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Normal Distribution
Def'n: NORMAL DISTRIBUTION is one type of
distribution of data, which describes how data
clusters around the mean.
The graph of a Normal Distribution is bell-shaped
and symmetric, with the peak at the mean.
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The EMPIRICAL RULE (also known as the 68 - 95 - 99.7
rule) applies to all Normal Distributions. It dictates the
following:
- 68% of all observations fall within one standard
deviation of the mean
- 95% of all observations fall within two standard
deviations of the mean
- 99.7% of all observations fall within three standard
deviations of the mean
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The Normal Distribution Curve:
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Ex #1
A normal distribution has a mean of 27 and a standard
deviation of 5. Find the probability that a randomly
selected x-value from the distribution is in the interval
of 17 and 37.
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Ex #2
A normal distribution has a mean of x and a standard
deviation of σ. Find the probability that a randomly
selected x-value from the distribution is in the interval
x < x - 2 σ.
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Ex #3
The distribution of heights of young women aged 18 to 24 is
approximately normal with a mean ( x ) of 64.5 inches and a
standard deviation ( σ ) of 2.5 inches.
a) Draw a Normal Distribution curve to represent this data,
clearly showing the application of the empirical rule.
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b) What percent of woman are taller than 69.5 inches?
c) Between what heights do the middle 95% of women
fall?
d) What percent of woman are shorter than 62 inches?
e) A height of 67 inches corresponds to what percentile
of adult female American heights?
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Ex #4
Scores for a professional exam are normally distributed
with a mean ( x ) of 650 and a st dev (σ) of 100.
a) Draw a Normal Distribution curve for the problem.
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b) What is the probability that a randomly selected test
score is between 450 and 850?
c) Out of 1000 randomly selected test scores, how many
would you expect to be between 450 and 850?
d) Out of 2300 randomly selected test scores, how many
would you expect to be between 650 and 950?
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Def'n: An OUTLIER is any value that is more than
2 standard deviations away from the mean
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Ex #5
Jesse is the manager of a guitar shop. He recorded the
number of guitars sold each week for a period of 10 weeks.
His data is shown in the following table:
week
# sold
1
12
2
15
3
20
4
8
5
15
6
18
7
17
8
21
9
10
10
24
a) What is the mean of Jesse's data?
b) What is the standard deviation of Jesse's data?
c) Which, if any, values in Jesse's data would be
considered an outlier?
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HOMEWORK
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