Modeling Modeling Continuous Continuous VariablesVariables

Lecture 19Lecture 19Section 6.1 - 6.3.1Section 6.1 - 6.3.1Fri, Feb 24, 2006Fri, Feb 24, 2006

ModelsModels Mathematical modelMathematical model – An – An

abstraction and, therefore, a abstraction and, therefore, a simplification of a real situation, one simplification of a real situation, one that retains the essential features.that retains the essential features.

Real situations are usually much to Real situations are usually much to complicated to deal with in all their complicated to deal with in all their details.details.

ExamplesExamples Economic models treat money as a Economic models treat money as a

continuous quantity, even though it is continuous quantity, even though it is discrete.discrete. This is an abstraction that is incorporated into the This is an abstraction that is incorporated into the

model to make it simpler.model to make it simpler. The “bell curve” is a model (an abstraction) The “bell curve” is a model (an abstraction)

of many populations.of many populations. Real populations have all sorts of bumps and Real populations have all sorts of bumps and

twists.twists. The bell curve is smooth and perfectly symmetric.The bell curve is smooth and perfectly symmetric.

ModelsModels No mathematical model is perfect.No mathematical model is perfect. A mathematical model is useful (and A mathematical model is useful (and

powerful) to the extent that it is a powerful) to the extent that it is a faithful representation of reality.faithful representation of reality.

Conversely, to the extent that is it Conversely, to the extent that is it not faithful to reality, it can lead to not faithful to reality, it can lead to false conclusions about the situation false conclusions about the situation that it is supposed to model.that it is supposed to model.

Example of a ModelExample of a Model Use a random number generator to Use a random number generator to

simulate how a pair of rolled dice will land.simulate how a pair of rolled dice will land. The possible totals range from 2 to 12.The possible totals range from 2 to 12. Using the TI-83, which is a correct model?Using the TI-83, which is a correct model?

Enter Enter randInt(2, 12)randInt(2, 12), that is, get a random , that is, get a random number from 2 to 12, ornumber from 2 to 12, or

Enter Enter 2*randInt(1, 6)2*randInt(1, 6), that is, double a random , that is, double a random number from 1 to 6, ornumber from 1 to 6, or

Enter Enter randInt(1, 6) + randInt(1, 6)randInt(1, 6) + randInt(1, 6), that is, add , that is, add two random numbers from 1 to 6.two random numbers from 1 to 6.

Histograms and AreaHistograms and Area If a histogram is drawn If a histogram is drawn

appropriately, then frequency is appropriately, then frequency is represented by area.represented by area.

Consider the following histogram of Consider the following histogram of test scores.test scores.

GradeGrade FrequencFrequencyy

60 – 6960 – 69 3370 – 7970 – 79 8880 – 8980 – 89 9990 – 9990 – 99 55

Histograms and AreaHistograms and Area

Grade

Frequency

60 70 80 90 1000

2

4

6

8

10

Histograms and AreaHistograms and Area In the histogram, we may replace In the histogram, we may replace

the frequency with the proportion the frequency with the proportion (of the total).(of the total).

GradeGrade FrequencyFrequency ProportionProportion60 – 6960 – 69 33 0.120.1270 – 7970 – 79 88 0.320.3280 – 8980 – 89 99 0.360.3690 – 9990 – 99 55 0.200.20

Histograms and AreaHistograms and Area

Grade

Proportion

60 70 80 90 1000

0.10

0.20

0.30

0.40

Histograms and AreaHistograms and Area

Grade

Proportion

60 70 80 90 1000

0.10

0.20

0.30

0.40

Histograms and AreaHistograms and Area

Furthermore, we may divide the Furthermore, we may divide the proportions by the width of the proportions by the width of the classes to get the classes to get the densitydensity..

GradeGrade FrequencyFrequency ProportionProportion DensityDensity60 – 6960 – 69 33 0.120.12 0.0120.01270 – 7970 – 79 88 0.320.32 0.0320.03280 – 8980 – 89 99 0.360.36 0.0360.03690 – 9990 – 99 55 0.200.20 0.0200.020

Histograms and AreaHistograms and Area

Grade

Density

60 70 80 90 1000

0.010

0.020

0.030

0.040

Histograms and AreaHistograms and Area The final histogram has the special The final histogram has the special

property that the property that the proportionproportion can be can be found by computing the found by computing the areaarea of the of the rectangle.rectangle.

