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Understanding Normal Probability Distribution in Business Statistics

This overview covers the essential concepts of normal probability distribution as taught in Business Statistics (BUSA 3101) by Dr. Lari H. Arjomand. It explains the characteristics of standard normal probability distribution, the conversion of normal distribution into standard form, and includes practical examples such as stock replenishment in a retail context. A focus is given to calculating stockout probabilities and reorder points using statistical methods. Learn how to apply these principles effectively in business decision-making scenarios.

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Understanding Normal Probability Distribution in Business Statistics

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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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