# Chap06 normal distributions & continous

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Modul Statistik Bisnis II

### Transcript of Chap06 normal distributions & continous

- 1. Statistics for Managers Using Microsoft Excel 4th Edition Chapter 6 The Normal Distribution and Other Continuous Distributions Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 6-1
- 2. Chapter Goals After completing this chapter, you should be able to: Describe the characteristics of the normal distribution Translate normal distribution problems into standardized normal distribution problems Find probabilities using a normal distribution table Evaluate the normality assumption Recognize when to apply the uniform and exponential distributions Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 6-2
- 3. Chapter Goals (continued) After completing this chapter, you should be able to: Define the concept of a sampling distribution Determine the mean and standard deviation for the _ sampling distribution of the sample mean, X Determine the mean and standard deviation for the sampling distribution of the sample proportion, ps Describe the Central Limit Theorem and its importance _ Statistics for Managers Using Apply sampling distributions for both X and p s Microsoft Excel, 4e 2004 Chap 6-3 Prentice-Hall, Inc.
- 4. Probability Distributions Probability Distributions Ch. 5 Discrete Probability Distributions Continuous Probability Distributions Binomial Normal Poisson Ch. 6 Uniform Statistics for Hypergeometric Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Exponential Chap 6-4
- 5. 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 Statisticsaccurately. Using for Managers Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 6-5
- 6. The Normal Distribution Probability Distributions Continuous Probability Distributions Normal Uniform Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Exponential Chap 6-6
- 7. The Normal Distribution Bell Shaped Symmetrical Mean, Median and Mode are Equal f(X) Location is determined by the mean, Spread is determined by the standard deviation, The random variable has an infinite theoretical range: Statistics for Managers Using + to Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Mean = Median = Mode Chap 6-7 X
- 8. Many Normal Distributions By varying the parameters and , we obtain different normal distributions Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 6-8
- 9. The Normal Distribution Shape f(X) Changing shifts the distribution left or right. Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Changing increases or decreases the spread. X Chap 6-9
- 10. The Normal Probability Density Function The formula for the normal probability density function is f(X) = Where 1 (1/2)[(X )/] 2 e 2 e = the mathematical constant approximated by 2.71828 = the mathematical constant approximated by 3.14159 = the population mean Statistics forManagers Using = the population standard deviation Microsoft Excel, 4e 2004 continuous variable X = any value of the Prentice-Hall, Inc. Chap 6-10
- 11. The Standardized Normal Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardized normal distribution (Z) Need to transform X units into Z units Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 6-11
- 12. Translation to the Standardized Normal Distribution Translate from X to the standardized normal (the Z distribution) by subtracting the mean of X and dividing by its standard deviation: X Z= Z always has mean = Statistics for Managers Using 0 and standard deviation = 1 Microsoft Excel, 4e 2004 Chap 6-12 Prentice-Hall, Inc.
- 13. The Standardized Normal Probability Density Function The formula for the standardized normal probability density function is f(Z) = Where 1 (1/2)Z 2 e 2 e = the mathematical constant approximated by 2.71828 = the mathematical constant approximated by 3.14159 Z = any value of the standardized normal distribution Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 6-13
- 14. The Standardized Normal Distribution Also known as the Z distribution Mean is 0 Standard Deviation is 1 f(Z) 1 0 Z Values above the mean have positive Z-values, Statistics for Managers Using values below the mean have negative Z-values Microsoft Excel, 4e 2004 Chap 6-14 Prentice-Hall, Inc.
- 15. Example If X is distributed normally with mean of 100 and standard deviation of 50, the Z value for X = 200 is X 200 100 Z= = = 2.0 50 This says that X = 200 is two standard deviations (2 increments of 50 units) above the Managers Using Statistics formean of 100. Microsoft Excel, 4e 2004 Prentice-Hall, Inc. Chap 6-15
- 16. Comparing X and Z units 100 0 200 2.0 X Z ( = 100, = 50) ( = 0, = 1) Note that the distribution is the same, only the scale has changed. We can express the problem in Statistics for Managers or in standardized units (Z) original units (X) Using Microsoft Excel, 4e 2004 Chap 6-16 Prentice-Hall, Inc.
- 17. Finding Normal Probabilities Probability is the Probability is measured area under the curve! under the curve f(X) by the area P (a X b) = P (a < X < b) (Note that the probability of any individual value is zero) a Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. b X Chap 6-17
- 18. Probability as Area Under the Curve The total area under the curve is 1.0, and the curve is symmetric, so half is above the mean, half is below f(X) P( < X < ) = 0.5 0.5 Statistics for Managers Using P( Microsoft Excel, 4e 2004 < Prentice-Hall, Inc. P( < X < ) = 0.5 0.5 X X < ) = 1.0 Chap 6-18
- 19. Empirical Rules What can we say about the distribution of values around the mean? There are some general rules: f(X) 1 encloses about 68% of Xs -1 +1 Statistics for Managers Using 68.26% Microsoft Excel, 4e 2004 Prentice-Hall, Inc. X Chap 6-19
- 20. The Empirical Rule (continued) 2 covers about 95% of Xs 3 covers about 99.7% of Xs 2 3 2 95.44% Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. x 3 x 99.72% Chap 6-20
- 21. The Standardized Normal Table The Standardized Normal table in the textbook (Appendix table E.2) gives the probability less than a desired value for Z (i.e., from negative infinity to Z) .9772 Example: P(Z < 2.00) = .9772 Statistics for Managers Using Microsoft Excel, 4e 2004 Prentice-Hall, Inc. 0 2.00 Z Chap 6-21
- 22.