1. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-1 Introduction to Statistics Chapter 6 Continuous Probability Distributions

2. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-2 Chapter Goals After completing this chapter, you should be able to: Understand the continuous random variables and there probability distribution Recognize when to apply the uniform Find probabilities using a normal distribution table and apply the normal distribution

3. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-3 Continuous Probability Distributions A continuous random variable is a variable that can assume any value on a continuum (can assume an uncountable number of values) thickness of an item time required to complete a task temperature of a solution height, in inches These can potentially take on any value, depending only on the ability to measure accurately.

4. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-4 The Uniform Distribution The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable

5. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-5 The Continuous Uniform Distribution: where f(x) = value of the density function at any x value a = lower limit of the interval b = upper limit of the interval The Uniform Distribution (continued) f(x) =

6. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-6 Uniform Distribution Example: Uniform Probability Distribution Over the range 2 ≤ x ≤ 6: 26.25 f(x) = =.25 for 2 ≤ x ≤ 6 6 - 2 1 x f(x)

7. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-7 The Normal Distribution ‘ Bell Shaped’ Symmetrical Mean, Median and Mode are Equal Location is determined by the mean, μ Spread is determined by the standard deviation, σ The random variable has an infinite theoretical range: +  to   Mean = Median = Mode x f(x) μ σ

8. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-8 By varying the parameters μ and σ, we obtain different normal distributions Many Normal Distributions

9. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-9 The Normal Distribution Shape x f(x) μ σ Changing μ shifts the distribution left or right. Changing σ increases or decreases the spread.

10. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-10 Finding Normal Probabilities Probability is the area under the curve! ab x f(x) Paxb( )  Probability is measured by the area under the curve

11. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-11 f(x) x μ Probability as Area Under the Curve 0.5 The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below

12. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-12 Empirical Rules μ ± 1 σ  encloses about 68% of x’s  f(x) x μ μ  σμ  σ What can we say about the distribution of values around the mean? There are some general rules: σσ 68.26%

13. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-13 The Empirical Rule μ ± 2σ covers about 95% of x’s μ ± 3σ covers about 99.7% of x’s xμ 2σ2σ2σ2σ xμ 3σ3σ3σ3σ 95.44%99.72% (continued)

14. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-14 Importance of the Rule If a value is about 2 or more standard deviations away from the mean in a normal distribution, then it is far from the mean The chance that a value that far or farther away from the mean is highly unlikely, given that particular mean and standard deviation

15. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-15 The Standard Normal Distribution Also known as the “z” distribution Mean is defined to be 0 Standard Deviation is 1 z f(z) 0 1 Values above the mean have positive z-values, values below the mean have negative z-values

16. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-16 The Standard Normal Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normal distribution (z) Need to transform x units into z units

17. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-17 Translation to the Standard Normal Distribution Translate from x to the standard normal (the “z” distribution) by subtracting the mean of x and dividing by its standard deviation:

18. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-18 Example If x is distributed normally with mean of 100 and standard deviation of 50, the z value for x = 250 is This says that x = 250 is three standard deviations (3 increments of 50 units) above the mean of 100.

19. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-19 Comparing x and z units z 100 3.00 250x Note that the distribution is the same, only the scale has changed. We can express the problem in original units (x) or in standardized units (z) μ = 100 σ = 50

20. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-20 The Standard Normal Table The Standard Normal table in the textbook (Appendix D) gives the probability from the mean (zero) up to a desired value for z z 02.00.4772 Example: P(0 < z < 2.00) =.4772

21. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-21 The Standard Normal Table The value within the table gives the probability from z = 0 up to the desired z value.4772 2.0 P(0 < z < 2.00) =.4772 The row shows the value of z to the first decimal point The column gives the value of z to the second decimal point 2.0...... (continued)

22. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-22 General Procedure for Finding Probabilities Draw the normal curve for the problem in terms of x Translate x-values to z-values Use the Standard Normal Table To find P(a < x < b) when x is distributed normally:

23. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-23 Z Table example Suppose x is normal with mean 8.0 and standard deviation 5.0. Find P(8 < x < 8.6) P(8 < x < 8.6) = P(0 < z < 0.12) Z0.12 0 x8.6 8 Calculate z-values:

24. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-24 Z Table example Suppose x is normal with mean 8.0 and standard deviation 5.0. Find P(8 < x < 8.6) P(0 < z < 0.12) z 0.12 0 x 8.6 8 P(8 < x < 8.6)  = 8  = 5  = 0  = 1 (continued)

25. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-25 Z 0.12 z.00.01 0.0.0000.0040.0080.0398.0438 0.2.0793.0832.0871 0.3.1179.1217.1255 Solution: Finding P(0 < z < 0.12).0478.02 0.1. 0478 Standard Normal Probability Table (Portion) 0.00 = P(0 < z < 0.12) P(8 < x < 8.6)

26. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-26 Finding Normal Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x < 8.6) Z 8.6 8.0

27. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-27 Finding Normal Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x < 8.6) (continued) Z 0.12.0478 0.00.5000 P(x < 8.6) = P(z < 0.12) = P(z < 0) + P(0 < z < 0.12) =.5 +.0478 =.5478

28. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-28 Upper Tail Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(x > 8.6) Z 8.6 8.0

29. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-29 Now Find P(x > 8.6)… (continued) Z 0.12 0 Z.0478 0.5000.50 -.0478 =.4522 P(x > 8.6) = P(z > 0.12) = P(z > 0) - P(0 < z < 0.12) =.5 -.0478 =.4522 Upper Tail Probabilities

30. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-30 Lower Tail Probabilities Suppose x is normal with mean 8.0 and standard deviation 5.0. Now Find P(7.4 < x < 8) Z 7.4 8.0

31. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-31 Lower Tail Probabilities Now Find P(7.4 < x < 8)… Z 7.4 8.0 The Normal distribution is symmetric, so we use the same table even if z-values are negative: P(7.4 < x < 8) = P(-0.12 < z < 0) =.0478 (continued).0478

32. Business Statistics: A Decision-Making Approach, 6e © 2005 Prentice-Hall, Inc. Chap 5-32 Chapter Summary Reviewed key continuous distributions uniform and normal, Found probabilities using formulas and tables Recognized when to apply different distributions Applied distributions to decision problems