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Nonstandard Normal Nonstandard Normal Distributions:Distributions:
Finding ProbabilitiesFinding ProbabilitiesSection 5-3 Section 5-3
M A R I O F. T R I O L ACopyright © 1998, Triola, Elementary Statistics
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0 1 z x Need to “standardize” these nonstandard distributionsWill use z-score formula
Nonstandard Normal Distributions
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0 1 z x Need to “standardize” these nonstandard distributionsWill use z-score formula
x – µz =
Nonstandard Normal Distributions
Formula 5-2
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Converting from Nonstandard to
Standard Normal Distribution
x 0
Figure 5-13
z
x – z =
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Probability of Height between 63.6 in. and 68.6 in.
63.6 68.6
z
0 2.00
z = 68.6 – 63.62.5
= 2.00
=2.5 = 63.6
Figure 5-14
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Nonstandard Normal Nonstandard Normal Distributions:Distributions:Finding ScoresFinding Scores
Section 5-4Section 5-4
M A R I O F. T R I O L ACopyright © 1998, Triola, Elementary Statistics
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Review of 5-2Standard normal distribution
finding z-scores when given the probability
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Finding z Scores When Given Probabilities
FIGURE 5-11 Finding the 95th Percentile
0
5% or 0.05
z
0.450.50
95% 5%
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FIGURE 5-12 Finding the 10th Percentile
Finding z Scores When Given Probabilities
Bottom 10%
10% 90%
0.400.10z 0
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Finding Scoreswhen
Given Probability for
Nonstandard Normal Distributions
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STEPS To Find Scores When Given Probability
1. Starting with a bell curve, enter the given probability (or percentage) in the appropriate region of the graph and identify the x value(s) being sought.
2. Use Table A-2 to find the z score corresponding to the region bounded by x and the centerline of 0.
Cautions: Refer to the BODY of Table A-2 to find the closest area, then identify the corresponding z score. Make the z score negative if it is located to the left of the centerline.
3. Using Formula 5-2, enter the values for µ, , and the z score found in step 2, then solve for x.
x = µ + (z • ) (Another form of Formula 5-2)
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.
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63.6
40%
x = ?
50%
90% 10%
10%
Finding P90 for Heights of Women
FIGURE 5-17 0 1.28
=2.5 = 63.6
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63.6
40%
x = 66.8
50%
Finding P90 for Heights of Women
FIGURE 5-17 0 1.28
x = 63.6 + (1.28 • 2.5) = 66.8
Finding P90 for Heights of Women
=2.5 = 63.6
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REMEMBER: z-Scores
BELOW THE MEAN are NEGATIVE
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REMEMBER: z-Scores
BELOW THE MEAN are NEGATIVE
.05.45
–1.645 0
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Finding the 5th Percentile for Eye-Contact Times
5%
Figure 5-18
A
x = ? 184Time (sec)
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Finding the 5th Percentile for Eye-Contact Times
5%
z = –1.645 0
.4500
x = 93.5 184Time (sec)
z
x = 184 + ( –1.645 • 55 ) = 93.5
Figure 5-18
=55 = 184