1 © 2008 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 7 Random Variables & Probability Distributions

* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Chapter 7Random Variables&Probability Distributions

2008 Brooks/Cole, a division of Thomson Learning, Inc.

* 2008 Brooks/Cole, a division of Thomson Learning, Inc.7.1 Random VariablesA numerical variable whose value depends on the outcome of a chance experiment is called a random variable. A random variable associates a numerical value with each outcome of a chance experiment.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.A random variable is discrete if its set of possible values is a collection of isolated points on the number line.Discrete and Continuous Random VariablesA random variable is continuous if its set of possible values includes an entire interval on the number line.We will use lowercase letters, such as x and y, to represent random variables.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Examples1.Experiment: A fair die is rolledRandom Variable: The number on the up faceType: Discrete2.Experiment: A pair of fair dice are rolledRandom Variable: The sum of the up facesType: DiscreteAnother random variable: The smaller of the up facesType: Discrete


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Examples3.Experiment: A coin is tossed until the 1st head turns upRandom Variable: The number of the toss that the 1st head turns upType: Discrete4.Experiment: Choose and inspect a number of parts Random Variable: The number of defective partsType: Discrete


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.5.Experiment: Inspect a precision ground mirror (Hubbell?)Random Variable: The number of defects on the surface of the mirrorType: Discrete6.Experiment: Measure the resistance of a '5' ohm resistorRandom Variable: The resistance (in ohms)Type: ContinuousExamples


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.7.Experiment: Measure the voltage in a outlet in your roomRandom Variable: The voltageType: Continuous8.Experiment: Observe the amount of time it takes a bank teller to serve a customerRandom Variable: The timeType: ContinuousExamples


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.9.Experiment: Measure the time until the next customer arrives at a customer service windowRandom Variable: The timeType: Continuous10.Experiment: Inspect a randomly chosen circuit board from a production lineRandom Variable:1, if the circuit board is defective0, if the circuit board is not defectiveType: DiscreteExamples


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.7.2 Discrete Probability DistributionsThe probability distribution of a discrete random variable x gives the probability associated with each possible x value. Each probability is the limiting relative frequency of occurrence of the corresponding x value when the experiment is repeatedly performed.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.ExampleSuppose that 20% of the apples sent to a sorting line are Grade A. If 3 of the apples sent to this plant are chosen randomly, determine the probability distribution of the number of Grade A apples in a sample of 3 apples. Consider the tree diagram


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.The Results in Table Formor


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Results in Graphical Form(Probability Histogram)For a probability histogram, the area of a bar is the probability of obtaining that value associated with that bar.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.The probabilities pi must satisfy1. 0 pi 1 for each i2. p1 + p2 + ... + pk = 1The probability P(X in A) of any event is found by summing the pi for the outcomes xi making up A.Probability Distributions


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.ExampleThe number of items a given salesman sells per customer is a random variable. The table below is for a specific salesman (Wilbur) in a clothing store in the mall. The probability distribution of X is given below:Note:0 p(x) 1 for each xp(x) = 1 (the sum is over all values of x)


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Example - continuedThe probability that he sells at least three items to a randomly selected customer isP(X 3) = 0.12 + 0.10 + 0.05 + 0.03 = 0.30The probability that he sells at most three items to a randomly selected customer isP(X 3) = 0.20 + 0.35 +0.15 + 0.12 = 0.82The probability that he sells between (inclusive) 2 and 4 items to a randomly selected customer is P(2 X 4) = 0.15 + 0.12 + 0.10 = 0.37


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Probability HistogramA probability histogram has its vertical scale adjusted in a manner that makes the area associated with each bar equal to the probability of the event that the random variable takes on the value describing the bar.


Chart1

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Probability Histogram

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xp(x)0123456

00.20.20.350.150.120.10.050.03

10.35

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60.03

* 2008 Brooks/Cole, a division of Thomson Learning, Inc.7.3 Continuous Probability DistributionsIf one looks at the distribution of the actual amount of water (in ounces) in one gallon bottles of spring water they might see something such as


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Probability Distribution for a Continuous Random VariableA probability distribution for a continuous random variable x is specified by a mathematical function denoted by f(x) which is called the density function. The graph of a density function is a smooth curve (the density curve). The following requirements must be met:f(x) 0 The total area under the density curve is equal to 1.The probability that x falls in any particular interval is the area under the density curve that lies above the interval.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Some IllustrationsNotice that for a continuous random variable x, P(x = a) = 0 for any specific value a because the area above a point under the curve is a line segment and hence has o area. Specifically this means P(x < a) = P(x a).


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Some IllustrationsNote:P(a < x < b) = P(a x < b) = P(a < x b) = P(a x b)


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Some Illustrations


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Example


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.ExampleFind the probability that during a given 1 hour period between 3 and 4.5 ounces of crumbs are left on the restaurant floor.The probability is represented by the shaded area in the graph. Since that shaded area is a rectangle, area = (base)(height)=(1.5)(.25) = .375


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Method of Probability CalculationThe probability that a continuous random variable x lies between a lower limit a and an upper limit b isP(a < x < b) = (cumulative area to the left of b) (cumulative area to the left of a) = P(x < b) P(x < a)


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.7.4 Mean & Standard DeviationThe mean value of a random variable x, denoted by mx, describes where the probability distribution of x is centered.The standard deviation of a random variable x, denoted by sx, describes variability in the probability distribution. When sx is small, observed values of x will tend to be close to the mean value and when sx is large, there will be more variability in observed values.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.IllustrationsTwo distributions with the same standard deviation with different means.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.IllustrationsTwo distributions with the same means and different standard deviation.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Mean of a Discrete Random VariableThe mean value of a discrete random variable x, denoted by mx, is computed by first multiplying each possible x value by the probability of observing that value and then adding the resulting quantities. Symbolically,


