The Normal Distribution Ch 9 Part b f

The Normal Distribution Ch. 9, Part b f (x ) x

Continuous Probability Distributions n n A continuous random variable can assume any real number value within some interval on the real line or in a collection of intervals. It is not possible to talk about the probability of the random variable assuming a particular value. Instead, we talk about the probability of the random variable assuming a value within a given interval. The probability of the random variable assuming a value within some given interval from x 1 to x 2 is defined to be the area under the graph of the probability density function between x 1 and x 2. ECO 3401 – B. Potter 2

Normal Probability Distribution n Graph of the Normal Probability Density Function f (x ) x 1 x 2 ECO 3401 – B. Potter x 3

Normal Probability Distribution Characteristics of the Normal Probability Distribution · The shape of the normal curve is often illustrated as a bell-shaped curve. · Two parameters Ø (mean) determines the location of the distribution. It can be any numerical value: negative, zero, or positive. Ø (standard deviation) determines the width of the curve: larger values result in wider, flatter curves. · The highest point on the normal curve is at the mean, which is also the median and mode. ECO 3401 – B. Potter 4

Normal Probability Distribution Characteristics of the Normal Probability Distribution · The normal curve is symmetric. · The total area under the curve is 1 (. 5 to the left of the mean and. 5 to the right). · Probabilities for the normal random variable are given by areas under the curve. ECO 3401 – B. Potter 5

Empirical Rule (68 – 95 – 99. 7 Rule) n % of Values in Some Commonly Used Intervals 68. 26% of values of a normal random variable are within +/- 1 standard deviation of its mean. · 95. 44% of values of a normal random variable are within +/- 2 standard deviations of its mean. · 99. 72% of values of a normal random variable are within +/- 3 standard deviations of its mean. · ECO 3401 – B. Potter 6

Empirical Rule (68 – 95 – 99. 7 Rule) f (x ) x 6 x 4 x 2 3 2 1 x 3 x 5 1 2 3 x σ 68. 26% ECO 3401 – B. Potter 7

Empirical Rule (68 – 95 – 99. 7 Rule) f (x ) x 6 x 4 x 2 3 2 1 x 3 x 5 1 2 3 x σ 95. 44% ECO 3401 – B. Potter 8

Empirical Rule (68 – 95 – 99. 7 Rule) f (x ) x 6 x 4 x 2 3 2 1 x 3 x 5 1 2 3 x σ 99. 72% ECO 3401 – B. Potter 9

Standard Normal Probability Distribution n A normal probability distribution with a mean of zero and a standard deviation of one. The letter z is commonly used to designate the standard normal random variable. Converting to the Standard Normal Distribution We can think of z as a measure of the number of. standard deviations x is from . n ECO 3401 – B. Potter 10

Standard Normal Probability Distribution P(z ) =0 =1 P = 1. 0 0 ECO 3401 – B. Potter z 11

Standard Normal Probability Distribution P(z ) =0 =1 P(z > 0) =. 5 P(z < 0) =. 5 0 ECO 3401 – B. Potter z 12

Standard Normal Probability Distribution P(z ) =0 =1 0 1 ECO 3401 – B. Potter z 13

Standard Normal Probability Table z - Table ECO 3401 – B. Potter 14

Standard Normal Probability Table z - Table ECO 3401 – B. Potter 15

Standard Normal Probability Distribution f (z ) =0 =1 0 1 ECO 3401 – B. Potter z 16

Standard Normal Probability Distribution f (z ) =0 =1 0 1 ECO 3401 – B. Potter z 17

Standard Normal Probability Distribution f (z ) =0 =1 0 1 ECO 3401 – B. Potter z 18

Standard Normal Probability Distribution f (z ) =0 =1 0 1 ECO 3401 – B. Potter z 19

Using the Standard Normal Probability Table ECO 3401 – B. Potter 20

n Using the Standard Normal Probability Table (Table 1) ECO 3401 – B. Potter 21

Using the Standard Normal Probability Table ECO 3401 – B. Potter 22

Example Annual salaries for sales associates from a particular store have a mean of $32, 500 and a standard deviation of $2, 500. a. Suppose that the distribution of annual salaries follows a normal distribution. Calculate and interpret the z-score for a sales associate who makes $36, 000. b. Calculate the probability that a randomly selected sales associate earns $30, 000 or less. ECO 3401 – B. Potter 23

Example Annual salaries for sales associates from a particular store have a mean of $32, 500 and a standard deviation of $2, 500. c. Calculate the percentage of sales associates with salaries between $35, 625 and $38, 750. ECO 3401 – B. Potter 24

Example: Pep Zone n 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. The store manager is concerned that sales are being lost due to stock-outs while waiting for an order. It has been determined that lead-time demand (x) 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 stock-out, P(x 20). ECO 3401 – B. Potter 25

Example: Pep Zone f (x ) = 15 =6 15 ECO 3401 – B. Potter x 26

Example: Pep Zone f (x ) = 15 =6 P(x 20) 15 20 0 ECO 3401 – B. Potter x z 27

Example: Pep Zone f (x ) = 15 =6 15 20 0. 83 ECO 3401 – B. Potter x z 28

Example: Pep Zone n Standard Normal Probability Distribution The Standard Normal table shows an area of. 7967 for the region below z =. 83. The shaded tail area is 1. 00. 7967 =. 2033. The probability of a stock-out is. 2033. Area =. 7967 Area = 1. 00 -. 7967 =. 2033 0. 83 ECO 3401 – B. Potter z 29

Example: Pep Zone n Standard Normal Probability Distribution If the manager of Pep Zone wants the probability of a stock-out to be no more than. 05, what should the reorder point be? Area =. 95 0 Area =. 05 z. 05 Let z. 05 represent the z value cutting the. 05 tail area. ECO 3401 – B. Potter 30

Example: Pep Zone n Using the Standard Normal Probability Table We now look-up the. 9500 area in the Standard Normal Probability table to find the corresponding z. 05 value. ECO 3401 – B. Potter 31

Example: Pep Zone n Using the Standard Normal Probability Table We now look-up the. 9500 area in the Standard Normal Probability table to find the corresponding z. 05 value. ECO 3401 – B. Potter 32

Example: Pep Zone n Using the Standard Normal Probability Table We now look-up the. 9500 area in the Standard Normal Probability table to find the corresponding z. 05 value. z. 05 = 1. 645 is a reasonable estimate. ECO 3401 – B. Potter 33

Example: Pep Zone n Standard Normal Probability Distribution If the manager of Pep Zone wants the probability of a stock-out to be no more than. 05, what should the reorder point be? Area =. 95 0 ECO 3401 – B. Potter Area =. 05 1. 645 z 34

Example: Pep Zone n Standard Normal Probability Distribution The corresponding value of x is given by n A reorder point of 24. 87 gallons will place the probability of a stock-out during lead-time at. 05. Perhaps Pep Zone should set the reorder point at 25 gallons to keep the probability under. 05. ECO 3401 – B. Potter 35

Now You Try n Annual salaries for sales associates from a particular store have a mean of $32, 500 and a standard deviation of $2, 500. How much does a sales associate make if no more than 5% of the sales associates make more than he or she? ECO 3401 – B. Potter 36

Now You Try n a) b) A certain type of light bulb has an average life of 500 hours with a standard deviation of 100 hours. The length of life of the bulb can be closely approximated by a normal curve. An amusement park buys and installs 10, 000 such bulbs. Find the total number that can be expected to last … between 650 and 780 hours. more than 300 hours. ECO 3401 – B. Potter 37

End of Chapter 9, Part b ECO 3401 – B. Potter 38