1. © 2005 McGraw-Hill Ryerson Ltd. 5-1 Statistics A First Course Donald H. Sanders Robert K. Smidt Aminmohamed Adatia Glenn A. Larson

2. © 2005 McGraw-Hill Ryerson Ltd. 5-2 Chapter 5 Probability Distributions

3. © 2005 McGraw-Hill Ryerson Ltd. 5-3 Chapter 5 - Topics Binomial Experiments Determining Binomial Probabilities The Poisson Distribution The Normal Distribution Normal Approximation of the Binomial

4. © 2005 McGraw-Hill Ryerson Ltd. 5-4 Binomial Experiments Properties of a Binomial Experiment –Same action (trial) is repeated a fixed number of times –Each trial is independent of the others –Two possible outcomes – success or failure –Constant probability of success for each trial

5. © 2005 McGraw-Hill Ryerson Ltd. 5-5 Determining Binomial Probabilities Combinations –Selection of r items from a set of n distinct objects without regard to the order in which r items are picked Combination Rule

6. © 2005 McGraw-Hill Ryerson Ltd. 5-6 Determining Binomial Probabilities Binomial Probability –Probability of correctly guessing exactly r items from a set of n distinct objects without regard to the order in which r items are picked Binomial Probability Formula

7. © 2005 McGraw-Hill Ryerson Ltd. 5-7 Our QuickQuiz probability distribution. Figure 5.1 (including table)

8. © 2005 McGraw-Hill Ryerson Ltd. 5-8

9. © 2005 McGraw-Hill Ryerson Ltd. 5-9 Variance of Binomial Distribution Formula Standard Deviation of Binomial Distribution Formula Expected Value (Mean) of Binomial Distribution Formula

10. © 2005 McGraw-Hill Ryerson Ltd. 5-10 The Poisson Distribution Discrete probability distribution Used to determine the number of specified occurrences that take place within a unit of time, distance, area, or volume Poisson Distribution Formula

11. © 2005 McGraw-Hill Ryerson Ltd. 5-11 The Normal Distribution Continuous probability distribution Used to investigate the probability that the variable assumes any value within a given interval of values

12. © 2005 McGraw-Hill Ryerson Ltd. 5-12 Normal Distribution. Figure 5.4

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15. © 2005 McGraw-Hill Ryerson Ltd. 5-15 Probability of breaking strength between 110 and 120. Figure 5.5

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17. © 2005 McGraw-Hill Ryerson Ltd. 5-17 Both intervals extend from the mean (z = 0) to 1 standard deviation above the mean (z = 1.00). Figure 5.6

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19. © 2005 McGraw-Hill Ryerson Ltd. 5-19 The probability that a z value selected at random will fall between 0 and 2.27 or between –2.27 and 0 is.4884. Figure 5.7 Calculating Probabilities for the Standard Normal Distribution

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21. © 2005 McGraw-Hill Ryerson Ltd. 5-21 The area under the normal curve between vertical lines drawn at z = –1.73 and z = +2.45 is.9511. Figure 5.8

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23. © 2005 McGraw-Hill Ryerson Ltd. 5-23 The area under the normal curve between a z value of –1.54 and a z value of –.76 is.1618. Figure 5.9

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25. © 2005 McGraw-Hill Ryerson Ltd. 5-25 The area under the normal curve to the left of a z value of –1.96 is.0250. Figure 5.10

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27. © 2005 McGraw-Hill Ryerson Ltd. 5-27 The area under the normal curve to the left of a z value of 1.42 is.9222. Figure 5.11

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29. © 2005 McGraw-Hill Ryerson Ltd. 5-29 The Normal Distribution Computing Probabilities for Any Normally Distributed Variable –z scores correspond to the number of standard deviations a data value is from the mean –Any value can be converted to a standard score (z score) Convert x value to z score formula

30. © 2005 McGraw-Hill Ryerson Ltd. 5-30 The z score interval corresponding to 70 < x < 130 Figure 5.13

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32. © 2005 McGraw-Hill Ryerson Ltd. 5-32 The Normal Distribution Finding Cut-off Scores for Normally Distributed Variables –Given the area under the standard normal curve, the z score method can be used to calculate the cut off point Convert z score to x value formula

33. © 2005 McGraw-Hill Ryerson Ltd. 5-33 90 th Percentile of z scores Figure 5.20

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35. © 2005 McGraw-Hill Ryerson Ltd. 5-35 Graph showing both the binomial probability histogram and the normal distribution Figure 5.13 The Normal Approximation of the Binomial

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37. © 2005 McGraw-Hill Ryerson Ltd. 5-37 The Normal Approximation of the Binomial Computing Probabilities for Any Normally Distributed Variable Method –Calculate mean and standard deviation –Apply continuity correction factor (±0.5) –Convert x values to z scores –Calculate area under standard normal curve

38. © 2005 McGraw-Hill Ryerson Ltd. 5-38 End of Chapter 5 Probability Distributions