Transcript Section 2.2 First Day Normal Curves, the

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Chapter 2: Modeling Distributions of Data
Section 2.2
Normal Distributions
The Practice of Statistics, 4th edition - For AP*
STARNES, YATES, MOORE
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The Empirical Rule
a.k.a. 68-95-99.7 Rule
 All
normal distributions follow this rule:
 68% of the observations are within one standard
deviation of the mean
 95% of the observations are within two standard
deviations of the mean
 99.7% of the observations are within three
standard deviations of the mean
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Yay, Math!
 IQ
scores on the WISC-IV are normally
distributed with a mean of 100 and a
standard deviation of 15.
up one σ and down one σ from 100
gives us the range from 85 to 115. 68% of
people have an IQ between 85 and 115.
 Going
 95%
of people have an IQ between ____
and ____.
 99.7%
of people have an IQ between ____
and ____.
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Try This
heights of women aged 18 – 24 is
approximately normally distributed with a
mean μ = 64.5 inches and a standard
deviation σ = 2.5 inches.
 The
 Between
what two heights does the middle 95%
fall?
 The tallest 2.5% of women are taller than what?
 What is the percentile for a woman who is 64.5
inches tall?
a)
Sketch the Normal density curve for this distribution.
b)
What percent of ITBS vocabulary scores are less than 3.74?
c)
What percent of the scores are between 5.29 and 9.94?
Normal Distributions
The distribution of Iowa Test of Basic Skills (ITBS) vocabulary
scores for 7th grade students in Gary, Indiana, is close to
Normal. Suppose the distribution is N(6.84, 1.55).
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Example, p. 113
All Normal distributions are the same if we measure in units
of size σ from the mean µ as center.
Definition:
The standard Normal distribution is the Normal distribution
with mean 0 and standard deviation 1.
If a variable x has any Normal distribution N(µ,σ) with mean µ
and standard deviation σ, then the standardized variable
z
x -
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has the standard Normal distribution, N(0,1).
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Normal Distributions

Standard Normal Distribution
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 The
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Standard Normal Table
Because all Normal distributions are the same when we
standardize, we can find areas under any Normal curve from
a single table.
Definition:
The Standard Normal Table
Table A is a table of areas under the standard Normal curve. The table
entry for each value z is the area under the curve to the left of z.
Suppose we want to find the
proportion of observations from the
standard Normal distribution that are
less than 0.81.
We can use Table A:
Z
.00
.01
.02
0.7
.7580
.7611
.7642
0.8
.7881
.7910
.7939
0.9
.8159
.8186
.8212
P(z < 0.81) = .7910
Normal Distributions
 The

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Example, p. 117
Finding Areas Under the Standard Normal Curve
Normal Distributions
Find the proportion of observations from the standard Normal distribution that
are between -1.25 and 0.81.
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Using Table A
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There are four types of questions I can ask with Table A.
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Less than: Find P(Z<0.42)
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Greater than: Find P(Z>0.42)
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Between: Find P(0.42 < Z < 1.3)
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Backwards: What Z-score marks the 15th percentile?
What Z-score has 62% of observations to the right?