Exercise: z-scores
n  Assume
you have a normal distribution.
Use the z-score table in Appendix A to
answer:
¨ 1)
What percent of observations lie below a zscore of 0?
¨ 2)
What percent of observations lie below a zscore of 1.72?
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Exercise: z-scores
¨ 3)
What percent of observations fall
BETWEEN z-scores of 0 and 1.72?
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5.2 Properties of the
Normal Distribution
part 3
n  Connecting
z-scores to probabilities.
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n  Example:
The Stanford-Binet IQ test is
normally distributed and scaled so that
scores have a mean of 100 and a standard
deviation of 16.
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n  Example:
The Stanford-Binet IQ test is
normally distributed and scaled so that
scores have a mean of 100 and a standard
deviation of 16.
¨ If
you draw someone at random, what is the
probability that they have an IQ score of 90 or
less?
¨ To
answer this, we just need to know what
percent of IQ scores are at 90 or lower.
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n  Example:
The Stanford-Binet IQ test is
normally distributed with a mean of 100
and standard deviation of 16.
Let X be an IQ score of a person.
Short-hand notation:
The 2 parameters needed to
define a normal distribution.
X ~ N(μ=100,σ=16)
Is distributed
Normal
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¨ If
you draw someone at random, what is the
probability that they have an IQ score of 90 or
less?
¨ We
need to answer:
When X ~ N(μ=100,σ=16), P(X ≤ 90) = ?
X is a data value (or IQ score in this case).
We will convert to a z-score…
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z = standard score =
¨ P(X
data value – mean
standard deviation
≤ 90) = P( X –μ ≤ 90 –100
σ
16
)
= P(z ≤ – 10/16)
= P(z ≤ – 0.63)
= 0.2643
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z = standard score =
¨ P(X
data value – mean
standard deviation
≤ 90) = P(z ≤ – 0.63) = 0.2643
Looked up
on z-table
An IQ score of 90 has a z-score of - 0.63
¨ The
probability of randomly drawing someone
with an IQ score of 90 or lower is 0.2643.
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QUICK-CHECK:
The Empirical Rule tells
me the percent that is
below an IQ of 90 has
to be between16% (to
the left of 1σ below the
mean) and 50% (to the
left of the mean itself).
IQ 90
So, 26.43% is totally in-line with my
Empirical Rule information because
being 0.63 standard deviations is
between 1 and 0 standard
deviations down from the mean.
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Exercise 1:
n  Let
X ~ N(μ=40,σ=5).
Find P(X < 51):
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Exercise 2:
n  Suppose
bowling scores are normally
distributed with a mean of 186 and a
standard deviation of 30. Find the
percentage of games with a score of 120
or HIGHER.
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