Transcript Ch. 4

Chapter 4
The Normal Curve
Copyright © 2012 by Nelson Education Limited.
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In this presentation
you will learn about:
• The Normal Curve
• Z scores
• The use of the Normal Curve table
(Appendix A)
• Finding areas above and below Z
scores
• Finding probabilities
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•
•
•
•
•
Bell Shaped
Unimodal
Symmetrical
Unskewed
Mode, Median,
and Mean are
same value
Frequency
Theoretical Normal
Curve
Scores
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Theoretical Normal Curve: Specific Areas
• Distances on horizontal axis always cut off the
same area. We can use this property to describe
areas above or below any point.
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Using the Normal Curve:
Z Scores
• To find areas,
• first compute a Z score. The formula for computing a Z
score is *
This formula changes a “raw” score (Xi ) to a standard
deviation or Z score.
• second, use Appendix A to find the area above or
below a Z score.
*Converting original scores in a population is done using the
same method.
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Using the Normal Curve:
Appendix A
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Using the Normal Curve:
Appendix A (continued)
• Appendix A has three columns.
– (a) = Z score
– (b) = areas between the mean and the Z score
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Using the Normal Curve:
Appendix A (continued)
– ( c) = areas beyond the Z score
c
c
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Using Appendix A to
Describe Areas Under
the Normal Curve
•The normal curve table can be used to find the:
area between a Z score and the mean. (Section 5.3)
2. area either above or below a Z score (5.4) *
3. area between two Z scores (5.5)
4. probability of randomly selected score (5.6) *
1.
* Only these are demonstrated in this presentation
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4-9
How to Find Area Above or
Below a (Positive) Z Score
• Find your Z score in
column (a).
• To find area below
a positive score:
– Add column (b) area
to 0.50.
• To find area above
a positive score
– Look in column (c).
(a)
(b)
(c)
.
.
.
1.66
0.4515
0.0485
1.67
0.4525
0.0475
1.68
0.4535
0.0465
.
.
.
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How to Find Area Below a
(Positive) Z Score: An Example
• A person has a height of 73 inches in a
distribution of height where, X = 68 inches and
s = 3 inches.
• The person’s score as a Z score is:
73  68
Z
 1.67
3
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How to Find Area Below a (Positive) Z
Score: An Example (continued)
• To find the area below a positive Z score, we consult the
normal curve table (Appendix A) to find the area between the
score and the mean (column b): 0.4525.
Normal curve with Z=+1.67
Then we add this area to
the area below the mean:
0.5000, or
0.4525 + 0.5000 = 0.9525.
• Areas can be expressed
as percentages: 95.25%.
The area below a Z score
of +1.67 is 95.25%. A person with a height of 73 inches is taller
than 95.25% of all persons.
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4-12
How to Find Area Above or
Below a (Negative) Z Score
• Find your Z score in
column (a).
• To find area below
a negative score:
– Look in column (c).
• To find area above
a negative score
– Add column (b) area
to 0.50
(a)
(b)
(c)
.
.
.
1.66 0.4515 0.0485
1.67 0.4525 0.0475
1.68 0.4535 0.0465
.
.
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How to Find Area Below a
(Negative) Z Score: An Example
• On the other hand, the Z score for a person with a
height of 63 is: -1.67.
Normal curve with Z=-1.67
• To find the area
below a negative
score we use column
c in Appendix A:
the area below a
Z score of -1.67 is
0.0475, or 4.75%. This person is taller than 4.75% of
all persons. Copyright © 2012 by Nelson Education Limited.
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Summary: Finding an Area Above
or Below a Z Score
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4-15
Finding Probabilities
• Areas under the curve can also be expressed
as probabilities.
• Probabilities are proportions and range from
0.00 to 1.00.
• The higher the value, the greater the
probability (the more likely the event).
• Probability is essential for understanding
inferential statistics in Part II of text.
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Finding Probabilities: An Example
If a distribution has: X =13 and s = 4,
what is the probability of randomly selecting a
score of 19 or more?
(a)
(b)
(c)
1.
2.
3.
Use the formula for
computing a Z score:
For Xi = 19, Z = 1.50
Find area above in
column (c).
Probability is 0.0668 of
randomly selecting a
score of 19 or more.
.
.
.
1.49
0.4319 0.0681
1.50
0.4332 0.0668
1.51
0.4345 0.0655
.
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