ECON 214 - The Normal Distribution.pdf

Slide 32 calculating probabilities using the standard

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Slide 32
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Calculating probabilities using the standard normal Another example : Suppose the time required to repair equipment by company maintenance personnel is normally distributed with mean of 50 minutes and standard deviation of 10 minutes. What is the probability that a randomly chosen equipment will require between 50 and 60 minutes to repair? Let X denote equipment repair time. We want to calculate the probability that X lies between 50 and 60. Or P(50 X 60) Slide 33
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Calculating probabilities using the standard normal Determine the Z values for 50 and 60. So we have P(0 ≤ Z ≤ 1.00) We read off the area under the standard normal curve from zero to 1.00 from the normal table. Slide 34 50 50 50 0 10 X X Z 60 50 60 1.00 10 X X Z
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Slide 35
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Calculating probabilities using the standard normal The answer is .3413 (read as 1.0 under 0.00). This was quite easy because the lower bound was at the mean (or zero). Most problems will not have the lower bound at the mean. Nevertheless the normal table can be used to calculate the relevant probabilities by the addition or subtraction of appropriate areas under the curve. For instance : Find the probability that more than 70 minutes will be required to repair the equipment. Slide 36
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Calculating probabilities using the standard normal We want P(X>70) = P[Z>(70-50)/10)=P(Z>2.00) Reading from the first table I introduced you to, P(Z>2.00) = 0.0228 But using the half table we can write this as: 0.5 – P(0 ≤ Z ≤ 2) =0.5 - 0.4772 = 0.0228 So depending on the type of table being used, we can calculate the required probability appropriately. Henceforth we shall use the half table. Slide 37
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Calculating probabilities using the standard normal Example : Find the probability that the equipment-repair time is between 35 and 50 minutes. We want P(35 X 50) Converting to Z values we have P(-1.5 ≤ Z ≤ 0) P(0 ≤ Z ≤ 1.5 ) = .4332, since the normal curve is symmetrical. Slide 38
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Calculating probabilities using the standard normal Example : Find the probability that the required equipment-repair time is between 40 and 70 minutes. P(40 X ≤ 7 0) P(-1 ≤ Z ≤ 2 ) = P(-1 ≤ Z ≤ 0 ) + P(0 ≤ Z ≤ 2 ) = .3413 + .4772 = .8185 Slide 39
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Calculating probabilities using the standard normal Example : Find the probability that the required equipment-repair time is either less than 25 minutes or greater than 75 minutes. P(X < 25) or P(X > 75) = P(X < 25) + P(X > 75) = P(Z < -2.5) + P(Z > 2.5) = [.5 – P(-2.5<Z<0)] + [.5 – P(0<Z<2.5)] = 1 – 2 P(0<Z<2.5) = 1 – 2(0.4938) = 1 - . 9876 = .0124 Slide 40
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Slide 41 Percentile points for normally distributed variables You discussed percentiles in under descriptive statistics. For example, the 90 th percentile is the value X such that 90% of observations are below this value and 10% above it.
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