The vertical scale has been adjusted so The vertical scale has been adjusted so that the total area is 1, or 100%.that the total area is 1, or 100%.

For example, what proportion of the For example, what proportion of the grades are less than 80?grades are less than 80? Compute: (10 Compute: (10 0.012) + (10 0.012) + (10 0.032) 0.032)

= 0.12 + 0.32 = 0.44 = 44%.= 0.12 + 0.32 = 0.44 = 44%.

Density FunctionsDensity Functions This is the fundamental property This is the fundamental property

that connects the graph of a that connects the graph of a continuous model to the population continuous model to the population that it represents, namely:that it represents, namely: The The area under the grapharea under the graph between two between two

numbers numbers aa and and bb on the on the xx-axis -axis represents the represents the proportion of the proportion of the populationpopulation that lies between that lies between aa and and bb..

AREA = PROPORTIONAREA = PROPORTION

Density FunctionsDensity Functions The area under the curve between The area under the curve between aa

and and bb is the proportion of the values is the proportion of the values of of xx that lie between that lie between aa and and bb..

a bx

Density FunctionsDensity Functions The area under the curve between The area under the curve between aa

and and bb is the proportion of the values is the proportion of the values of of xx that lie between that lie between aa and and bb..

a bx

Density FunctionsDensity Functions The area under the curve between The area under the curve between aa

and and bb is the proportion of the values is the proportion of the values of of xx that lie between that lie between aa and and bb..

a b

Area = Proportion

x

Density FunctionsDensity Functions A consequence of this is that the A consequence of this is that the

total area under the curve must be total area under the curve must be 1, representing a proportion of 1, representing a proportion of 100%.100%.

a bx

Density FunctionsDensity Functions A consequence of this is that the A consequence of this is that the

total area under the curve must be total area under the curve must be 1, representing a proportion of 1, representing a proportion of 100%.100%.

a b

100%x

The Normal DistributionThe Normal Distribution Normal distributionNormal distribution – The – The

statistician’s name for the bell curve.statistician’s name for the bell curve. It is a density function in the shape It is a density function in the shape

of a bell (sort of).of a bell (sort of). Symmetric.Symmetric. Unimodal.Unimodal. Extends over the entire real line (no Extends over the entire real line (no

endpoints).endpoints). ““Main part” lies within Main part” lies within 33 of the mean. of the mean.

The Normal DistributionThe Normal Distribution The curve has a bell shape, with The curve has a bell shape, with

infinitely long tails in both infinitely long tails in both directions.directions.

The Normal DistributionThe Normal Distribution The mean The mean is located in the center, is located in the center,

at the peak.at the peak.

The Normal DistributionThe Normal Distribution The width of the “main” part of the The width of the “main” part of the

curve is 6 standard deviations wide curve is 6 standard deviations wide (3 standard deviations each way (3 standard deviations each way from the mean).from the mean).

– 3 + 3

The Normal DistributionThe Normal Distribution The area under the entire curve is 1.The area under the entire curve is 1. (The area outside of (The area outside of 3 st. dev. is 3 st. dev. is

approx. 0.0027.)approx. 0.0027.)

Area = 1

– 3 + 3

The Normal DistributionThe Normal Distribution The normal distribution with mean The normal distribution with mean

and standard deviation and standard deviation is denoted is denoted NN((, , ).).

For example, if For example, if XX is a variable whose is a variable whose distribution is normal with mean 30 distribution is normal with mean 30 and standard deviation 5, then we and standard deviation 5, then we say that “say that “XX is is NN(30, 5).”(30, 5).”

