Unformatted text preview: Chapter 6 (6.2) Normal Probability Distribution
s s Continuous Probability Distributions Normal Probability Distribution f( x ) µ x Slide 1 Continuous Probability Distributions
s s s s A continuous random variable can assume any value in an 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 x1 to x2 is defined to be the area under the graph of the probability density function between x1 and x2. Slide 2 Normal Probability Distribution
s Graph of the Normal Probability Density Function f( x ) µ x Slide 3 Normal Probability Distribution
s Characteristics of the Normal Probability Distribution • The shape of the normal curve is often illustrated as a bellshaped curve. • Two parameters, µ (mean) and σ (standard deviation), determine the location and shape of the distribution. • The highest point on the normal curve is at the mean, which is also the median and mode. • The mean can be any numerical value: negative, zero, or positive. … continued Slide 4 Normal Probability Distribution
s Characteristics of the Normal Probability Distribution • The normal curve is symmetric. • The standard deviation determines the width of the curve: larger values result in wider, flatter curves. • 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. Slide 5 Normal Probability Distribution
s % 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. Slide 6 Normal Probability Distribution
s Normal Probability Density Function 1 − ( x − µ )2 / 2σ2 f ( x) = e 2 πσ
where: µ = mean σ = standard deviation π = 3.14159 e = 2.71828 Slide 7 Standard Normal Probability Distribution
s s s A random variable that has a normal distribution with a mean of zero and a standard deviation of one is said to have a standard normal probability distribution. The letter z is commonly used to designate this normal random variable. Converting to the Standard Normal Distribution x−µ z= σ
s We can think of z as a measure of the number of standard deviations x is from µ . Slide 8 Example: Pep Zone
Standard Normal Probability Distribution Pep Zone sells auto parts and supplies including a popular multigrade 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 stockouts while waiting for an order. It has been determined that leadtime demand 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). s Slide 9 Example: Pep Zone
Standard Normal Probability Distribution The Standard Normal table shows an area of .2967 for the region between the z = 0 and z = .83 lines below. The shaded tail area is .5 .2967 = .2033. The probability of a stockout is .2033. z = (x µ )/σ Area = .2967 = (20 15)/6 = .83
s Area = .5 .2967 = .2033 Area = .5 0 .83 z Slide 10 Example: Pep Zone
s Using the Standard Normal Probability Table
.00 .01 .02 .03 .04 .05 .06 .07 .08 .09 z .0 .0000 .0040 .0080 .0120 .0160 .0199 .0239 .0279 .0319 .0359 .1 .0398 .0438 .0478 .0517 .0557 .0596 .0636 .0675 .0714 .0753 .2 .0793 .0832 .0871 .0910 .0948 .0987 .1026 .1064 .1103 .1141 .3 .1179 .1217 .1255 .1293 .1331 .1368 .1406 .1443 .1480 .1517 .4 .5 .6 .7 .8 .9 .1554 .1591 .1628 .1664 .1700 .1736 .1772 .1808 .1844 .1879 .1915 .1950 .1985 .2019 .2054 .2088 .2123 .2157 .2190 .2224 .2257 .2291 .2324 .2357 .2389 .2422 .2454 .2486 .2518 .2549 .2580 .2612 .2642 .2673 .2704 .2734 .2764 .2794 .2823 .2852 .2881 .2910 .2939 .2967 .2995 .3023 .3051 .3078 .3106 .3133 .3159 .3186 .3212 .3238 .3264 .3289 .3315 .3340 .3365 .3389 Slide 11 Example: Pep Zone
s Standard Normal Probability Distribution If the manager of Pep Zone wants the probability of a stockout to be no more than .05, what should the reorder point be? Area = .05 Area = .5 Area = .45 z.05 0 Let z.05 represent the z value cutting the .05 tail area. Slide 12 Example: Pep Zone
s Using the Standard Normal Probability Table We now lookup the .4500 area in the Standard Normal Probability table to find the corresponding z.05 value.
z . .00 . .01 . .02 . .03 . .04 . .05 . .06 . .07 . .08 . .09 . 1.5 .4332 .4345 .4357 .4370 .4382 .4394 .4406 .4418 .4429 .4441 1.6 .4452 .4463 .4474 .4484 .4495 .4505 .4515 .4525 .4535 .4545 1.7 .4554 .4564 .4573 .4582 .4591 .4599 .4608 .4616 .4625 .4633 1.8 .4641 .4649 .4656 .4664 .4671 .4678 .4686 .4693 .4699 .4706 1.9 .4713 .4719 .4726 .4732 .4738 .4744 .4750 .4756 .4761 .4767 . . . . . . . . . . . z.05 = 1.645 is a reasonable estimate. Slide 13 Example: Pep Zone
s Standard Normal Probability Distribution The corresponding value of x is given by x = µ + z.05σ = 15 + 1.645(6) = 15 + 1.645(6) = 24.87 A reorder point of 24.87 gallons will place the probability of a stockout during leadtime at .05. Perhaps Pep Zone should set the reorder point at 25 gallons to keep the probability under .05. Slide 14 ...
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 Spring '08
 Priya
 25 gallons, 6 gallons, 20 gallons, 15 gallons, .07%, .2967

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