1. Continuous Distributions Chapter 6 MSIS 111 Prof. Nick Dedeke

2. Learning Objectives Appreciate the importance of the normal distribution. Recognize normal distribution problems, and know how to solve them. Decide when to use the normal distribution to approximate binomial distribution problems, and know how to work them. Decide when to use the exponential distribution to solve problems in business, and know how to work them.

3. Probabilities If a man had six children and he wanted to go a Red Sox game with one of them, what is the likelihood that he will choose Buba? A.1 B.6 C.1/6 D. I have no idea!

4. Probability Example 1 If a man had six children and he wanted to go a Red Sox game with one of them, what is the likelihood that he will choose Buba? Probability of event A = Number of ways event A could occur Number of events in the sample space Events that could occur (EVENTS SAMPLE SPACE)  He takes Buba  He takes Maria  He takes Bakuba  He takes Melinda  He takes Susan  He takes Elisha Probability (A= He takes Buba) = 1/6

5. Probability Example 2 If a man had ten bills in his pocket. Two $20 bills, three $100 bills and five $ 50 bills. He wanted to buy a baseball for $20. What is the probability that if he drew a bill out of his pocket it would be a $50 bill? Probability of event A = Number of ways event A could occur Number of events in the sample space Events that could occur (EVENTS SAMPLE SPACE)  He draws out a $20 bill  He draws out a $50 bill  He draws out a $100 bill Probability (A= It is a $50 ) = 5/(5+3+2) = 0.5

6. Probability Example 3 If a man had ten bills in his pocket. Two $20 bills, three $100 bills and five $ 50 bills. He wanted to buy a baseball for $20. What is the probability that if he drew a bill out of his pocket it would be a $20 bill? Probability of event A = Number of ways event A could occur Number of events in the sample space Events that could occur (EVENTS SAMPLE SPACE)  He draws out a $20 bill  He draws out a $50 bill  He draws out a $100 bill Probability (A= It is a $20 ) = 2/(5+3+2) = 0.2

7. Probability (Freq. Table) Example 3 If a man had ten bills in his pocket. Two $20 bills, three $100 bills and five $ 50 bills. He wanted to buy a baseball for $20. What is the probability that if he drew a bill out of his pocket it would be a $50 bill? Probability of event A = Number of ways event A could occur Number of events in the sample space Xi Fi Rel. Freq or P(x) $20 2 2/10 = 0.2 $50 5 5/10 = 0.5 $100 3 3/10 = 0.3 N 10 1 Probability (x = Bill is a $20 ) = 0.2 Probability (x = Bill is $50 or less) = 0.2 + 0.5 = 0.7 Probability (x = Bill is > $50) = 1 - Probability (x = Bill is $50 or less) = 1 – 0.7 = 0.3

8. Probability and Areas & Curve Example 5 Xi FiRel. Freq or P(x) $2022/10 = 0.2 $5055/10 = 0.5 $10033/10 = 0.3 10 1.0 Probability (B= Bill is $50 or less) = 0.2 + 0.5 = 0.7 P(x) 1.0 $100 $50 $20 0.5

9. Normal Distribution Probably the most widely known and used of all distributions is the normal distribution. It fits many human characteristics, such as height, weigh, length, speed, IQ scores, scholastic achievements, and years of life expectancy, among others. Many things in nature such as trees, animals, insects, and others have many characteristics that are normally distributed. It is used for measured data not counted ones

10. Normal Distribution Many variables in business and industry are also normally distributed. For example variables such as the annual cost of household insurance, the cost per square foot of renting warehouse space, and managers’ satisfaction with support from ownership on a five-point scale, amount of fill in soda cans, etc. Because of the many applications, the normal distribution is an extremely important distribution.

11. Normal Distribution Discovery of the normal curve of errors is generally credited to mathematician and astronomer Karl Gauss (1777 – 1855), who recognized that the errors of repeated measurement of objects are often normally distributed. Thus the normal distribution is sometimes referred to as the Gaussian distribution or the normal curve of errors.

12. Properties of the Normal Distribution The normal distribution exhibits the following characteristics: It is a continuous distribution. It is symmetric about the mean. It is asymptotic to the horizontal axis. It is unimodal. It is a family of curves. Area under the curve is 1. It is bell-shaped.

13. Graphic Representation of the Normal Distribution

14. Probability Density of the Normal Distribution

15. Family of Normal Curves

16. Standardized Normal Distribution Since there is an infinite number of combinations for  and , then we can generate an infinite family of curves. Because of this, it would be impractical to deal with all of these normal distributions. z distribution Fortunately, a mechanism was developed by which all normal distributions can be converted into a single distribution called the z distribution. standardized normal distribution This process yields the standardized normal distribution (or curve).

17. Standardized Normal Distribution The conversion formula for any x value of a given normal distribution is given below. It is called the z-score. A z-score gives the number of standard deviations that a value x, is above or below the mean.

18. Standardized Normal Distribution If x is normally distributed with a mean of  and a standard deviation of , then the z- score will also be normally distributed with a mean of 0 and a standard deviation of 1. Since we can covert to this standard normal distribution, tables have been generated for this standard normal distribution which will enable us to determine probabilities for normal variables. The tables in the text are set up to give the probabilities between z = 0 and some other z value, z 0 say, which is depicted on the next slide.

19. Standardized Normal Distribution

20. Z Table Second Decimal Place in Z Z0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.000.00000.00400.00800.01200.01600.01990.02390.02790.03190.0359 0.100.03980.04380.04780.05170.05570.05960.06360.06750.07140.0753 0.200.07930.08320.08710.09100.09480.09870.10260.10640.11030.1141 0.300.11790.12170.12550.12930.13310.13680.14060.14430.14800.1517 0.900.31590.31860.32120.32380.32640.32890.33150.33400.33650.3389 1.000.34130.34380.34610.34850.35080.35310.35540.35770.35990.3621 1.100.36430.36650.36860.37080.37290.37490.37700.37900.38100.3830 1.200.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015 2.000.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817 3.000.49870.49870.49870.49880.49880.49890.49890.49890.49900.4990 3.400.49970.49970.49970.49970.49970.49970.49970.49970.49970.4998 3.500.49980.49980.49980.49980.49980.49980.49980.49980.49980.4998

21. Applying the Z Formula Z0.00 0.01 0.02 0.000.00000.00400.0080 0.100.03980.04380.0478 1.000.34130.34380.3461 1.100.36430.36650.3686 1.200.38490.38690.3888 What is the probability that a measured value would be less than or equal to 600 but greater than or equal to 485?

22. Applying the Z Formula 0.5 + 0.2123 = 0.7123 What is the probability that a measured value would be less than or equal to 550?

23. Applying the Z Formula 0.5 – 0.4803 = 0.0197

24. Applying the Z Formula 0.4738+ 0.3554 = 0.8292

25. Applying the Z Formula

26. Exercise 1 The weekly demand for bicycles has a mean of 5,000 and a standard deviation of 600. Assume normal distribution. What is the probability that the firm will sell between 3,000 and 6,000 in a week? There is 1% chance that the firm will sell more than what number of bicycles per week?

27. Exercise 2 A furniture store has to decide if the current range of prices of its goods is appropriate for its strategy. The mean amount that customers spend in the store is $ 3,500. The standard deviation is $ 550. Assuming that the distribution is normal, the firm wants its prices to cover persons about 40% of the customers around the mean. The current price range of goods is $4200 and $ 2500. Does the price range match the expectation?

28. Exercises