The Normal Distribution

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The Normal Distribution

• Chapter 2

2
Continuous Random Variable

• Represented by a function/graph.
• Area under the curve represents the proportions
  of the observations
• Total area is exactly 1.

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Example 1
A
.4(1)
0.4
0.4
0.5
What percent of the observations lie below 0.4?
40
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Example 2
A
1.4(.5)
0.7
0.6
What proportion of the observations lie above 0.6?
Notice, to find proportion for observation above,
we can use the complement rule.
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Mean and Median

• The median of a continuous random variable is the
  equal-areas point, the point that divides the
  area under the curve in half.
• The mean of a continuous random variable is the
  balance point, at which the curve would balance
  if made of solid material.
• The median and mean are the same for a symmetric
  continuous random variable.

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Normal Distributions

• Symmetric, single-peaked, and mound-shaped
  distributions are called normal distributions
• Normal curves
• Mean median
• The mean m and standard deviation s completely
  determine the shape
• Fathom

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Normal Distribution
8

• Inflection points

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Place in Statistics

• Normal distributions are good descriptions for
  real data
• Normal distributions are good approximations to
  the results of many kinds of chance outcomes
• Much of statistical inference (in this course)
  procedures area based on normal distributions
• FYI many distributions arent normal

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68 of observations fall within 1s of m
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95 of observations fall within 2s of m
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99.7 of observations fall within 3s of m
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68-95-99.7 Rule
Applet
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68-95-99.7 Rule
34
34
13.5
13.5
2.35
2.35
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Ilianas Grade

• After 5 weeks of class Iliana must transfer from
  a stat class at Lanier to this class. Last week
  was the chapter 1 test in both classes. Iliana
  scored a 61 out of 70. Lets say our test was out
  of 100 points. What score should she be given?

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Ilianas claim

• Iliana claims that her test at Lanier was harder
  than our test.
• Does your previous method of assigning a grade
  take in consideration difficulty?
• If we have all of the data, what important facts
  can we utilize to improve our assignment of
  Ilianas grade?

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Important Facts

• Maximum possible on our test was 100 pts while
  Laniers test was 70 pts.
• Mean score on Laniers test was 50.5 pts while
  our test was 77.2 pts.
• Standard deviation on Laniers test was 8.8
  pts while ours was 9.7 pts.
• How will we fairly assign Ilianas score?

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Relative Position

• How can we find Ilianas relative position?

Ilianas score
class average
standard deviation
zscore
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1. Suppose as student has taken two quizzes in a
statistics course. On the first quiz the mean
score was 32, the standard deviation was 8, and
the student received a 44. The student obtained
a 28 on the second quiz, for which the mean was
23 and the standard deviation was 3. On which
quiz did the student perform better relative to
the rest of the class?
First quiz
Second quiz
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3. A married couple is employed by the same
company. The husband works in a department for
which the mean hourly rate is 12.80 and the
standard deviation is 1.20. His wife is
employed in a department where the mean rate is
13.50 and the standard deviation is 1.80.
Relative to their departments, which is better
paid if the husband earns 14.60 and the wife
earns 15.75?
Husband
Wife
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What percentile is the husband located in his
department?
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What percent of employees in the wifes
department earn better than her?
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What would the wife need to earn to match her
husbands relative position?
Husband
Wife
The wife would need to earn 16.20 to match the
husbands relative position.
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If the husband wanted to earn in the 95th
percentile, how much should he earn per hour?
Need a z-score of 1.65!
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The husband will need to earn at least 14.78 to
be in the 95th percentile.
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(No Transcript)
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89
44.5
z-score 1.60
z-score 1.60
The middle 89 of the data ranges from 18.81 to
55.03 ppb.
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• The Beanstalk Club is limited to women and men
  who are very tall. If womens heights have a
  mean of 63.6 inches and a standard deviation of
  2.5 inches. Find the cut off heights for the
  following chapters.
• The Travis County chapter wants to take women who
  are taller than 1.75 standard deviations above
  the mean.
• The Austin chapter wants to take only women in
  the top 8.
• The Central Texas chapter wants to take women in
  the 88th percentile.