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.ExampleA professor regularly gives multiple choice quizzes with 5 questions. Over time, he has found the distribution of the number of wrong answers on his quizzes is as follows


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.ExampleMultiply each x value by its probability and add the results to get mx.mx = 1.41


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Variance and Standard Deviation of a Discrete Random Variable


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Previous Example - continued


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.The Mean & Variance of a Linear Function


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.ExampleSuppose x is the number of sales staff needed on a given day. If the cost of doing business on a day involves fixed costs of $255 and the cost per sales person per day is $110, find the mean cost (the mean of x or mx) of doing business on a given day where the distribution of x is given below.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Example continuedWe need to find the mean of y = 255 + 110x


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Example continuedWe need to find the variance and standard deviation of y = 255 + 110x


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Means and Variances for Linear CombinationsIf x1, x2, , xn are random variables and a1, a2, , an are numerical constants, the random variable y defined as y = a1x1 + a2x2 + + anxnis a linear combination of the xis.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.1.my = a1m1 + a2m2 + + anmn (This is true for any random variables with no conditions.)Means and Variances for Linear Combinations


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.ExampleA distributor of fruit baskets is going to put 4 apples, 6 oranges and 2 bunches of grapes in his small gift basket. The weights, in ounces, of these items are the random variables x1, x2 and x3 respectively with means and standard deviations as given in the following table.Find the mean, variance and standard deviation of the random variable y = weight of fruit in a small gift basket.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Example continuedIt is reasonable in this case to assume that the weights of the different types of fruit are independent.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Another ExampleSuppose 1 lb boxes of Sugar Treats cereal have a weight distribution with a mean T=1.050 lbs and standard deviation T=.051 lbs and 1 lb boxes of Sour Balls cereal have a weight distribution with a mean B=1.090 lbs and standard deviation B=.087 lbs. If a promotion is held where the customer is sold a shrink wrapped package containing 1 lb boxes of both Sugar Treats and Sour Balls cereals, what is the mean and standard deviation for the distribution of promotional packages.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Another Example - continued


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.7.5 The Binomial DistributionProperties of a Binomial ExperimentIt consists of a fixed number of observations called trials.Each trial can result in one of only two mutually exclusive outcomes labeled success (S) and failure (F).Outcomes of different trials are independent.The probability that a trial results in S is the same for each trial.The binomial random variable x is defined as x = number of successes observed when experiment is performedThe probability distribution of x is called the binomial probability distribution.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.The Binomial DistributionLet n = number of independent trials in a binomial experimentp = constant probability that any particular trial results in a success.Then


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.ExampleThe adult population of a large urban area is 60% black. If a jury of 12 is randomly selected from the adults in this area, what is the probability that precisely 7 jurors are black.Clearly, n=12 and p=.6, so


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Example - continuedThe adult population of a large urban area is 60% black. If a jury of 12 is randomly selected from the adults in this area, what is the probability that less than 3 are black.Clearly, n = 12 and = 0.6, so


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Another Examplea)What is the probability that exactly two will respond favorably to this sales pitch?On the average, 1 out of 19 people will respond favorably to a certain telephone solicitation. If 25 people are called,


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Another Example continuedOn the average, 1 out of 19 people will respond favorably to a certain telephone sales pitch. If 25 people are called, b)What is the probability that at least two will respond favorably to this sales pitch?


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Mean & Standard Deviation of a Binomial Random VariableThe mean value and the standard deviation of a binomial random variable are, respectively,


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.ExampleA professor routinely gives quizzes containing 50 multiple choice questions with 4 possible answers, only one being correct. Occasionally he just hands the students an answer sheet without giving them the questions and asks them to guess the correct answers. Let x be a random variable defined byx = number of correct answers on such an examFind the mean and standard deviation for x


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Example - solutionThe random variable is clearly binomial with n = 50 and p = . The mean and standard deviation of x are


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.The Geometric DistributionSuppose an experiment consists of a sequence of trials with the following conditions:The trials are independent.Each trial can result in one of two possible outcomes, success and failure.The probability of success is the same for all trials.

A geometric random variable is defined asx = number of trials until the first success is observed (including the success trial)

The probability distribution of x is called the geometric probability distribution.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.The Geometric DistributionIf x is a geometric random variable with probability of success = p for each trial, then p(x) = (1 )x-1 x = 1, 2, 3,


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.ExampleOver a very long period of time, it has been noted that on Fridays 25% of the customers at the drive-in window at the bank make deposits.

What is the probability that it takes 4 customers at the drive-in window before the first one makes a deposit.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Example - solutionThis problem is a geometric distribution problem with = 0.25.

