Normal Distributions Sections 3.3, 3.4, 3.5, 3.6 Lecture 9 Robb T. Koether

Normal Distributions
Sections 3.3, 3.4, 3.5, 3.6
Lecture 9
Robb T. Koether
Hampden-Sydney College
Thu, Jan 28, 2016
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
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Outline
1
The Normal Density Curve
2
Examples
3
The 68-95-99.7 Rule
4
z-Scores
5
Assignment
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
2 / 19
Outline
1
The Normal Density Curve
2
Examples
3
The 68-95-99.7 Rule
4
z-Scores
5
Assignment
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
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The Normal Density Curve
Definition (The Normal Density Curve)
A normal density curve has a very specific shape.
It is symmetric.
It has a single, central peak.
The curve drops steadily to the left and right of the peak.
The curve extends forever in both directions.
The “main part” of the curves lies between 3 standard deviations
below the mean and 3 standard deviations above the mean.
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
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Outline
1
The Normal Density Curve
2
Examples
3
The 68-95-99.7 Rule
4
z-Scores
5
Assignment
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
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Tossing a Coin
Suppose a coin is tossed 10,000 times and the number of heads
is counted.
What is the distribution of the number of heads?
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
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Tossing a Coin
Suppose a coin is tossed 10,000 times and the number of heads
is counted.
What is the distribution of the number of heads?
It is normal with mean µ = 5, 000 and σ = 50.
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
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Tossing a Coin
Suppose a coin is tossed 10,000 times and the number of heads
is counted.
What is the distribution of the number of heads?
It is normal with mean µ = 5, 000 and σ = 50.
Sketch the shape of that distribution, including the scale.
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
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The Normal Density Curve
0.008
0.006
0.004
0.002
4900
4950
5000
5050
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
5100
5150
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Rolling a Die
Suppose a die is rolled 720 times and the number of sixes is
counted.
What is the distribution of the number of sixes?
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
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Rolling a Die
Suppose a die is rolled 720 times and the number of sixes is
counted.
What is the distribution of the number of sixes?
It is normal with mean µ = 120 and σ = 10.
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
8 / 19
Rolling a Die
Suppose a die is rolled 720 times and the number of sixes is
counted.
What is the distribution of the number of sixes?
It is normal with mean µ = 120 and σ = 10.
Sketch the shape of that distribution, including the scale.
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
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The Normal Density Curve
0.04
0.03
0.02
0.01
100
110
120
130
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
140
150
Thu, Jan 28, 2016
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IQ Scores
IQ scores have an approximately normal distribution with µ = 100
and σ = 15.
Sketch the shape of that distribution, including the scale.
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
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The Normal Density Curve
0.025
0.020
0.015
0.010
0.005
80
100
120
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
140
Thu, Jan 28, 2016
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Outline
1
The Normal Density Curve
2
Examples
3
The 68-95-99.7 Rule
4
z-Scores
5
Assignment
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
12 / 19
The 68-95-99.7 Rule
The 68-95-99.7 Rule
The 68-95-99.7 Rule says that
Approximately 68% of the observations fall within σ of µ. That is,
between µ − σ and µ + σ.
Approximately 95% of the observations fall within 2σ of µ. That is,
between µ − 2σ and µ + 2σ.
Approximately 99.7% of the observations fall within 3σ of µ. That
is, between µ − 3σ and µ + 3σ.
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
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The 68-95-99.7 Rule
Apply this rule to the coin-tossing, die-rolling, and IQ examples.
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
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The 68-95-99.7 Rule
What proportion of the observations lie
Between the µ and µ + σ?
Between the µ and µ + 2σ?
Between the µ + σ and µ + 2σ?
Greater than µ + σ?
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
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Outline
1
The Normal Density Curve
2
Examples
3
The 68-95-99.7 Rule
4
z-Scores
5
Assignment
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
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z-Scores
Definition (z-Score)
If x is an observation from a distribution that has mean µ and standard
deviation σ, then the standardized value, or z-score, of x is
z=
x −µ
.
σ
The z-score is a measure of the number of standard deviations
the observation is above or below average.
z-scores greater than 2 or less than −2 are rare. (How rare?)
z-scores greater than 3 or less than −3 are very rare. (How rare?)
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
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Outline
1
The Normal Density Curve
2
Examples
3
The 68-95-99.7 Rule
4
z-Scores
5
Assignment
Robb T. Koether (Hampden-Sydney College)Normal DistributionsSections 3.3, 3.4, 3.5, 3.6
Thu, Jan 28, 2016
18 / 19
Assignment
Assignment
Read Sections 3.3 - 3.6.
Apply Your Knowledge: 5, 6, 7, 8, 9.
Check Your Skills: 16, 17, 18, 21.
Exercises: 26, 27.
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Thu, Jan 28, 2016
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