#### Notes on Empirical Rule and Using the Z

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Notes on Empirical Rule and Using the Z

Using the Empirical
Rule
Remember: Normal Distributions
These are special density curves.
They have the same overall shape
Symmetric
Single-Peaked
Bell-Shaped
They are completely described by giving its
mean () and its standard deviation ().
We abbreviate it N(,)
Normal Curves….
•Changing the mean without changing the standard
deviation simply moves the curve horizontally.
•The Standard deviation controls the spread of a Normal
Curve.
Standard Deviation
It’s the natural measure of spread for Normal
distributions.
It can be located by eye on a Normal curve.
It’s
the point at which the curve changes from concave
down to concave up.
Empirical Rule (68-95-99.7 Rule)
In the Normal distribution with mean ()
and standard deviation ():
1 of ≈ 68% of the observations
Within 2 of ≈ 95% of the observations
Within 3 of ≈ 99.7% of the observations
Within
Scores on the Wechsler adult Intelligence Scale (a
standard IQ test) for the 20 to 34 age group are
approximately Normally distributed. N(110, 25).
What percent are between 85 and 135?
What percent are below 185?
What percent are below 100?
Standard Normal Distribution
It is the Normal distribution with mean 0 and
standard deviation 1.
If a variable x has any Normal distribution N(,
), then the standardized variable
z
x
has the standard Normal distribution.
A standard Normal table give the area
under the curve to the left of z. Find the
area to the left of z = 0.21
Using the chart
Find P(z<1.23)
Find P(z > 2.01)
More examples
Find P(z< -0.13)
Find P(z > -1.72)
More examples
Find P(-1.56 < z < 1.01)
Find P(-2.23 < z < -0.27)
Try the following:
P(z < 1.39)
P(z > -2.15)
P(-0.56 < z < 1.81)
Find the z-score that correlates with the
20th percentile.
For what z-score are 45% of all
observations greater than z?
Homework
Correction to Assignment Sheet
Page
131 (43-52)