Let x = number of customers at the drive-in window before a customer makes a deposit.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.7.6 Normal DistributionsTwo characteric values (numbers) completely determine a normal distributionMean - Standard deviation -


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Normal Distributions


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m=2, s=1

m=3, s=1

Normal Distributions s = 1

Data Sigma=1

xm=0, s=1m=-1, s=1m=1, s=1m=2, s=1m=3, s=10-1123

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0.50.350.130.350.130.02

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0.90.270.070.400.220.04

1.00.240.050.400.240.05

1.10.220.040.400.270.07

1.20.190.040.390.290.08

1.30.170.030.380.310.09

1.40.150.020.370.330.11

1.50.130.020.350.350.13

1.60.110.010.330.370.15

1.70.090.010.310.380.17

1.80.080.010.290.390.19

1.90.070.010.270.400.22

2.00.050.000.240.400.24

2.10.040.000.220.400.27

2.20.040.000.190.390.29

2.30.030.000.170.380.31

2.40.020.000.150.370.33

2.50.020.000.130.350.35

2.60.010.000.110.330.37

2.70.010.000.090.310.38

2.80.010.000.080.290.39

2.90.010.000.070.270.40

3.00.000.000.050.240.40

3.10.000.000.040.220.40

3.20.000.000.040.190.39

3.30.000.000.030.170.38

3.40.000.000.020.150.37

3.50.000.000.020.130.35

3.60.000.000.010.110.33

3.70.000.000.010.090.31

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3.90.000.000.010.070.27

4.00.000.000.000.050.24

4.10.000.000.000.040.22

4.20.000.000.000.040.19

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4.50.000.000.000.020.13

4.60.000.000.000.010.11

4.70.000.000.000.010.09

4.80.000.000.000.010.08

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5.00.000.000.000.000.05

5.10.000.000.000.000.04

5.20.000.000.000.000.04

5.30.000.000.000.000.03

5.40.000.000.000.000.02

5.50.000.000.000.000.02

5.60.000.000.000.000.01

5.70.000.000.000.000.01

5.80.000.000.000.000.01

5.90.000.000.000.000.01

6.00.000.000.000.000.00

Graphs Mean-0

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m=0, s=1

m=0, s=0.5

m=0, s=0.25

m=0, s=2

m=0, s=3

Normal Distributionsm=0

Data Mean=0

xm=0, s=1m=0, s=0.5m=0, s=0.25m=0, s=2m=0, s=300000

-5.00.000.000.000.010.0310.50.2523

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0.90.270.160.000.180.13

1.00.240.110.000.180.13

1.10.220.070.000.170.12

1.20.190.040.000.170.12

1.30.170.030.000.160.12

1.40.150.020.000.160.12

1.50.130.010.000.150.12

1.60.110.000.000.140.12

1.70.090.000.000.140.11

1.80.080.000.000.130.11

1.90.070.000.000.130.11

2.00.050.000.000.120.11

2.10.040.000.000.110.10

2.20.040.000.000.110.10

2.30.030.000.000.100.10

2.40.020.000.000.100.10

2.50.020.000.000.090.09

2.60.010.000.000.090.09

2.70.010.000.000.080.09

2.80.010.000.000.070.09

2.90.010.000.000.070.08

3.00.000.000.000.060.08

3.10.000.000.000.060.08

3.20.000.000.000.060.08

3.30.000.000.000.050.07

3.40.000.000.000.050.07

3.50.000.000.000.040.07

3.60.000.000.000.040.06

3.70.000.000.000.040.06

3.80.000.000.000.030.06

3.90.000.000.000.030.06

4.00.000.000.000.030.05

4.10.000.000.000.020.05

4.20.000.000.000.020.05

4.30.000.000.000.020.05

4.40.000.000.000.020.05

4.50.000.000.000.020.04

4.60.000.000.000.010.04

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4.90.000.000.000.010.04

5.00.000.000.000.010.03

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5.20.000.000.000.010.03

5.30.000.000.000.010.03

5.40.000.000.000.010.03

5.50.000.000.000.000.02

5.60.000.000.000.000.02

5.70.000.000.000.000.02

5.80.000.000.000.000.02

5.90.000.000.000.000.02

6.00.000.000.000.000.02

Sheet3

* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Normal Distributions


Graphs Sigma=1

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0.00172256890.00003853520.02832703770.1713685920.3813878155

