Download Introduction to Normal Introduction to Normal Distributions and the

Introduction to Normal Introduction
to Normal
Distributions and the Standard Normal Distribution
Properties of a Normal Distribution
Properties of a Normal Distribution
• A
A normal distribution is a continuous normal distribution is a continuous
probability for a random variable x. • The mean, median and mode are equal
The mean median and mode are equal
• The normal curve is a bell shaped and is symmetric about the mean
i b
h
• The total area under the curve is equal to one
• The normal curve approaches but never touches the x‐axis
Normal Curve
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0
μ‐σ
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μ+σ
μ
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A
0
B
Normal Curve
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μ=3.5, σ=1.5
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0
C
Normal Curve
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μ=3.5, σ=0.7
Curve A and Curve B have the same mean,
C
Curve B and Curve C have the same standard deviation,
B dC
Ch
th
t d d d i ti
Each Curve has a total area of 1.
0
Normal Curve
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μ=1.5, σ=0.7
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A
B
Normal Curve
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Normal Curve
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• Which normal curve has a greater mean?
• Curve A C
A
• Which normal curve has a greater standard deviation?
• Curve B
The Standard Normal Distribution
The Standard Normal Distribution
Standard Normal Curve
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The Standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.
Properties of the Standard Normal Distribution
b
• The
The cumulative area is close to 0 for z
cumulative area is close to 0 for z‐scores
scores close to z = ‐3.49
• The cumulative area increases as the z‐scores The cumulative area increases as the z scores
increase
• The cumulative are for z=0 is 0.500
Th
l i
f
0 i 0 500
• The cumulative are is close to 1 for z‐scores close to z = 3.49
Using the Standard Normal Table
Using the Standard Normal Table
Find the cumulative area that corresponds to a z‐score
corresponds to a z
score of 1.15
of 1.15
Find the cumulative area that corresponds to a z‐score
corresponds to a z
score of of ‐0.24
0.24
Standard Normal Curve
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.8749
8749
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Standard Normal Curve
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.4052
4052
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Finding Area Under the Standard Normal Curve
l
Find the area under the standard normal curve to the left of z = ‐0.99, then draw and shade the area under the curve.
From the standard normal table the area is equal to 0.1611
Standard Normal Curve
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Finding Area Under the Standard Normal Curve
l
Find the area under the standard normal curve to the left of z = 2.13, then draw and shade the area under the curve.
From the standard normal table the area is equal to 0.9834
Standard Normal Curve
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Finding Area Under the Standard Normal Curve
Normal Curve
Find the area under the standard normal curve to the right of z = 1.06, then draw and shade the area under the curve.
From the standard normal table the area is equal to 0.8554, so to the right it is 1 ‐ 0.8554 = 0.1446
Standard Normal Curve
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Finding Area Under the Standard Normal Curve
Normal Curve
Find the area under the standard normal curve to the right of z = ‐2.16, then draw and shade the area under the curve.
From the standard normal table the area is equal to 0.0154, so to the right it is 1 ‐ 0.0154 = 0.9846
Standard Normal Curve
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Finding Area Under the Standard Normal Curve
Normal Curve
Find the area under the standard normal curve between z = ‐1.5 and z = 1.25, then draw and shade the area under the curve.
From the standard normal table the area’s are 0.0668 and 0.8944, so the are between is 0.8944 – 0.0668 = 0.8276
Standard Normal Curve
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Finding Area Under the Standard Normal Curve
Normal Curve
Find the area under the standard normal curve between z = ‐2.16 and z = ‐1.35, then draw and shade the area under the curve.
From the standard normal table the area’s are 0.0154 and 0.0885, so the are between is 0.0885 – 0.0154 = 0.0731
Standard Normal Curve
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