Copyright 1998, Triola, Elementary Statistics Addison Wesley Longman 1 Nonstandard Normal Distributions: Finding Probabilities Section 5-3 M A R I O

Copyright © 1998, Triola, Elementary Statistics

Addison Wesley Longman 1

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

Addison Wesley Longman



0 1 z x Need to “standardize” these nonstandard distributionsWill use z-score formula

Nonstandard Normal Distributions



0 1 z x Need to “standardize” these nonstandard distributionsWill use z-score formula

x – µz =


Formula 5-2



Converting from Nonstandard to

Standard Normal Distribution

x 0

Figure 5-13

z

x – z =



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



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

Addison Wesley Longman



Review of 5-2Standard normal distribution

finding z-scores when given the probability



Finding z Scores When Given Probabilities

FIGURE 5-11 Finding the 95th Percentile

0

5% or 0.05

z

0.450.50

95% 5%



FIGURE 5-12 Finding the 10th Percentile

Finding z Scores When Given Probabilities

Bottom 10%

10% 90%

0.400.10z 0



Finding Scoreswhen

Given Probability for




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.



63.6

40%

x = ?

50%

90% 10%

10%

Finding P90 for Heights of Women

FIGURE 5-17 0 1.28

=2.5 = 63.6



63.6

40%

x = 66.8

50%


FIGURE 5-17 0 1.28

x = 63.6 + (1.28 • 2.5) = 66.8


=2.5 = 63.6



REMEMBER: z-Scores

BELOW THE MEAN are NEGATIVE



REMEMBER: z-Scores

BELOW THE MEAN are NEGATIVE

.05.45

–1.645 0



Finding the 5th Percentile for Eye-Contact Times

5%

Figure 5-18

A

x = ? 184Time (sec)



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

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Copyright 1998, Triola, Elementary Statistics Addison Wesley Longman 1 Nonstandard Normal Distributions: Finding Probabilities Section 5-3 M A R I O