0.00123221920.00002494250.02239453030.14972746560.3682701403

0.00087268270.00001598370.01752830050.12951759570.3520653268

0.00061190190.00001014090.01358296920.11092083470.3332246029

0.00042478030.00000636980.01042093480.09404907740.3122539334

0.00029194690.00000396130.00791545160.07895015830.2896915528

0.00019865550.0000024390.00595253240.06561581480.2660852499

0.00013383020.00000148670.00443184840.05399096650.2419707245

0.00008926170.00000089720.00326681910.0439835960.217852177

0.00005894310.00000053610.00238408820.03547459280.194186055

0.00003853520.00000031710.00172256890.02832703770.171368592

0.00002494250.00000018570.00123221920.02239453030.1497274656

0.00001598370.00000010770.00087268270.01752830050.1295175957

0.00001014090.00000006180.00061190190.01358296920.1109208347

0.00000636980.00000003510.00042478030.01042093480.0940490774

0.00000396130.00000001980.00029194690.00791545160.0789501583

0.0000024390.0000000110.00019865550.00595253240.0656158148

0.00000148670.00000000610.00013383020.00443184840.0539909665

0.00000089720.00000000330.00008926170.00326681910.043983596

0.00000053610.00000000180.00005894310.00238408820.0354745928

0.00000031710.0000000010.00003853520.00172256890.0283270377

0.00000018570.00000000050.00002494250.00123221920.0223945303

0.00000010770.00000000030.00001598370.00087268270.0175283005

0.00000006180.00000000010.00001014090.00061190190.0135829692

0.00000003510.00000000010.00000636980.00042478030.0104209348

0.000000019800.00000396130.00029194690.0079154516

0.00000001100.0000024390.00019865550.0059525324

0.000000006100.00000148670.00013383020.0044318484

m=0, s=1

m=-1, s=1

m=1, s=1

m=2, s=1

m=3, s=1

Normal Distributions: s=1

Data Sigma=1

xm=0, s=1m=-1, s=1m=1, s=1m=2, s=1m=3, s=10-1123

-5.00.000.000.000.000.0011111

-4.90.000.000.000.000.00

-4.80.000.000.000.000.00

-4.70.000.000.000.000.00

-4.60.000.000.000.000.00

-4.50.000.000.000.000.00

-4.40.000.000.000.000.00

-4.30.000.000.000.000.00

-4.20.000.000.000.000.00

-4.10.000.000.000.000.00

-4.00.000.000.000.000.00

-3.90.000.010.000.000.00

-3.80.000.010.000.000.00

-3.70.000.010.000.000.00

-3.60.000.010.000.000.00

-3.50.000.020.000.000.00

-3.40.000.020.000.000.00

-3.30.000.030.000.000.00

-3.20.000.040.000.000.00

-3.10.000.040.000.000.00

-3.00.000.050.000.000.00

-2.90.010.070.000.000.00

-2.80.010.080.000.000.00

-2.70.010.090.000.000.00

-2.60.010.110.000.000.00

-2.50.020.130.000.000.00

-2.40.020.150.000.000.00

-2.30.030.170.000.000.00

-2.20.040.190.000.000.00

-2.10.040.220.000.000.00

-2.00.050.240.000.000.00

-1.90.070.270.010.000.00

-1.80.080.290.010.000.00

-1.70.090.310.010.000.00

-1.60.110.330.010.000.00

-1.50.130.350.020.000.00

-1.40.150.370.020.000.00

-1.30.170.380.030.000.00

-1.20.190.390.040.000.00

-1.10.220.400.040.000.00

-1.00.240.400.050.000.00

-0.90.270.400.070.010.00

-0.80.290.390.080.010.00

-0.70.310.380.090.010.00

-0.60.330.370.110.010.00

-0.50.350.350.130.020.00

-0.40.370.330.150.020.00

-0.30.380.310.170.030.00

-0.20.390.290.190.040.00

-0.10.400.270.220.040.00

-0.00.400.240.240.050.00

0.10.400.220.270.070.01

0.20.390.190.290.080.01

0.30.380.170.310.090.01

0.40.370.150.330.110.01

0.50.350.130.350.130.02

0.60.330.110.370.150.02

0.70.310.090.380.170.03

0.80.290.080.390.190.04

0.90.270.070.400.220.04

1.00.240.050.400.240.05

1.10.220.040.400.270.07

1.20.190.040.390.290.08

1.30.170.030.380.310.09

1.40.150.020.370.330.11

1.50.130.020.350.350.13

1.60.110.010.330.370.15

1.70.090.010.310.380.17

1.80.080.010.290.390.19

1.90.070.010.270.400.22

2.00.050.000.240.400.24

2.10.040.000.220.400.27

2.20.040.000.190.390.29

2.30.030.000.170.380.31

2.40.020.000.150.370.33

2.50.020.000.130.350.35

2.60.010.000.110.330.37

2.70.010.000.090.310.38

2.80.010.000.080.290.39

2.90.010.000.070.270.40

3.00.000.000.050.240.40

3.10.000.000.040.220.40

3.20.000.000.040.190.39

3.30.000.000.030.170.38

3.40.000.000.020.150.37

3.50.000.000.020.130.35

3.60.000.000.010.110.33

3.70.000.000.010.090.31

3.80.000.000.010.080.29

3.90.000.000.010.070.27

4.00.000.000.000.050.24

4.10.000.000.000.040.22

4.20.000.000.000.040.19

4.30.000.000.000.030.17

4.40.000.000.000.020.15

4.50.000.000.000.020.13

4.60.000.000.000.010.11

4.70.000.000.000.010.09

4.80.000.000.000.010.08

4.90.000.000.000.010.07

5.00.000.000.000.000.05

5.10.000.000.000.000.04

5.20.000.000.000.000.04

5.30.000.000.000.000.03

5.40.000.000.000.000.02

5.50.000.000.000.000.02

5.60.000.000.000.000.01

5.70.000.000.000.000.01

5.80.000.000.000.000.01

5.90.000.000.000.000.01

6.00.000.000.000.000.00

Graphs Mean-0

0.0000014867000.00876415020.0331590463

0.000002439000.00991867720.0350338791

0.0000039613000.01119726510.0369736116

0.0000063698000.012609110.0389774095

0.0000101409000.01416351890.041044174

0.0000159837000.01586982590.0431725319

0.0000249425000.01773729640.0453608275

0.0000385352000.01977502080.0476071155

0.0000589431000.0219917980.0499091552

0.0000892617000.02439600930.0522644061

0.0001338302000.02699548330.0546700249

0.0001986555000.0297973530.057122864

0.0002919469000.03280790740.0596194722

0.0004247803000.03603243720.0621560962

0.0006119019000.03947507920.064728685

0.0008726827000.04313865940.0673328952

0.00123221920.000000000100.04702453870.0699640987

