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Converting to a Standard Normal Distribution

Converting to a Standard Normal Distribution. Think of me as the measure of the distance from the mean, measured in standard deviations. is used to compute the cumulative probability given a z value. NORMSDIST.

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Converting to a Standard Normal Distribution

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

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

  3. Using Excel to ComputeStandard Normal Probabilities • Formula Worksheet

  4. Using Excel to ComputeStandard Normal Probabilities • Value Worksheet

  5. Using Excel to ComputeStandard Normal Probabilities • Formula Worksheet

  6. Using Excel to ComputeStandard Normal Probabilities • Value Worksheet

  7. Pep Zone 5w-20 Motor Oil 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.

  8. Pep Zone 5w-20 Motor Oil Example: Pep Zone • 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).

  9. Pep Zone 5w-20 Motor Oil Solving for Stockout Probability 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.

  10. Pep Zone 5w-20 Motor Oil Solving for Stockout Probability:Step 2 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.

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

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

  13. Pep Zone 5w-20 Motor Oil Example: Pep Zone • Solving for the Stockout Probability Area = 1 - .7967 = .2033 Area = .7967 z 0 .83

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

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

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

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

  18. Pep Zone 5w-20 Motor Oil 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

  19. Using Excel to ComputeNormal Probabilities • 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. NORMINV is used to compute the x value given a cumulative probability.

  20. Pep Zone 5w-20 Motor Oil Using Excel to ComputeNormal Probabilities • Formula Worksheet

  21. Pep Zone 5w-20 Motor Oil Using Excel to ComputeNormal Probabilities • Value Worksheet 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.

  22. 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. • What is the probability a subscriber will spend at least one hour reading the Journal? • What is the probability a reader will spend no more than 30 minutes reading the Journal? • For the 10 percent who spend the most time reading the Journal, how much time do they spend?

  23. Exercise 18, p. 261 • 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. • Convert x to the standard normal distribution

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

  25. Exercise 18, p. 261 (c)

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