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Understanding Normal Distribution Properties

Explore the bell-shaped curve and properties of the normal distribution, including the Empirical Rule and its application in calculating probabilities. Learn to use control charts to track statistical control and detect outliers.

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Understanding Normal Distribution Properties

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  1. Chapter SixNormal Curves and Sampling Probability Distributions

  2. Chapter 6 Section 1 Graphs of Normal Probability Distributions

  3. Properties of TheNormal Distribution The curve is bell-shaped with the highest point over the mean, μ.

  4. Properties of The Normal Distribution The curve is symmetrical about a vertical line through μ.

  5. Properties of The Normal Distribution The curve approaches the horizontal axis but never touches or crosses it.

  6. Properties of The Normal Distribution The transition points between cupping upward and downward occur above μ + σ and μ – σ .

  7. The Empirical Rule Approximately 68.2% of the data values lie within one standard deviation of the mean.

  8. The Empirical Rule Approximately 95.4% of the data values lie within two standard deviations of the mean.

  9. The Empirical Rule Almost all (approximately 99.7%) of the data values will be within three standard deviations of the mean.

  10. The Empirical Rule Percentages of data that lies between given values

  11. Application of the Empirical Rule

  12. Application of the Empirical Rule Question: What is the probability that a light bulb of this type will last between 1000 and 1200 hours? The life of a particular type of light bulb is normally distributed with a mean of 1100 hours and a standard deviation of 100 hours. Answer:Approximately 0.6820

  13. Application of the Empirical Rule Question: What is the probability that a light bulb of this type will last between 950 and 1325 hours? The life of a particular type of light bulb is normally distributed with a mean of 1100 hours and a standard deviation of 75 hours. Answer:Approximately 0.9755

  14. Application of the Empirical Rule Answer:Approximately 0.3410

  15. Application of the Empirical Rule Answer:Approximately 0.8180

  16. Application of the Empirical Rule Answer:Approximately 0.0215

  17. Application of the Empirical Rule Answer:Approximately 0.0230

  18. Application of the Empirical Rule Answer:Approximately 0.0015

  19. Control Chart A statistical tool to track data over a period of equally spaced time intervals or in some sequential order.

  20. Statistical Control A random variable is in statistical control if it can be described by the same probability distribution when it is observed at successive points in time.

  21. To Construct aControl Chart • Draw a center horizontal line at μ. • Draw dashed lines (control limits) at μ±2σ and μ±3σ. • The values of μ and σ may be target values or may be computed from past data when the process was in control. • Plot the variable being measured using time on the horizontal axis.

  22. Control Chart μ+3σ μ+2σ μ μ-2σ μ-3σ

  23. Out-Of-ControlWarning Signals • One point beyond the 3σ level. • A run of nine consecutive points on one side of the center line. • At least two of three consecutive points beyond the 2σ level on the same side of the center line.

  24. Is the Process in Control? μ+3σ μ+2σ μ μ-2σ μ-3σ

  25. Is the Process in Control? μ+3σ μ+2σ μ μ-2σ μ-3σ

  26. Is the Process in Control? μ+3σ μ+2σ μ μ-2σ μ-3σ

  27. Is the Process in Control? μ+3σ μ+2σ μ μ-2σ μ-3σ

  28. Is the Process in Control?

  29. Is the Process in Control? μ+3σ μ+2σ μ μ-2σ μ-3σ

  30. Is the Process in Control? • Do any points fall beyond the LCL and UCL three standard deviation limits? • No points fall beyond the LCL and the UCL three standard • deviations limit. • Is there a run of nine consecutive points on one side of the center line? • There is no run of nine consecutive points on one side of the • center line. • Is there an instance of two out of three points beyond the two standard deviation limits on the same side of the center line? • There is no instance of two out of three points beyond the two standard deviation limits on the same side of the center line.

  31. Uniform Probability Distributions

  32. Uniform Probability Distributions

  33. Uniform Probability Distributions

  34. Uniform Probability Distributions

  35. Uniform Probability Distributions

  36. Uniform Probability Distributions

  37. Exponential Probability Distributions

  38. Uniform Probability Distributions

  39. Uniform Probability Distributions

  40. Uniform Probability Distributions

  41. Uniform Probability Distributions

  42. Uniform Probability Distributions

  43. THE END OF SECTION 1 Homework Assignments Pages: 259 - 266 Exercises: #1 - 19, odd Exercises: #2 - 20, even

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