0.00172256890.000000000300.05113246230.0726173923

0.00238408820.00000000100.05546041730.0752876092

0.00326681910.000000003600.06000450030.0779693319

0.00443184840.000000012200.06475879780.0806569082

0.00595253240.000000039500.06971528320.0833444679

0.00791545160.000000123700.07486373280.086025942

0.01042093480.000000371500.08019166370.0886950833

0.01358296920.000001072200.0856842960.0913454891

0.01752830050.000002973400.09132454270.0939706251

0.02239453030.000007922600.09709302750.0965638509

0.02832703770.000020281700.10296813440.0991184469

0.03547459280.000049884900.10892608850.1016276423

0.0439835960.000117886100.11494107030.1040846445

0.05399096650.000267660500.12098536230.1064826685

0.06561581480.000583893900.12702952820.1088149683

0.07895015830.001223803900.13304262490.1110748676

0.09404907740.00246443830.00000000010.13899244310.1132557914

0.11092083470.00476817640.0000000020.14484577640.1153512977

0.12951759570.00886369680.00000002430.15056871610.1173551089

0.14972746560.01583090320.00000024730.15612696670.1192611431

0.1713685920.02716593850.00000214440.16148617980.1210635444

0.1941860550.04478906060.00001584520.16661230140.1227567134

0.2178521770.07094918570.00009976990.17147192750.1243353356

0.24197072450.1079819330.00053532090.17603266340.1257944092

0.26608524990.15790031660.00244760770.18026348120.1271292718

0.28969155280.22184166940.00953635280.18413507020.1283356249

0.31225393340.29945493130.03166180630.18762017350.1294095569

0.33322460290.388372110.08957812120.19069390770.1303475647

0.35206532680.4839414490.21596386610.19333405840.131146572

0.36827014030.57938310550.44368333870.1955213470.131803947

0.38138781550.66644920580.77674421990.19723966550.1323175158

0.3910426940.73654028061.1587662110.19847627370.1326855754

0.39695254750.7820853881.47308056120.1992219570.1329069025

0.39894228040.79788456081.59576912160.19947114020.1329807601

0.39695254750.7820853881.47308056120.1992219570.1329069025

0.3910426940.73654028061.1587662110.19847627370.1326855754

0.38138781550.66644920580.77674421990.19723966550.1323175158

0.36827014030.57938310550.44368333870.1955213470.131803947

0.35206532680.4839414490.21596386610.19333405840.131146572

0.33322460290.388372110.08957812120.19069390770.1303475647

0.31225393340.29945493130.03166180630.18762017350.1294095569

0.28969155280.22184166940.00953635280.18413507020.1283356249

0.26608524990.15790031660.00244760770.18026348120.1271292718

0.24197072450.1079819330.00053532090.17603266340.1257944092

0.2178521770.07094918570.00009976990.17147192750.1243353356

0.1941860550.04478906060.00001584520.16661230140.1227567134

0.1713685920.02716593850.00000214440.16148617980.1210635444

0.14972746560.01583090320.00000024730.15612696670.1192611431

0.12951759570.00886369680.00000002430.15056871610.1173551089

0.11092083470.00476817640.0000000020.14484577640.1153512977

0.09404907740.00246443830.00000000010.13899244310.1132557914

0.07895015830.001223803900.13304262490.1110748676

0.06561581480.000583893900.12702952820.1088149683

0.05399096650.000267660500.12098536230.1064826685

0.0439835960.000117886100.11494107030.1040846445

0.03547459280.000049884900.10892608850.1016276423

0.02832703770.000020281700.10296813440.0991184469

0.02239453030.000007922600.09709302750.0965638509

0.01752830050.000002973400.09132454270.0939706251

0.01358296920.000001072200.0856842960.0913454891

0.01042093480.000000371500.08019166370.0886950833

0.00791545160.000000123700.07486373280.086025942

0.00595253240.000000039500.06971528320.0833444679

0.00443184840.000000012200.06475879780.0806569082

0.00326681910.000000003600.06000450030.0779693319

0.00238408820.00000000100.05546041730.0752876092

0.00172256890.000000000300.05113246230.0726173923

0.00123221920.000000000100.04702453870.0699640987

0.0008726827000.04313865940.0673328952

0.0006119019000.03947507920.064728685

0.0004247803000.03603243720.0621560962

0.0002919469000.03280790740.0596194722

0.0001986555000.0297973530.057122864

0.0001338302000.02699548330.0546700249

0.0000892617000.02439600930.0522644061

0.0000589431000.0219917980.0499091552

0.0000385352000.01977502080.0476071155

0.0000249425000.01773729640.0453608275

0.0000159837000.01586982590.0431725319

0.0000101409000.01416351890.041044174

0.0000063698000.012609110.0389774095

0.0000039613000.01119726510.0369736116

0.000002439000.00991867720.0350338791

0.0000014867000.00876415020.0331590463

0.0000008972000.00772467360.0313496925

0.0000005361000.00679148460.0296061537

0.0000003171000.00595612180.0279285343

0.0000001857000.00521046740.0263167194

0.0000001077000.00454678130.0247703879

0.0000000618000.00395772580.0232890254

0.0000000351000.00343638330.0218719383

0.0000000198000.00297626620.0205182671

0.000000011000.00257132050.0192270005

0.0000000061000.00221592420.0179969888

m=0, s=1

m=0, s=2

m=0, s=3

m=0, s=0.5

m=0, s=0.25

m=0, s=1

m=0, s=0.5

m=0, s=0.25

m=0, s=2

m=0, s=3

Graphs Mean = 0

0.0000014867000.00876415020.0331590463

0.000002439000.00991867720.0350338791

0.0000039613000.01119726510.0369736116

0.0000063698000.012609110.0389774095

0.0000101409000.01416351890.041044174

0.0000159837000.01586982590.0431725319

0.0000249425000.01773729640.0453608275

0.0000385352000.01977502080.0476071155

0.0000589431000.0219917980.0499091552

