Download Using the Empirical Rule

11/30/2014
Using the Empirical
Rule
Remember: Normal Distributions
These are special density curves.
They have the same overall shape
Symmetric
Single-Peaked
Bell-Shaped
The mean describes where the
curve is centered.
The st. dev. describes how much
the curve spreads out around that
center.
They are completely described by giving its
mean (µ) and its standard deviation (σ).
We abbreviate it N(µ,σ)
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Normal Curves….
When a st. dev. is small, a larger area is concentrated
near the center of the curve and the chance of observing
a value near the mean is much greater.
•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.
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Why is the Normal Curve Important?
They are good descriptions for some real data
such as
Test
scores like SAT, IQ
Repeated careful measurements of the same quantity
Characteristics of biological populations (height)
They are good approximations to the results of
many kinds of chance outcomes
They are used in many statistical inference
procedures.
Empirical Rule (68-95-99.7 Rule)
In the Normal distribution with mean (µ)
and standard deviation (σ):
Within
1σ of µ ≈ 68% of the observations
Within 2σ of µ ≈ 95% of the observations
Within 3σ of µ ≈ 99.7% of the observations
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The distribution of batting average (proportion of hits) for the 432
Major League Baseball players with at least 100 plate appearances
in the 2009 season is normally distributed defined N(0.261, 0.034).
Sketch a Normal density curve for this distribution of batting
averages. Label the points that are 1, 2, and 3 standard
deviations from the mean.
What percent of the batting averages are above 0.329?
What percent are between 0.227 and .295?
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 60?
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Standard Normal Distribution
It is the Normal distribution with mean 0 and
standard deviation 1.
If a variable x has any Normal distribution N(µ,
σ), then the standardized variable
z=
x−µ
σ
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
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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)
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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)
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Find the z-score that correlates with the
20th percentile.
For what z-score are 45% of all
observations greater than z?
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Homework
Page 131 (43-52)
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