1. © 2002 Thomson / South-Western Slide 6-1 Chapter 6 Continuous Probability Distributions

2. © 2002 Thomson / South-Western Slide 6-2 Learning Objectives Understand concepts of the uniform distribution. Appreciate the importance of the normal distribution. Recognize normal distribution problems, and know how to solve them. Decide when to use the normal distribution to approximate binomial distribution problems, and know how to work them. Decide when to use the exponential distribution to solve problems in business, and know how to work them.

3. © 2002 Thomson / South-Western Slide 6-3 Uniform Distribution Area = 1 ab The uniform distribution is a continuous distribution in which the same height, of f(X), is obtained over a range of values.

4. © 2002 Thomson / South-Western Slide 6-4 Example: Uniform Distribution of Lot Weights Area = 1

5. © 2002 Thomson / South-Western Slide 6-5 Example: Uniform Distribution,continued Mean and Standard Deviation

6. © 2002 Thomson / South-Western Slide 6-6 Example: Uniform Distribution Probability, continued Area = 0.5

7. © 2002 Thomson / South-Western Slide 6-7 The Normal Distribution A widely known and much-used distribution that fits the measurements of many human characteristics and most machine-produced items. Many other variable in business and industry are normally distributed. The normal distribution and its associated probabilities are an integral part of statistical quality control

8. © 2002 Thomson / South-Western Slide 6-8 Characteristics of the Normal Distribution Continuous distribution Symmetrical distribution Asymptotic to the horizontal axis Unimodal A family of curves Total area under the curve sums to 1. Area to right of mean is 1/2. Area to left of mean is 1/2. 1/2

9. © 2002 Thomson / South-Western Slide 6-9 Probability Density Function of the Normal Distribution X

10. © 2002 Thomson / South-Western Slide 6-10 Normal Curves for Different Means and Standard Deviations 2030405060708090100110120

11. © 2002 Thomson / South-Western Slide 6-11 Standardized Normal Distribution A normal distribution with –a mean of zero, and –a standard deviation of one Z Formula –standardizes any normal distribution Z Score –computed by the Z Formula –the number of standard deviations which a value is away from the mean

12. © 2002 Thomson / South-Western Slide 6-12 Z Table Second Decimal Place in Z Z0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.000.00000.00400.00800.01200.01600.01990.02390.02790.03190.0359 0.100.03980.04380.04780.05170.05570.05960.06360.06750.07140.0753 0.200.07930.08320.08710.09100.09480.09870.10260.10640.11030.1141 0.300.11790.12170.12550.12930.13310.13680.14060.14430.14800.1517 0.900.31590.31860.32120.32380.32640.32890.33150.33400.33650.3389 1.000.34130.34380.34610.34850.35080.35310.35540.35770.35990.3621 1.100.36430.36650.36860.37080.37290.37490.37700.37900.38100.3830 1.200.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015 2.000.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817 3.000.49870.49870.49870.49880.49880.49890.49890.49890.49900.4990 3.400.49970.49970.49970.49970.49970.49970.49970.49970.49970.4998 3.500.49980.49980.49980.49980.49980.49980.49980.49980.49980.4998

13. © 2002 Thomson / South-Western Slide 6-13 -3-20123 Table Lookup of a Standard Normal Probability Z0.00 0.01 0.02 0.000.00000.00400.0080 0.100.03980.04380.0478 0.200.07930.08320.0871 1.000.34130.34380.3461 1.100.36430.36650.3686 1.200.38490.38690.3888

14. © 2002 Thomson / South-Western Slide 6-14 Applying the Z Formula: Example, Assume…. Z0.00 0.01 0.02 0.000.00000.00400.0080 0.100.03980.04380.0478 1.000.34130.34380.3461 1.100.36430.36650.3686 1.200.38490.38690.3888

15. © 2002 Thomson / South-Western Slide 6-15 Normal Approximation of the Binomial Distribution The normal distribution can be used to approximate binomial probabilities Procedure –Convert binomial parameters to normal parameters –Does the interval lie between 0 and n? If so, continue; otherwise, do not use the normal approximation. –Correct for continuity –Solve the normal distribution problemms±3

16. © 2002 Thomson / South-Western Slide 6-16 Using the Normal Distribution to Work Binomial Distribution Problems The normal distribution can be used to approximate the probabilities in binomial distribution problems that involve large values of n. To work a binomial problem by the normal distribution requires conversion of the n and p of the binomial distribution to the µ and  of the normal distribution.

17. © 2002 Thomson / South-Western Slide 6-17 Conversion equations Conversion example: Normal Approximation of Binomial: Parameter Conversion

18. © 2002 Thomson / South-Western Slide 6-18 Normal Approximation of Binomial: Interval Check 0102030405060 n 70

19. © 2002 Thomson / South-Western Slide 6-19 Normal Approximation of Binomial: Correcting for Continuity Values Being Determined Correction XXXXXXXXXXXX +.50 -.50 +.05 -.50 and +.50 +.50 and -.50

20. © 2002 Thomson / South-Western Slide 6-20 Normal Approximation of Binomial: Graphs 0 0.02 0.04 0.06 0.08 0.10 0.12 681012141618202224262830

21. © 2002 Thomson / South-Western Slide 6-21 Normal Approximation of Binomial: Computations 25 26 27 28 29 30 31 32 33 Total 0.0167 0.0096 0.0052 0.0026 0.0012 0.0005 0.0002 0.0001 0.0000 0.0361 XP(X)

22. © 2002 Thomson / South-Western Slide 6-22 Exponential Distribution Continuous Family of distributions Skewed to the right X varies from 0 to infinity Apex is always at X = 0 Steadily decreases as X gets larger Probability function

23. © 2002 Thomson / South-Western Slide 6-23 Graphs of Selected Exponential Distributions 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 012345678    

24. © 2002 Thomson / South-Western Slide 6-24 Exponential Distribution Example: Probability Computation 0.0 0.2 0.4 0.6 0.8 1.0 1.2 012345 