The Normal DistributionThe Normal Distribution If If XX is is NN(30, 5), then the distribution (30, 5), then the distribution

of of XX looks like this: looks like this:

3015 45

Some Normal Some Normal DistributionsDistributions

0 1 2 3 4 5 6 87

N(3, 1)

Some Normal Some Normal DistributionsDistributions

0 1 2 3 4 5 6 87

N(3, 1)N(5, 1)

Some Normal Some Normal DistributionsDistributions

0 1 2 3 4 5 6 87

N(3, 1)

N(2, ½)

N(5, 1)

Some Normal Some Normal DistributionsDistributions

0 1 2 3 4 5 6 87

N(3, 1)

N(2, ½)

N(5, 1)N(3½, 1½)

Bag Bag AA vs. Bag vs. Bag BB Suppose we have two bags, Bag A and Suppose we have two bags, Bag A and

Bag B.Bag B. Each bag contains millions of vouchers.Each bag contains millions of vouchers. In Bag In Bag AA, the values of the vouchers , the values of the vouchers

have distribution have distribution NN(50, 10).(50, 10). Normal with Normal with = $50 and = $50 and = $10.= $10.

In Bag In Bag BB, the values of the vouchers , the values of the vouchers have distribution have distribution NN(80, 15).(80, 15). Normal with Normal with = $80 and = $80 and = $15. = $15.

Bag Bag AA vs. Bag vs. Bag BB

30 40 50 60 70 9080 100 110

H0: Bag A

H1: Bag B

Bag Bag AA vs. Bag vs. Bag BB We are presented with one of the We are presented with one of the

bags.bags. We select one voucher at random We select one voucher at random

from that bag.from that bag.

30 40 50 60 70 9080 100 110

H0: Bag A

H1: Bag B

Bag Bag AA vs. Bag vs. Bag BB If its value is less than or equal to If its value is less than or equal to

$65, then we will decide that it was $65, then we will decide that it was from Bag from Bag AA..

30 40 50 60 70 9080 100 110

H0: Bag A

H1: Bag B

65

Bag Bag AA vs. Bag vs. Bag BB If its value is less than or equal to If its value is less than or equal to

$65, then we will decide that it was $65, then we will decide that it was from Bag from Bag AA..

30 40 50 60 70 9080 100 110

H0: Bag A

H1: Bag B

65

Acceptance Region

Bag Bag AA vs. Bag vs. Bag BB If its value is less than or equal to If its value is less than or equal to

$65, then we will decide that it was $65, then we will decide that it was from Bag from Bag AA..

30 40 50 60 70 9080 100 110

H0: Bag A

H1: Bag B

65

Acceptance Region Rejection Region

Bag Bag AA vs. Bag vs. Bag BB What is What is ??

30 40 50 60 70 9080 100 110

H0: Bag A

H1: Bag B

65

Bag Bag AA vs. Bag vs. Bag BB What is What is ??

30 40 50 60 70 9080 100 110

H0: Bag A

H1: Bag B

65

Bag Bag AA vs. Bag vs. Bag BB What is What is ??

30 40 50 60 70 9080 100 110

H0: Bag A

H1: Bag B

65

Bag Bag AA vs. Bag vs. Bag BB What is What is ??

30 40 50 60 70 9080 100 110

H0: Bag A

H1: Bag B

65

Bag Bag AA vs. Bag vs. Bag BB If the distributions are very close If the distributions are very close

together, then together, then and and will be large. will be large.

30 40 50 60 70 9080 100 110

H0: Bag A

H1: Bag B

65

Bag Bag AA vs. Bag vs. Bag BB If the distributions are very similar, If the distributions are very similar,

then then and and will be large. will be large.

30 40 50 60 70 9080 100 110

H0: Bag A

H1: Bag B

65

Bag Bag AA vs. Bag vs. Bag BB If the distributions are very similar, If the distributions are very similar,

then then and and will be large. will be large.

30 40 50 60 70 9080 100 110

H0: Bag A

H1: Bag B

65

Bag Bag AA vs. Bag vs. Bag BB Similarly, if the distributions are far Similarly, if the distributions are far

apart, then apart, then and and will both be very will both be very small.small.

Bag Bag AA vs. Bag vs. Bag BB What is the What is the pp-value of $75?-value of $75?

30 40 50 60 70 9080 100 110

H0: Bag A

H1: Bag B

65 75

Bag Bag AA vs. Bag vs. Bag BB What is the What is the pp-value of $75?-value of $75?

30 40 50 60 70 9080 100 110

H0: Bag A

H1: Bag B

65 75

p-value