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ELEMENTARY
Section 5-4
STATISTICS
Normal Distributions: Finding Values
EIGHTH
Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman EDITION
MARIO F. TRIOLA
1
Cautions to keep in mind
1. Don’t confuse z scores and areas.
Z scores are distances away from the mean
along the horizontal scale, but areas are
regions under the normal curve.
2. Use invNorm() to find a z-score for a given
percentage to the left(percentile).
3. A z score must be negative whenever it
is located to the left of the centerline of 0.
Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
2
Finding z Scores when Given Probabilities
95%
z = invNorm(.95)
0.95
0
1.64
(z score will be positive)
FIGURE 5-11
Finding the 95th Percentile
Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
3
Finding z Scores when Given Probabilities
80%
20%
Bottom 20%
0.80
0.20
-.84
0
z = invNorm(.20)
(z score will be negative)
Finding the top 80%
Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
4
Procedure for Finding Values (x)
1. Sketch a normal distribution curve, enter the given probability or
percentage in the appropriate region of the graph, and identify the
value(s) being sought.
2.
x
Use invNorm() to find the z score corresponding to the region bounded
by x. Cautions:
•
You must input the percentile (area below) into invNorm()
•
If the area(%) is above you must enter its complement into invNorm()
•
If the area(%) is the middle you must enter area(%) in the tail into
invNorm(). You will have two opposite z-scores to define the interval.
3. Enter the values for µ, , and the z score found in step 2, then solve for
x.
x = µ + (z • )
(z-score formula solved for x)
4. Refer to the sketch of the curve to verify that the solution makes sense in
the context of the graph and the context of the problem.
Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
5
Example
What weight denotes the 10th
percentile of women’s weight?
Assume women’s weights are
normally distributed with a mean of
143 pounds and standard deviation
of 29 pounds.
Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
6
Finding P10 for Weights of Women
10%
.10
x=?
143
Weight
FIGURE 5-17
Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
7
Finding P10 for Weights of Women
z = invNorm(.10) = -1.28
0.10
x=?
143
Weight
z
-1.28
0
Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
8
Finding P10 for Weights of Women
x = + (z• )
x = 143 + (-1.28 • 29) = 105.88
0.10
x=?
143
-1.28
0
Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
Weight
9
Finding P10 for Weights of Women
The weight of 106 lb (rounded) separates
the lowest 10% from the highest 90%.
0.10
x = 106
FIGURE 5-17
-1.28
143
Weight
0
Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
10
Forgot to make z score negative???
x = 143 + (1.28 • 29) = 180
0.10
x = 180
1.28
143
Weight
0
Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
11
Forgot to make z score negative???
UNREASONABLE ANSWER!
0.10
x = 180
1.28
143
Weight
0
Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
12
z
x
x
normalcdf(zleft, zright)

z
x = +z
%
invNormal(% to Left)
Chapter 5. Section 5-4. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman
13