Normal Distribution

Percent Voting for G.W. Bush among U.S. Counties – 2004

Makes use of Z-Tables with upper-tail probabilities (1-F(z))

Data

• Percent Voting for G.W. Bush (among those voting for Bush, Kerry, Nader)

• 3109 Counties/Cities in Conterminous U.S. (Note: Virginia has several “independent cities”, St. Louis, Baltimore

Descriptive Statistics

Percentile PctBush0 9.315 39.1210 44.7315 47.9620 50.5625 52.7330 54.7135 56.2940 58.0845 59.6650 61.1755 62.6760 64.3865 65.9370 67.5275 69.3780 71.3085 73.8690 76.8895 80.49

100 92.83

m 60.67s 12.61

Range Values Observed Expected0 m s 0.00 35.44 99 70.8852

m s m s 35.44 48.05 374 422.5131m s m 48.05 60.67 1029 1061.1017m ms 60.67 73.28 1111 1061.1017

ms ms 73.28 85.89 461 422.5131ms 85.89 1.00 35 70.8852

Frequency Distribution of Pct Bush by County - 2004 Presidential Election-Cell Width = 2.5%

0

50

100

150

200

250

300

1.25

3.75

6.25

8.75

11.2

513

.7516

.2518

.7521

.2523

.7526

.2528

.7531

.2533

.7536

.2538

.7541

.2543

.7546

.2548

.7551

.2553

.7556

.2558

.7561

.2563

.7566

.2568

.7571

.2573

.7576

.2578

.7581

.2583

.7586

.2588

.7591

.2593

.7596

.2598

.75

Percent Voting For Bush

Fre

qu

en

cy

Problems Involving Probabilities

• Y ~ N(60.7,12.6) Approximately• Probability that a randomly selected county

has at least 75% Vote for GWB?

• That a Majority favors GWB?

• That fewer than 40% favor GWB?

• Between 40 and 60% favor GWB?

P(Y≥75)Normal Density (Mean=60.7, SD=12.6) - Shaded Area Above 75

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Percent Bush

De

ns

ity

751292.13.113.1

13.16.12

3.14

6.12

7.6075

7575

)6.12,7.60(~

YPZPZP

ZZ

YYY

NY

UL

UL

sm

z 0.02 0.03 0.041 0.1539 0.1515 0.1492

1.1 0.1314 0.1292 0.12711.2 0.1112 0.1093 0.1075

P(Y>50)Normal Density (Mean=60.7, SD=12.6) - Shaded Area Above 50

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Percent Bush

De

ns

ity

8023.1977.185.0185.0185.0

85.06.12

7.6050

50

)6.12,7.60(~

ZPZPZP

ZZ

YY

NY

UL

UL

z 0.04 0.05 0.060.7 0.2296 0.2266 0.22360.8 0.2005 0.1977 0.19490.9 0.1736 0.1711 0.1685

P(Y<40)Normal Density (Mean=60.7, SD=12.6) - Shaded Area Below 40

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Percent Bush

De

ns

ity

400505.64.164.1

64.16.12

7.6040

40

6.12,7.60~

YPZPZP

ZZ

YY

NY

UL

UL

sm

z 0.03 0.04 0.051.5 0.0630 0.0618 0.06061.6 0.0516 0.0505 0.04951.7 0.0418 0.0409 0.0401

P(40≤Y≤60)Normal Density (Mean=60.7, SD=12.6) - Shaded Area Between 40 and 60

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Percent Bush

De

ns

ity

4256.0505.4761.64.106.0

64.106.006.064.1

06.06.12

7.606064.1

6.12

7.6040

6040

ZPZP

ZPZP

ZZ

YY

UL

UL

Problems Involving Percentiles

• Y ~ N(60.7,12.6) Approximately• Above what percentage do the highest

10% fall?

• Below what percentage do the lowest 5% fall?

• Between what percentages do the middle 50% fall?

Upper Tail Percentiles (>50%)Finding Upper Tail Percentiles (100p>50 or p>.50 )

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-3 0 3

z

No

rma

l De

ns

ity

1-p

p

zp

Look up a=1-p in “body” of Z-table, obtain zp=za from the “edge” of table

Lower Tail Percentiles (<50%)

Look up a=p in “body” of Z-table, obtain zp=-za from the “edge” of table

Finding Lower Tail Percentiles (100p<50 or p<.50 )

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-3 0 3

z

No

rma

l De

ns

ity

p

1-p

zp

90th-PercentileFinding 90th Percentile (100p=90 or p=.90 )

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-3 -2 -1 0 1 2 3

z

No

rma

l De

ns

ity

z(.10)=1.28

.10

.90

8.761.167.60)6.12)(28.1(7.60

282.1

10.1)50.90.( Percentile-90 :Target

6.12,7.60~

90.

10.

th

Y

zzz

pap

NY

ap

sm

z .07 .08 .09

1.1 .1210 .1190 .1170

1.2 .1020 .1003

.0985

1.3 .0853 .0838 .0823

5th-Percentile

0.407.207.60)6.12)(645.1(7.60

645.1

05.)50.05.( Percentile-5 :Target

6.12,7.60~

05.

05.

th

Y

zzz

pap

NY

ap

sm

z .04 .051.5 .0618 .0606

1.6 .0505 .0495

1.7 .0409 .0401

Finding 5th Percentile (100p=5 or p=.05 )

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

-3 -2 -1 0 1 2 3

z

No

rma

l De

ns

ity

-z(.05)=-1.645

.95

.05

Range Containing Middle 50% of Counties (60.7 +/- 0.675(12.6))

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

Percent Bush

De

ns

ity

Percent Voting for GWB vs LN(Total Votes Cast)

0

10

20

30

40

50

60

70

80

90

100

4 6 8 10 12 14 16