0.0000892617000.02439600930.0522644061

0.0001338302000.02699548330.0546700249

0.0001986555000.0297973530.057122864

0.0002919469000.03280790740.0596194722

0.0004247803000.03603243720.0621560962

0.0006119019000.03947507920.064728685

0.0008726827000.04313865940.0673328952

0.00123221920.000000000100.04702453870.0699640987

0.00172256890.000000000300.05113246230.0726173923

0.00238408820.00000000100.05546041730.0752876092

0.00326681910.000000003600.06000450030.0779693319

0.00443184840.000000012200.06475879780.0806569082

0.00595253240.000000039500.06971528320.0833444679

0.00791545160.000000123700.07486373280.086025942

0.01042093480.000000371500.08019166370.0886950833

0.01358296920.000001072200.0856842960.0913454891

0.01752830050.000002973400.09132454270.0939706251

0.02239453030.000007922600.09709302750.0965638509

0.02832703770.000020281700.10296813440.0991184469

0.03547459280.000049884900.10892608850.1016276423

0.0439835960.000117886100.11494107030.1040846445

0.05399096650.000267660500.12098536230.1064826685

0.06561581480.000583893900.12702952820.1088149683

0.07895015830.001223803900.13304262490.1110748676

0.09404907740.00246443830.00000000010.13899244310.1132557914

0.11092083470.00476817640.0000000020.14484577640.1153512977

0.12951759570.00886369680.00000002430.15056871610.1173551089

0.14972746560.01583090320.00000024730.15612696670.1192611431

0.1713685920.02716593850.00000214440.16148617980.1210635444

0.1941860550.04478906060.00001584520.16661230140.1227567134

0.2178521770.07094918570.00009976990.17147192750.1243353356

0.24197072450.1079819330.00053532090.17603266340.1257944092

0.26608524990.15790031660.00244760770.18026348120.1271292718

0.28969155280.22184166940.00953635280.18413507020.1283356249

0.31225393340.29945493130.03166180630.18762017350.1294095569

0.33322460290.388372110.08957812120.19069390770.1303475647

0.35206532680.4839414490.21596386610.19333405840.131146572

0.36827014030.57938310550.44368333870.1955213470.131803947

0.38138781550.66644920580.77674421990.19723966550.1323175158

0.3910426940.73654028061.1587662110.19847627370.1326855754

0.39695254750.7820853881.47308056120.1992219570.1329069025

0.39894228040.79788456081.59576912160.19947114020.1329807601

0.39695254750.7820853881.47308056120.1992219570.1329069025

0.3910426940.73654028061.1587662110.19847627370.1326855754

0.38138781550.66644920580.77674421990.19723966550.1323175158

0.36827014030.57938310550.44368333870.1955213470.131803947

0.35206532680.4839414490.21596386610.19333405840.131146572

0.33322460290.388372110.08957812120.19069390770.1303475647

0.31225393340.29945493130.03166180630.18762017350.1294095569

0.28969155280.22184166940.00953635280.18413507020.1283356249

0.26608524990.15790031660.00244760770.18026348120.1271292718

0.24197072450.1079819330.00053532090.17603266340.1257944092

0.2178521770.07094918570.00009976990.17147192750.1243353356

0.1941860550.04478906060.00001584520.16661230140.1227567134

0.1713685920.02716593850.00000214440.16148617980.1210635444

0.14972746560.01583090320.00000024730.15612696670.1192611431

0.12951759570.00886369680.00000002430.15056871610.1173551089

0.11092083470.00476817640.0000000020.14484577640.1153512977

0.09404907740.00246443830.00000000010.13899244310.1132557914

0.07895015830.001223803900.13304262490.1110748676

0.06561581480.000583893900.12702952820.1088149683

0.05399096650.000267660500.12098536230.1064826685

0.0439835960.000117886100.11494107030.1040846445

0.03547459280.000049884900.10892608850.1016276423

0.02832703770.000020281700.10296813440.0991184469

0.02239453030.000007922600.09709302750.0965638509

0.01752830050.000002973400.09132454270.0939706251

0.01358296920.000001072200.0856842960.0913454891

0.01042093480.000000371500.08019166370.0886950833

0.00791545160.000000123700.07486373280.086025942

0.00595253240.000000039500.06971528320.0833444679

0.00443184840.000000012200.06475879780.0806569082

0.00326681910.000000003600.06000450030.0779693319

0.00238408820.00000000100.05546041730.0752876092

0.00172256890.000000000300.05113246230.0726173923

0.00123221920.000000000100.04702453870.0699640987

0.0008726827000.04313865940.0673328952

0.0006119019000.03947507920.064728685

0.0004247803000.03603243720.0621560962

0.0002919469000.03280790740.0596194722

0.0001986555000.0297973530.057122864

0.0001338302000.02699548330.0546700249

0.0000892617000.02439600930.0522644061

0.0000589431000.0219917980.0499091552

0.0000385352000.01977502080.0476071155

0.0000249425000.01773729640.0453608275

0.0000159837000.01586982590.0431725319

0.0000101409000.01416351890.041044174

0.0000063698000.012609110.0389774095

0.0000039613000.01119726510.0369736116

0.000002439000.00991867720.0350338791

0.0000014867000.00876415020.0331590463

0.0000008972000.00772467360.0313496925

0.0000005361000.00679148460.0296061537

0.0000003171000.00595612180.0279285343

0.0000001857000.00521046740.0263167194

0.0000001077000.00454678130.0247703879

0.0000000618000.00395772580.0232890254

0.0000000351000.00343638330.0218719383

0.0000000198000.00297626620.0205182671

0.000000011000.00257132050.0192270005

0.0000000061000.00221592420.0179969888

m=0, s=1

m=0, s=0.5

m=0, s=0.25

m=0, s=2

m=0, s=3

Normal Distributions m = 0

Data Mean=0

xm=0, s=1m=0, s=0.5m=0, s=0.25m=0, s=2m=0, s=300000

