1. Introduction to the Normal Curve
Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc.
All rights reserved.
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Section 6.1
Introduction to the Normal Curve

2. A normal curve is symmetric and bell-shaped.
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Continuous Random Variables
6.1 Introduction to the Normal Curve
Normal Distribution:
A continuous probability distribution for a given random variable, X, that is completely defined by it’s mean and standard deviation.
Properties of a Normal Distribution:
A normal curve is symmetric and bell-shaped.
A normal curve is completely defined by its mean, , and standard deviation, .
The total area under a normal curve equals 1.
The x-axis is a horizontal asymptote for a normal curve.

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Continuous Random Variables
6.1 Introduction to the Normal Curve
Symmetric and Bell-Shaped:

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Continuous Random Variables
6.1 Introduction to the Normal Curve
Completely Defined by its Mean and Standard Deviation:
An inflection point is a point on the curve where the curvature of the line changes. The inflection points are located at  - and  + .

5. 0.84 HAWKES LEARNING SYSTEMS math courseware specialists
Continuous Random Variables
6.1 Introduction to the Normal Curve
Total Area Under the Curve = 1:
0.84
16% %
σ=1

6. HAWKES LEARNING SYSTEMS
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Continuous Random Variables
6.1 Introduction to the Normal Curve
The x-Axis is a Horizontal Asymptote:

7. Birth weights of 75 babies. Normal Ages of 250 students in 10th grade.
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Continuous Random Variables
6.1 Introduction to the Normal Curve
Determine if the following is a normal distribution:
Birth weights of 75 babies.
Normal
Ages of 250 students in 10th grade.
No, this would be uniform
Heights of 100 adult males.
Frequency of outcomes from rolling a die.
No, because the data is discrete
Weights of 50 fully grown tigers.

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Continuous Random Variables
6.1 Introduction to the Normal Curve
How Many Normal Curves are there?
Because there are an infinite number of possibilities for  and , there are an infinite number of normal curves.

9. The standard normal curve is symmetric and bell-shaped.
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Continuous Random Variables
6.1 Introduction to the Normal Curve
Standard Normal Distribution:
A standard normal distribution has the same properties as the normal distribution; in addition, it has a mean of 0 and a standard deviation of 1.
Properties of a Standard Normal Distribution:
The standard normal curve is symmetric and bell-shaped.
It is completely defined by its mean and standard deviation,  = 0 and  = 1.
The total area under a standard normal curve equals 1.
The x-axis is a horizontal asymptote for a standard normal curve.

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Continuous Random Variables
6.1 Introduction to the Normal Curve
Converting to the Standard Normal Curve:
Standard Score Formula (z-score):
When calculating the z-score, round your answers to two decimal places.

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Continuous Random Variables
6.1 Introduction to the Normal Curve
Draw a Normal Curve:
Given  = 40 and  = 5, indicate the mean, each of the inflections points, and where each given value of x will appear on the curve.
x1 = 33 and x2 = 51
Solution: 40
68% 35 45 33 51

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Continuous Random Variables
6.1 Introduction to the Normal Curve
Convert to the Standard Normal Curve:
Given  = 40 and  = 5, calculate the standard score for each x value and indicate where each would appear on the standard normal curve.
x1 = 33 and x2 = 51
Solution: -1 1
-1.4
2.2

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Continuous Random Variables
6.1 Introduction to the Normal Curve
Convert to the Standard Normal Curve:
Given  = 48 and  = 5, convert to a normal curve and indicate where a score of x = 45 would appear on each standard normal curve.
Solution: 48 43 -1 53 1 45
-0.6

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