Transcript Normal Distributions

Using the Empirical
Rule
Normal Distributions

These are special density curves.
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They have the same overall shape
 Symmetric
 Single-Peaked
 Bell-Shaped
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They are completely described by giving its
mean () and its standard deviation ().
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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
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It’s the natural measure of spread for Normal
distributions.
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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.
Why is the Normal Curve Important?
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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
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They are used in many statistical inference
procedures.
Empirical Rule
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
What percent do you think……
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Empirical Rule (68-95-99.7 Rule)
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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
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.
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What percent of the batting averages are above 0.329?
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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).
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What percent are between 85 and 135?
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What percent are below 185?
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What percent are below 60?
Standard Normal Distribution

It is the Normal distribution with mean 0 and
standard deviation 1.
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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
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Find P(z<1.23)
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Find P(z > 2.01)
More examples
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Find P(z< -0.13)
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Find P(z > -1.72)
More examples
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Find P(-1.56 < z < 1.01)
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Find P(-2.23 < z < -0.27)
Find the z-score that correlates with the
20th percentile.
Normal Distribution
Calculations
When Tiger Woods hits his driver, the distance the ball travels
follows a Normal Distribution with mean 304 yards and standard
deviation 8 yards. What percent of Tiger’s drives travel at least
290 yards?
Let x = the distance Tiger’s ball travels and it has a
distribution N(304, 8). We want P(x  290).
290  304 

P( x  290)  P z 
  P z  1.75  1  0.0401  0.9599
5
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
About 96% of Tigers drives travel at least 290 yards.
Scores for a test have a mean of 100 and standard
deviation of 15. Find the probability that a score is
below 112.
Every month, American households generate an
average of 28 pounds of newspaper for garbage or
recycling. Assume =2 pounds. If a household is
selected at random, find the probability that it
generates between 27 and 31 pounds per month.
An exclusive college desires to accept only the top 10% of all
graduating seniors based on the results of a national placement
test. This test has a mean of 500 and a standard deviation of 100.
Find the cutoff score for the exam.
For a medical study a researcher wishes to select people in the
middle 60% of the population based on blood pressure. If the
mean systolic blood pressure is 120 and the standard deviation is
8, find the upper and lower reading that would qualify a person to
be in the study.
Homework
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