-5.00.000.000.000.010.0310.50.2523

-4.90.000.000.000.010.04

-4.80.000.000.000.010.04

-4.70.000.000.000.010.04

-4.60.000.000.000.010.04

-4.50.000.000.000.020.04

-4.40.000.000.000.020.05

-4.30.000.000.000.020.05

-4.20.000.000.000.020.05

-4.10.000.000.000.020.05

-4.00.000.000.000.030.05

-3.90.000.000.000.030.06

-3.80.000.000.000.030.06

-3.70.000.000.000.040.06

-3.60.000.000.000.040.06

-3.50.000.000.000.040.07

-3.40.000.000.000.050.07

-3.30.000.000.000.050.07

-3.20.000.000.000.060.08

-3.10.000.000.000.060.08

-3.00.000.000.000.060.08

-2.90.010.000.000.070.08

-2.80.010.000.000.070.09

-2.70.010.000.000.080.09

-2.60.010.000.000.090.09

-2.50.020.000.000.090.09

-2.40.020.000.000.100.10

-2.30.030.000.000.100.10

-2.20.040.000.000.110.10

-2.10.040.000.000.110.10

-2.00.050.000.000.120.11

-1.90.070.000.000.130.11

-1.80.080.000.000.130.11

-1.70.090.000.000.140.11

-1.60.110.000.000.140.12

-1.50.130.010.000.150.12

-1.40.150.020.000.160.12

-1.30.170.030.000.160.12

-1.20.190.040.000.170.12

-1.10.220.070.000.170.12

-1.00.240.110.000.180.13

-0.90.270.160.000.180.13

-0.80.290.220.010.180.13

-0.70.310.300.030.190.13

-0.60.330.390.090.190.13

-0.50.350.480.220.190.13

-0.40.370.580.440.200.13

-0.30.380.670.780.200.13

-0.20.390.741.160.200.13

-0.10.400.781.470.200.13

-0.00.400.801.600.200.13

0.10.400.781.470.200.13

0.20.390.741.160.200.13

0.30.380.670.780.200.13

0.40.370.580.440.200.13

0.50.350.480.220.190.13

0.60.330.390.090.190.13

0.70.310.300.030.190.13

0.80.290.220.010.180.13

0.90.270.160.000.180.13

1.00.240.110.000.180.13

1.10.220.070.000.170.12

1.20.190.040.000.170.12

1.30.170.030.000.160.12

1.40.150.020.000.160.12

1.50.130.010.000.150.12

1.60.110.000.000.140.12

1.70.090.000.000.140.11

1.80.080.000.000.130.11

1.90.070.000.000.130.11

2.00.050.000.000.120.11

2.10.040.000.000.110.10

2.20.040.000.000.110.10

2.30.030.000.000.100.10

2.40.020.000.000.100.10

2.50.020.000.000.090.09

2.60.010.000.000.090.09

2.70.010.000.000.080.09

2.80.010.000.000.070.09

2.90.010.000.000.070.08

3.00.000.000.000.060.08

3.10.000.000.000.060.08

3.20.000.000.000.060.08

3.30.000.000.000.050.07

3.40.000.000.000.050.07

3.50.000.000.000.040.07

3.60.000.000.000.040.06

3.70.000.000.000.040.06

3.80.000.000.000.030.06

3.90.000.000.000.030.06

4.00.000.000.000.030.05

4.10.000.000.000.020.05

4.20.000.000.000.020.05

4.30.000.000.000.020.05

4.40.000.000.000.020.05

4.50.000.000.000.020.04

4.60.000.000.000.010.04

4.70.000.000.000.010.04

4.80.000.000.000.010.04

4.90.000.000.000.010.04

5.00.000.000.000.010.03

5.10.000.000.000.010.03

5.20.000.000.000.010.03

5.30.000.000.000.010.03

5.40.000.000.000.010.03

5.50.000.000.000.000.02

5.60.000.000.000.000.02

5.70.000.000.000.000.02

5.80.000.000.000.000.02

5.90.000.000.000.000.02

6.00.000.000.000.000.02

Sheet3

* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Major PrincipleThe proportion or percentage of a normally distributed population that is in an interval depends only on how many standard deviations the endpoints are from the mean.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Standard Normal DistributionA normal distribution with mean 0 and standard deviation 1, is called the standard (or standardized) normal distribution.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.NormalTables


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Using the Normal TablesFor any number z* between 3.89 and 3.89 and rounded to two decimal places, Appendix Table II gives (Area under z curve to the left of z*) = P(z < z*) = P(z z*)where the letter z is used to represent a random variable whose distribution is the standard normal distribution


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Using the Normal TablesTo find this probability, locate the following:The row labeled with the sign of z* and the digit to either side of the decimal point The column identified with the second digit to the right of the decimal point in z*The number at the intersection of this row and column is the desired probability, P(z < z*).


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Using the Normal TablesFind P(z < 0.46)


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Using the Normal TablesFind P(z < -2.74)


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Sample Calculations Using the Standard Normal DistributionUsing the standard normal tables, find the proportion of observations (z values) from a standard normal distribution that satisfy each of the following:


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Sample Calculations Using the Standard Normal DistributionUsing the standard normal tables, find the proportion of observations (z values) from a standard normal distribution that satisfies each of the following:


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Symmetry PropertyNotice from the preceding examples it becomes obvious that P(z > z*) = P(z < -z*)P(z > -2.18) = P(z < 2.18) = 0.9854


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Sample Calculations Using the Standard Normal DistributionUsing the standard normal tables, find the proportion of observations (z values) from a standard normal distribution that satisfies -1.37 < z < 2.34, that is find P(-1.37 < z < 2.34).


