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

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 2   0  1 z  x  Need to “standardize” these nonstandard distributions  Will use z -score formula Nonstandard Normal Distributions

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 3   0  1 z  x  Need to “standardize” these nonstandard distributions  Will use z -score formula x – µx – µ  z = Nonstandard Normal Distributions Formula 5-2

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 4 Converting from Nonstandard to Standard Normal Distribution x 0 Figure 5-13  z x –   z =

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 5 Probability of Height between 63.6 in. and 68.6 in z z = 68.6 – = 2.00  =  2.5  = 63.6 Figure 5-14

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 6 Nonstandard Normal Distributions: Finding Scores Section 5-4 M A R I O F. T R I O L A Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 7 Review of 5-2 Standard normal distribution finding z-scores when given the probability

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 8 Finding z Scores When Given Probabilities FIGURE 5-11 Finding the 95th Percentile 0 5% or z %5%

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 9 FIGURE 5-12 Finding the 10th Percentile Finding z Scores When Given Probabilities Bottom 10% 10%90% z 0

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 10 Finding Scores when Given Probability for Nonstandard Normal Distributions

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 11 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.

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman % x = ? 50% 90%10% Finding P 90 for Heights of Women FIGURE  =  2.5  = 63.6

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman % x = % Finding P 90 for Heights of Women FIGURE x = ( ) = 66.8 Finding P 90 for Heights of Women  =  2.5  = 63.6

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 14 REMEMBER: z -Scores BELOW THE MEAN are NEGATIVE

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 15 REMEMBER: z -Scores BELOW THE MEAN are NEGATIVE –

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 16 Finding the 5th Percentile for Eye-Contact Times 5% Figure 5-18 A x = ?184 Time (sec)

Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 17 Finding the 5th Percentile for Eye-Contact Times 5% z = – x = Time (sec) z x = ( – ) = 93.5 Figure 5-18  =  55  = 184