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

Continuous Probability Distributions 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. Potter2

Normal Probability Distribution Graph of the Normal Probability Density Function  x f(x)f(x)f(x)f(x) x1x1 x2x2 ECO 3401 – B. Potter3

Normal Probability Distribution  The shape of the normal curve is often illustrated as a bell-shaped curve.  Two parameters   (mean) determines the location of the distribution. It can be any numerical value: negative, zero, or positive.   (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. Characteristics of the Normal Probability Distribution ECO 3401 – B. Potter4

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. Characteristics of the Normal Probability Distribution ECO 3401 – B. Potter5

Empirical Rule (68 – 95 – 99.7 Rule)  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. % of Values in Some Commonly Used Intervals ECO 3401 – B. Potter6

 x f(x)f(x)f(x)f(x) σ x2x2 x4x4 x6x6 x1x1 x3x3 x5x % Empirical Rule (68 – 95 – 99.7 Rule) ECO 3401 – B. Potter7

 x f(x)f(x)f(x)f(x) σ x2x2 x4x4 x6x6 x1x1 x3x3 x5x Empirical Rule (68 – 95 – 99.7 Rule) ECO 3401 – B. Potter %

 x f(x)f(x)f(x)f(x) σ x2x2 x4x4 x6x6 x1x1 x3x3 x5x Empirical Rule (68 – 95 – 99.7 Rule) ECO 3401 – B. Potter %

Standard Normal Probability Distribution 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 . We can think of z as a measure of the number of. standard deviations x is from . ECO 3401 – B. Potter10

Standard Normal Probability Distribution  z  = 0  = 1 P = 1.0 ECO 3401 – B. Potter11 P(z)P(z)P(z)P(z)

Standard Normal Probability Distribution  z P(z)P(z)P(z)P(z)  = 0  = 1 P(z > 0) =.5 P(z < 0) =.5 ECO 3401 – B. Potter12

z    = 0  = 1 Standard Normal Probability Distribution ECO 3401 – B. Potter13 P(z)P(z)P(z)P(z)

Standard Normal Probability Table z - Table ECO 3401 – B. Potter14

Standard Normal Probability Table z - Table ECO 3401 – B. Potter15

z    = 0  = 1 f(z)f(z)f(z)f(z) Standard Normal Probability Distribution ECO 3401 – B. Potter16

z    = 0  = 1 f(z)f(z)f(z)f(z) Standard Normal Probability Distribution ECO 3401 – B. Potter17

z    = 0  = 1 f(z)f(z)f(z)f(z) Standard Normal Probability Distribution ECO 3401 – B. Potter18

z    = 0  = 1 f(z)f(z)f(z)f(z) Standard Normal Probability Distribution ECO 3401 – B. Potter19

Using the Standard Normal Probability Table ECO 3401 – B. Potter20

n Using the Standard Normal Probability Table (Table 1) ECO 3401 – B. Potter21

Using the Standard Normal Probability Table ECO 3401 – B. Potter22

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. Potter23

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. Potter24

Example: Pep Zone 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. Potter25

 x f(x)f(x)f(x)f(x)  = 15  = 6 Example: Pep Zone ECO 3401 – B. Potter26

 x f(x)f(x)f(x)f(x)  = 15  = 6  P ( x  20) Example: Pep Zone z  27ECO 3401 – B. Potter

 x f(x)f(x)f(x)f(x)  = 15  = 6  Example: Pep Zone z  ECO 3401 – B. Potter28

Standard Normal Probability Distribution The Standard Normal table shows an area of.7967 for the region below z =.83. The shaded tail area is = The probability of a stock-out is Area =.7967 Area = =.2033 =.2033 z Example: Pep Zone ECO 3401 – B. Potter29

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? Example: Pep Zone Area =.05 Area =.95 0 z.05 ECO 3401 – B. Potter30 Let z.05 represent the z value cutting the.05 tail area.

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. Example: Pep Zone ECO 3401 – B. Potter31

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. Example: Pep Zone ECO 3401 – B. Potter32

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 = is a reasonable estimate. Example: Pep Zone ECO 3401 – B. Potter33

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? Example: Pep Zone Area =.05 Area = z ECO 3401 – B. Potter34

Standard Normal Probability Distribution The corresponding value of x is given by A reorder point of 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. Example: Pep Zone ECO 3401 – B. Potter35

Now You Try 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. Potter36

Now You Try 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 … a) between 650 and 780 hours. b) more than 300 hours. ECO 3401 – B. Potter37

End of Chapter 9, Part b ECO 3401 – B. Potter38