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

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

© 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

© 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

© 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

© 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

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

© 2005 McGraw-Hill Ryerson Ltd. 5-8

© 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

© 2005 McGraw-Hill Ryerson Ltd 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

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

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

© 2005 McGraw-Hill Ryerson Ltd. 5-13

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

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

© 2005 McGraw-Hill Ryerson Ltd. 5-18

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

© 2005 McGraw-Hill Ryerson Ltd. 5-20

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

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

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

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

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© 2005 McGraw-Hill Ryerson Ltd 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

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

© 2005 McGraw-Hill Ryerson Ltd. 5-31

© 2005 McGraw-Hill Ryerson Ltd 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

© 2005 McGraw-Hill Ryerson Ltd th Percentile of z scores Figure 5.20

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

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© 2005 McGraw-Hill Ryerson Ltd 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

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