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Converting to a Standard Normal Distribution Think of me as the measure of the distance from the mean, measured in standard deviations

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is used to compute the z value is used to compute the z value given a cumulative probability. given a cumulative probability. is used to compute the z value is used to compute the z value given a cumulative probability. given a cumulative probability. NORMSINVNORMSINV NORM S INV is used to compute the cumulative is used to compute the cumulative probability given a z value. probability given a z value. is used to compute the cumulative is used to compute the cumulative probability given a z value. probability given a z value.NORMSDISTNORMSDIST NORM S DIST Using Excel to Compute Standard Normal Probabilities n Excel has two functions for computing probabilities and z values for a standard normal distribution: (The “S” in the function names reminds us that they relate to the standard normal probability distribution.)

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n Formula Worksheet Using Excel to Compute Standard Normal Probabilities

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n Value Worksheet Using Excel to Compute Standard Normal Probabilities

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n Formula Worksheet Using Excel to Compute Standard Normal Probabilities

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n Value Worksheet Using Excel to Compute Standard Normal Probabilities

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Example: Pep Zone Standard Normal Probability Distribution Pep Zone sells auto parts and supplies including a popular multi-grade motor oil. When the stock of this oil drops to 20 gallons, a replenishment order is placed. Pep Zone 5w-20 Motor Oil

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Example: Pep Zone n Standard Normal Probability Distribution The store manager is concerned that sales are being lost due to stockouts while waiting for an order. It has been determined that demand during replenishment lead time is normally distributed with a mean of 15 gallons and a standard deviation of 6 gallons. The manager would like to know the probability of a stockout, P ( x > 20). Pep Zone 5w-20 Motor Oil

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Solving for Stockout Probability Pep Zone 5w-20 Motor Oil Step 1: Convert x to the standard normal distribution Thus 20 gallons sold during the replenishment lead time would be.83 standard deviations above the average of 15.

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Solving for Stockout Probability: Step 2 Pep Zone 5w-20 Motor Oil Now we need to find the area under the curve to the left of z =.83. This will give us the probability that x ≤ 20 gallons.

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n Cumulative Probability Table for the Standard Normal Distribution Example: Pep Zone Pep Zone 5w-20 Motor Oil P ( z <.83)

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P ( z >.83) = 1 – P ( z.83) = 1 – P ( z <.83) = 1-.7967 = 1-.7967 =.2033 =.2033 P ( z >.83) = 1 – P ( z.83) = 1 – P ( z <.83) = 1-.7967 = 1-.7967 =.2033 =.2033 n Solving for the Stockout Probability Example: Pep Zone Step 3: Compute the area under the standard normal curve to the right of z =.83. curve to the right of z =.83. Step 3: Compute the area under the standard normal curve to the right of z =.83. curve to the right of z =.83. Pep Zone 5w-20 Motor Oil Probability of a stockout of a stockout P ( x > 20)

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n Solving for the Stockout Probability Example: Pep Zone 0.83 Area =.7967 Area = 1 -.7967 =.2033 =.2033 z Pep Zone 5w-20 Motor Oil

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If the manager of Pep Zone wants the probability of a stockout to be no more than.05, what should the reorder point be? Example: Pep Zone Pep Zone 5w-20 Motor Oil

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n Solving for the Reorder Point Example: Pep Zone Pep Zone 5w-20 Motor Oil 0 Area =.9500 Area =.0500 z z.05

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n Solving for the Reorder Point Example: Pep Zone Pep Zone 5w-20 Motor Oil Step 1: Find the z -value that cuts off an area of.05 in the right tail of the standard normal in the right tail of the standard normal distribution. distribution. Step 1: Find the z -value that cuts off an area of.05 in the right tail of the standard normal in the right tail of the standard normal distribution. distribution. We look up the complement of the tail area (1 -.05 =.95)

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Solving for the Reorder Point Step 2: Convert z.05 to the corresponding value of x : Pep Zone 5w-20 Motor Oil

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Solving for the Reorder Point So if we raising our reorder point from 20 to 25 gallons, we reduce the probability of a stockout from about.20 to less than.05 Pep Zone 5w-20 Motor Oil

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Using Excel to Compute Normal Probabilities n Excel has two functions for computing cumulative probabilities and x values for any normal distribution: NORMDIST is used to compute the cumulative probability given an x value. NORMDIST is used to compute the cumulative probability given an x value. NORMINV is used to compute the x value given a cumulative probability. NORMINV is used to compute the x value given a cumulative probability.

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n Formula Worksheet Using Excel to Compute Normal Probabilities Pep Zone 5w-20 Motor Oil

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n Value Worksheet Using Excel to Compute Normal Probabilities Note: P( x > 20) =.2023 here using Excel, while our previous manual approach using the z table yielded.2033 due to our rounding of the z value. Pep Zone 5w-20 Motor Oil

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Exercise 18, p. 261 The average time a subscriber reads the Wall Street Journal is 49 minutes. Assume the standard deviation is 16 minutes and that reading times are normally distributed. a)What is the probability a subscriber will spend at least one hour reading the Journal? b)What is the probability a reader will spend no more than 30 minutes reading the Journal? c)For the 10 percent who spend the most time reading the Journal, how much time do they spend?

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Exercise 18, p. 261 a)Convert x to the standard normal distribution: Thus one who read 560 minutes would be.69 from the mean. Now find P(z ≤.6875). P(z ≤.69)=.7549. Thus P(x > 60 minutes) = 1 -.7549 =.2541. b)Convert x to the standard normal distribution

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P(x ≤ 30 minutes) 0 z 1.19-1.19 Red-shaded area is equal to blue shaded area Thus: P(x < 30 minutes) =.117

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Exercise 18, p. 261 (c)

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