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Probability is area under curve!

Normal Probability Distribution. Probability is area under curve!. Normal Probabilities. Standard Normal Probability Distribution. A random variable having a normal distribution with a mean of 0 and a standard deviation of 1 is said to have a standard normal probability

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Probability is area under curve!

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  1. Normal Probability Distribution Probability is area under curve! Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  2. Normal Probabilities Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  3. Standard Normal Probability Distribution A random variable having a normal distribution with a mean of 0 and a standard deviation of 1 is said to have a standard normal probability distribution. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  4. Standard Normal Probability Distribution The letter z is used to designate the standard normal random variable. s = 1 z 0 Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  5. Standard Normal Probability Distribution • Converting to the Standard Normal Distribution We can think of z as a measure of the number of standard deviations x is from . We use the above equation to convert normaldistribution into standard normal distribution. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  6. StandardNormal Probability Distribution Pep Zone 5w-20 Motor Oil • Example: Pep Zone 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. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  7. Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution • Example: Pep Zone 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). Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  8. Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution • Solving for the Stockout Probability Step 1: Convert x to the standard normal distribution. z = (x - )/ = (20 - 15)/6 = .83 Step 2: Find the area under the standard normal curve to the left of z = .83 see next slide Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  9. Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution • Cumulative Probability Table for the Standard Normal Distribution P(z< .83) Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  10. Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution • 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) Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  11. Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution • Solving for the Stockout Probability Area = 1 - .7967 P (x > 20)= .2033 Area = .7967 z 0 .83 Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  12. Example (Finding the X value): If the manager of Pep Zone wants the probability of a stockout to be no more than .05, what should the reorder point be? Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  13. Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution • Solving for the Reorder Point Area = 0.95 Area = .05 z 0 z.05 Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  14. Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution • 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) Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  15. Pep Zone 5w-20 Motor Oil • Solving for the Reorder Point Step 2: Convert z.05 to the corresponding value of x. x =  + z.05  = 15 + 1.645(6) = 24.87 or 25 A reorder point of 25 gallons will place the probability of a stockout during lead-time at (slightly less than) .05. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  16. Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution • Solving for the Reorder Point: Some Observation By raising the reorder point from 20 gallons to 25 gallons on hand, the probability of a stockout decreases from about .20 to .05. This is a significant decrease in the chance that Pep Zone will be out of stock and unable to meet a customer’s desire to make a purchase. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  17. Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution • Using half of the Normal Table to solve for the Reorder Point Area = 0.5 - .05 Area = .05 0.45 z 0 z.05 Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  18. Pep Zone 5w-20 Motor Oil Standard Normal Probability DistributionExample Continued • The question is: P ( X = ? ) = 0.05. In another word, we need to find the value of X. The equation is: • From the problem, we know that s =6, m = 15. The z value for probability of 0.45 from the table is (1.64 + 1.65)/2 = 1.645. • Thus, 1.645 = ( X – 15 )/ 6 = 24.87 or X = 25. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

  19. End of Chapter 6 Business Statistics (BUSA 3101). Dr.Lari H. Arjomand

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