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

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

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

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

5. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand n Converting to the Standard Normal Distribution Standard Normal Probability 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 normal distribution into standard normal distribution.

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

7. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand The store manager is concerned that sales are being 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). Standard Normal Probability Distribution Pep Zone 5w-20 Motor Oil n Example: Pep Zone

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

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

10. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand 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 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) Standard Normal Probability Distribution

11. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand n Solving for the Stockout Probability 0.83 Area =.7967 z Pep Zone 5w-20 Motor Oil Standard Normal Probability Distribution Area = 1 -.7967 P (x > 20)=.2033

12. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand 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?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

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

14. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand n Solving for the Reorder Point 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) Standard Normal Probability Distribution

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

16. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand n Solving for the Reorder Point: Some Observation Pep Zone 5w-20 Motor Oil By raising the reorder point from 20 gallons to 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 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. Standard Normal Probability Distribution

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

18. Business Statistics (BUSA 3101). Dr.Lari H. Arjomand Standard Normal Probability Distribution Example Continued Pep Zone 5w-20 Motor Oil  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  =6,  = 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.

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