std. dev. σ is the standard normal’s st.dev = 1
• Two ways to find the areas
1. With a printed table of values
2. With the TI-84 normalcdf( ) function
There are three basic problem types
1. “What is the area to the LEFT of z = _____ ?”
– This is the same as probability P(z < ___ )
2. “What is the area to the RIGHT of z = ____ ?”
– This is the same as probability P(z > ___ )
3. “What is the area BETWEEN z = __ and __?”
– This is the same as probability P( ___ < z < ___ )
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.2 Reading a Normal Curve Table
Standard Normal Distribution Table:
Standard Normal Distribution Table from – to positive z
z
0.00
0.01
0.02
0.03
0.04
0.0
0.5000
0.5040
0.5080
0.5120
0.5160
0.1
0.5398
0.5438
0.5478
0.5517
0.5557
0.2
0.5793
0.5832
0.5871
0.5910
0.5948
0.3
0.6179
0.6217
0.6255
0.6293
0.6331
0.4
0.6554
0.6591
0.6628
0.6664
0.6700
0.5
0.6915
0.6950
0.6985
0.7019
0.7054
0.6
0.7257
0.7291
0.7324
0.7357
0.7389
0.7
0.7580
0.7611
0.7642
0.7673
0.7704
0.8
0.7881
0.7910
0.7939
0.7967
0.7995
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.2 Reading a Normal Curve Table
Standard Normal Distribution Table (continued):
1. The standard normal tables reflect a z-value that is
rounded to two decimal places.
2. The first decimal place of the z-value is listed
down the left-hand column.
3. The second decimal place is listed across the top
row.
4. Where the appropriate row and column intersect,
we find the amount of area under the standard
normal curve to the left of that particular z-value.
When calculating the area under the curve, round your
answers to four decimal places.
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.2 Reading a Normal Curve Table
Area to the Left of z:
“Area to left”: P(z < 1.69); P(z < -2.03)
With the printed table
For P(z < 1.69)
• You should know to expect
something > 0.5000
• Look down to row 1.6
• Look across to column 0.09
For P(z < -2.03)
• You should expect < .5000
• Look down to row -2.0
• Look across to column 0.03
With the TI-84
“Area to the left of z=0”: P(z < 0)
• You should know
instantly that it’s .5000
because of
– Total area = 1.00000000
– Symmetry
• But just confirm it with
table and TI-84 for now
• Note insignificant
rounding error in TI-84
Area to the left of z = 4.2, z = -4.2
• Very very little area way
out in the extremities of
the tails
• Almost 100% to the left
of z = 4.2
• Almost 0% to the left of
z = -4.2
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.2 Reading a Normal Curve Table
Area to the Right of z:
Finding area to the right of some z
Probability P(z > ___ )
With the printed table
1. Find the area to the LEFT of
that z value
2. Subtract 1.0000 total area
minus area to the left
equals area to the right
With the TI-84
• It’s just normalcdf again
• Your z value is the low z
• Except this time it’s positive
infinity for the high z
Find area to right of z = 3.02; z=-1.70
With the printed table
• Lookup area to the left of z
= 3.02 is _____
• So area to the right of
z = 3.02 is
1.0000 - _____ = _____
• Lookup area to the left of z
= -1.70 is _____
• So area to the right of
z = -1.70 is
1.0000 - _____ = _____
With the TI-84
Find area to the right of z = 0, z = 5.1
• P(z > 0) should be instantly known as 0.5000
• P(z > 5.1) should be instantly known as ≈0
• How about area to right of z = -5.1 ?
HAWKES LEARNING SYSTEMS
Continuous Random Variables
math courseware specialists
6.2 Reading a Normal Curve Table
Area Between z1 and z2:
Area between z = 1.16 and z = 2.31
With the printed table
• Area to the left of the
higher z, ______
• Minus area to the left of the
lower z, ______
• Equals the area between
the two z values, ______
With the TI-84
Area between z = -2.76 and z = 0.31
With the printed table
• Area to the left of the
higher z, ______
• Minus area to the left of the
lower z, ______
• Equals the area between
the two z values, ______
With the TI-84
Area between z = -3.01 and z = -1.33
With the printed table
• Area to the left of the
higher z, ______
• Minus area to the left of the
lower z, ______
• Equals the area between
the two z values, ______
With the TI-84
Area in two tails,
outside of z=1.25 and z = 2.31
With the printed tables
• 1.0000 minus area between
the two z values
• Or another way, area to left
of lower z + area to right of
higher z
With the TI-84
Special: z = -1 and z = +1
• Agrees with The Empirical Rule value of ____%
• So the area in the two tails is ____ %
• And the area in each is tail is ____%
Special: z = -2 and z = +2
• Agrees with The Empirical Rule value of ____%
• And the area in the two tails is _____ %
• Therefore ____ % in each tail.
Special: z = -3 and z = +3
• Agrees with The Empirical Rule value of ____%
• And the area in the two tails is _____ %
• Therefore ____ % in each tail.