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DG5 --- 20 minutes
Accel Math III
Unit #1: Data Analysis
Lesson #8: Normal Distribution
EQ: What are the characteristics of a
normal distribution and how is
probability calculated using this type of
distribution?
Recall: Three Types of Distributions
Binomial
Normal
Geometric
Normal Distributions --- created from
continuous random variables
Characteristics of a Normal Distribution:
1. Symmetric, Bell-Shaped Curve and
Uni-modal.
2. Mean, Median, Mode are equal and located
at the middle of the distribution. Symmetric
about the mean. Not skew.
3. The curve is continuous, no gaps or holes.
The curve never touches or crosses the
x-axis.
4.The total area under the curve equals 1.
Recall: Empirical Rule
Normal Distribution --- each has its own
mean and standard deviation.
What are µ and σ in this normal distribution ?
50
10
Standard Normal Distribution --- mean is
always 0 and standard deviation is always 1
STANDARDIZE
z-score --- the number of standard
deviations above or below the mean
z = observed – mean
or
standard deviation
Correlates to area
under the curve.
z=X-µ
σ
Ex. In a study of bone brittleness, the ages of
people at the onset of osteoporosis followed a
normal distribution with a mean age of 71 and a
standard deviation of 2.8 years. What z-score
would an age of 65 represent in this study?
65  71
z
 2.14
2.8
| ||
|
|
|
|
62.6 65.4 68.2 71 73.8 76.6 79.4
65
| ||
|
-3
-1
-2
-2.14
0
|
|
|
1
2
3
Using Table A to Finding the Area under
A Standard Normal Curve
Ex. Find the area under the curve
to the left of z = -2.18.
P( z  2.18)  0.0146  1.46%
|
-2.18
Ex. Find the area under the curve to the
left of z = 1.35.
P( z  1.35)  0.9115  91.15%
|
1.35
Ex. Find the area under the curve
to the right of z = 0.75.
P( z  0.75)  1  0.7734  0.2266  22.66%
WHY??
|
.75
Ex Find the area under the curve between
z = -1.36 and z = 0.42.
Ex. Find the area under the curve
between z = 1.60 and z = 3.3.
In Class Practice Worksheet:
Area Under the Standard
Normal Curve
#1 - 11
 What about finding a z-score when
given area under the curve?
Ex. Determine the z-score that would give
this area under the curve.
Ex. Determine the z-score that would give
this area under the curve.
Ex. Determine the z-score that would give
this area under the curve.
In Class Practice Worksheet:
Area Under the Standard
Normal Curve
#12
Practice Worksheet Calculating Area
Using z-scores
Quiz 2 --- 30 minutes
Using the Graphing Calculator with Normal
Distributions
Command and Arguments:
I. When given a z-score, you are looking
for area under the curve.
normalcdf(low bound, high bound)
Ex. Find the area under the curve to the
left of z = 1.35.
Ex Find the area under the curve between
z = -1.36 and z = 0.42.
Assignment:
Example Handout: Area Under the
Standard Normal Curve #1 – 11
Go back and rework these using the
calculator function.
II. When given area, you’re looking
for a z-score.
Use invnorm function under
distributions.
invnorm(% to the left)
***Remember invnorm and normcdf
are calculator jargon
Do not write these functions on
assessments as “work”.
Assignment: Redo WS Calculating Area
Using z-scores