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Sample Calculations Using the Standard Normal DistributionUsing the standard normal tables, find the proportion of observations (z values) from a standard normal distribution that satisfies 0.54 < z < 1.61, that is find P(0.54 < z < 1.61).


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Sample Calculations Using the Standard Normal DistributionUsing the standard normal tables, find the proportion of observations (z values) from a standard normal distribution that satisfy -1.42 < z < -0.93, that is find P(-1.42 < z < -0.93).


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Using the standard normal tables, in each of the following, find the z values that satisfy :

Example Calculation

The closest entry in the table to 0.9800 is 0.9798 corresponding to a z value of 2.05(a)The point z with 98% of the observations falling below it.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Using the standard normal tables, in each of the following, find the z values that satisfy :(b)The point z with 90% of the observations falling above it.Example CalculationThe closest entry in the table to 0.1000 is 0.1003 corresponding to a z value of -1.28


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Finding Normal ProbabilitiesTo calculate probabilities for any normal distribution, standardize the relevant values and then use the table of z curve areas. More specifically, if x is a variable whose behavior is described by a normal distribution with mean m and standard deviation s, then

P(x < b) = p(z < b*)P(x > a) = P(a* < z) = P(z > a*)

where z is a variable whose distribution is standard normal and


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Standard Normal Distribution RevisitedThis is called the standard normal distribution.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Conversion to N(0,1)Where is the true population mean and is the true population standard deviation


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Example 1A Company produces 20 ounce jars of a picante sauce. The true amounts of sauce in the jars of this brand sauce follow a normal distribution.Suppose the companies 20 ounce jars follow a N(20.2,0.125) distribution curve. (i.e., The contents of the jars are normally distributed with a true mean = 20.2 ounces with a true standard deviation = 0.125 ounces.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Example 1What proportion of the jars are under-filled (i.e., have less than 20 ounces of sauce)?

Looking up the z value -1.60 on the standard normal table we find the value 0.0548.The proportion of the sauce jars that are under-filled is 0.0548 (i.e., 5.48% of the jars contain less than 20 ounces of sauce.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.What proportion of the sauce jars contain between 20 and 20.3 ounces of sauce.

Looking up the z values of -1.60 and 0.80 we find the areas (proportions) 0.0548 and 0.7881Example 1


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.99% of the jars of this brand of picante sauce will contain more than what amount of sauce?When we try to look up 0.0100 in the body of the table we do not find this value. We do find the following

Since the relationship between the real scale (x) and the z scale is z=(x-)/ we solve for x getting x = + zx=20.2+(-2.33)(0.125)= 19.90875=19.91The entry closest to 0.0100 is 0.0099 corresponding to the z value -2.33Example 1


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.The weight of the cereal in a box is a random variable with mean 12.15 ounces, and standard deviation 0.2 ounce. What percentage of the boxes have contents that weigh under 12 ounces? Example 2


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.If the manufacturer claims there are 12 ounces in a box, does this percentage cause concern? If so, what could be done to correct the situation? The machinery could be reset with a higher or larger mean.The machinery could be replaced with machinery that has a smaller standard deviation of fills.The label on the box could be changed.Which would probably be cheaper immediately but might cost more in the long run?In the long run it might be cheaper to get newer more precise equipment and not give away as much excess cereal.Example 2


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Example 3a)What proportion of these printers will fail before 1,200 hours?The time to first failure of a unit of a brand of ink jet printer is approximately normally distributed with a mean of 1,500 hours and a standard deviation of 225 hours.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.b)What proportion of these printers will not fail within the first 2,000 hours?Example 3


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.c)What should be the guarantee time for these printers if the manufacturer wants only 5% to fail within the guarantee period?Notice that when you look for a proportion of 0.0500 in the body of the normal table you do not find it. However, you do find the value 0.0505 corresponding to a z score of -1.64 and value 0.0495 corresponding to the z score of -1.65. Since 0.0500 is exactly 1/2 way between -1.64 and -1.65, the z value we want is -1.645.Example 3


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.The guarantee period should be 1130 hours.Example 3


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.7.7 Normal PlotsA normal probability plot is a scatter plot of the (normal score*, observation) pairs.A substantially linear pattern in a normal probability plot suggests that population normality is plausible.A systematic departure from a straight-line pattern (such as curvature in the plot) casts doubt on the legitimacy of assuming a normal population distribution.*There are different techniques for determining the normal scores. Typically, we use a statistical package such as Minitab, SPSS or SAS to create a normal probability plot.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Normal Probability Plot ExampleTen randomly selected shut-ins were each asked to list how many hours of television they watched per week. The results are826690847588809411091Minitab obtained the normal probability plot on the following slide.


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Notice that the points all fall nearly on a line so it is reasonable to assume that the population of hours of tv watched by shut-ins is normally distributed.Normal Probability Plot Example


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Normal Probability Plot ExampleA sample of times of 15 telephone solicitation calls (in seconds) was obtained and is given below. 510712356514514322011685616 Minitab obtained the normal probability plot on the next slide


* 2008 Brooks/Cole, a division of Thomson Learning, Inc.Normal Probability Plot ExampleClearly the points do not fall nearly on a line. Specifically the pattern has a distinct nonlinear appearance. It is not reasonable to assume that the population of hours of tv watched by shut-ins is normally distributed. One would most assuredly say that the distribution of lengths of calls is not normal.


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Documents

1 © 2008 Brooks/Cole, a division of Thomson Learning, Inc. Chapter 7 Random Variables & Probability Distributions