Estatística

Estatstica Descritiva

a parte da estatstica que visa, apenas, a descrever, apresentar e analisar os dados sem aquela preocupao de fazer generalizaes ou tirar concluses para o conjunto de onde foram retirados tais dados. Introduo EstatsticaEstatstica Indutiva

a estatstica que partindo da anlise dos resultados de pequenos conjuntos, denominados amostras, chega-se a concluses extensivas a conjuntos maiores que contm os primeiros e que sero denominados de populao ou universo.Conceitos bsicos

Esquematicamente, temos: Introduo Estatstica

Estatstica

um mtodo que cuida da observao, classificao formal e anlise dos fenmenos coletivos.

um mtodo cientifico de obter, apurar, apresentar e interpretar dados numricos, relativos variao de um ou mais fenmenos.

Introduo Estatstica

Coleo de unidades individuais, que podem ser pessoas ou resultados experimentais, com uma ou mais caractersticas comuns, que se pretendem estudar. Populao ou universoIntroduo Estatstica

Nem sempre possvel estudar exaustivamente todos os elementos da populao!Porqu?- Pode a populao ter dimenso infinita Exemplo: Populao constituda pelas presses atmosfricas, nos diferentes pontos de uma cidade. - Pode o estudo da populao levar destruio da populao Exemplo: Populao dos fsforos de uma caixa. - Pode o estudo da populao ser muito dispendioso Exemplo: Sondagens exaustivas de todos os eleitores, sobre determinado candidato. Introduo Estatstica

Quando no possvel estudar, exaustivamente, todos os elementos da populao, estudam-se s alguns elementos, a que damos o nome de Amostra. Conjunto de dados ou observaes, recolhidos a partir de um subconjunto da populao, que se estuda com o objetivo de tirar concluses para a populao de onde foi recolhida. Amostra Introduo Estatstica

Amostra

A amostra uma parcela ou uma frao de uma populao.A diferena bsica entre conceitos de AMOSTRA e POPULAO , ento, que a amostra representa uma parte do todo, enquanto a Populao representa o todo. A amostra considerada parte representativa do todo, ou seja, todas as caractersticas devem nela estar contidas e, por isso, as concluses a respeito da Amostra podem ser consideradas como pertencentes s da Populao.Introduo Estatstica

Justificam-se os trabalhos com amostras no lugar de estudar toda a populao pelos seguintes fatores (vantagens): Introduo EstatsticaCusto: as despesas com operacionalizao estatstica da populao so geralmente bem maiores que com a averiguao amostral. Velocidade: as pesquisas realizadas em amostras so mais rpidas em virtude de conter bem menor nmero de unidades. Praticabilidade: conforme o prprio conceito. s vezes, a dimenso da populao torna as pesquisas impraticveis.A grande desvantagem consiste em se trabalhar com pequenos conjuntos o que nos conduzir a no se ter certeza absoluta da validade das estimativas, anlise, concluses.

Experimento aleatrio

Os experimentos aleatrios so aqueles cujos resultados no so sempre os mesmos, apesar de se repetirem vrias vezes em condies semelhantes. O experimento aleatrio um fenmeno que apresenta resultados imprevisveis. O lanamento de moedas e dados, bem como sorteios, extraes lotricas so fenmenos aleatrios.Introduo Estatstica

IntroduoDas medidas, ou estatsticas que iremos definir, para caracterizar os dados, destacam-se asAs que localizam o centro da amostra (mdia, moda e mediana) Que indicam a variao entre os valores.

Medidas de tendncia central MdiaA mdia, que se representa por uma medida de localizao do centro da amostra, e obtm-se a partir da seguinte expresso: onde x1, x2, ..., xn representam os elementos da amostra e n a sua dimenso.

Mdia Tpica

quando os valores da srie esto concentradas em torno da mdia. X = 18 + 20 + 20 + 22 = 80 = 20 4 4

20 considerada mdia tpica, pela concentrao dos valores em torno da mdia.Mdia Atpica

quando os valores da srie no esto concentrados em torno da mdia. A mdia atpica representa mal a srie de valores. Observe que no conjunto abaixo, o valor 100 discrepante, conduzindo a falsa anlise do conjunto. X = 2 + 4 + 5 + 6 + 100 = 117 = 23,4 5 5 O nmero 23,4 representa mal os valores da srie. uma mdia atpica.Medidas de tendncia central Mdia

Medidas de tendncia central MdiaVantagens da Mdia Aritmtica

obtida atravs de clculos relativamente simples. a mais conhecida de todas as mdias.Pela facilidade com que determinada a de mais fcil compreenso.Desvantagens da Mdia Aritmtica

Para a obteno da X necessrio ter todos os valores do grupo; basta no ter um desses valores e a mdia no pode ser determinada. excessivamente afetada por valores ou muito grandes ou muito pequenos.

Medidas de tendncia central MedianaA mediana, m, definida como: Ordenados os elementos da amostra, a mediana o valor (pertencente ou no amostra) que a divide ao meio, isto , 50% dos elementos da amostra so menores ou iguais mediana e os outros 50% so maiores ou iguais mediana Para a sua determinao utiliza-se a seguinte regra, depois de ordenada a amostra de n elementos: Se n mpar, a mediana o elemento mdio. Se n par, a mediana a semi-soma dos dois elementos mdios.

Medidas de tendncia central Medianaento uma expresso para o clculo da mediana ser: Como medida de tendncia central, a mediana mais robusta do que a mdia, pois no to sensvel aos dados ! Sejam os elementos da amostra ordenada com a seguinte notao: X1, X2, ... , Xn

Introduo

A mdia uma forma de resumir os dados de um grupo e est representada pelo Ponto de equilbrio. Porm, muitas das vezes necessrio se conhecer a variao obtida entre os valores. Consideremos as sries abaixo com suas respectivas mdias.I) 20; 20; 20; 20; 20; X = 20II) 18; 19; 20; 21; 22; X = 20III) 5; 25; 15; 15; 40 X = 20Nos dois primeiros casos, a mdia representa um ponto de equilbrio entre os valores. Percebam que no existe uma significativa disperso entre os valores. Medidas de Disperso Introduo

Medidas de Disperso IntroduoDesvio padro Amplitude Varincia Coeficiente de variaoJ no terceiro caso, embora a mdia nos d uma idia da localizao dos valores na srie, ela no suficiente para descrev-los completamente.

Ao comparar os valores da srie 1 e 2 com os valores da srie 3 vemos que possuem a mesma mdia mas os valores da srie 3 se apresentam mais espalhados que os das sries 1 e 2.

Desta forma conclu-se que para descrever uma srie de valores necessrio um indicador de espalhamento ou disperso, alm da mdia.

Para medir o grau de disperso ou concentrao dos valores em torno da mdia, utilizamos as medidas de disperso; a saber:

Amplitude de variao ( R ):

A amplitude definida como sendo o intervalo entre o maior e o menor item do grupo:

R = (maior medida) - (menor medida). Medidas de Disperso AmplitudeR(I) = 20 - 20 R = 0

R(I) = 22 - 18 R = 4

R(I) = 40 - 5 R = 35

Desvio mdio ( DM ):

a mdia dos desvios em relao mdia.

Exemplo: 12; 10; 9; 8; 10; 8; 6

A mdia dos valores = 9

Desvio de cada elemento em relao mdia:

12 - 9 = 3 ; 10 9 = 1; ... Medidas de Disperso Desvio mdio

Medidas de Disperso Desvio mdio

Define-se a varincia, e representa-se por s2, como sendo a medida que se obtm somando os quadrados dos desvios das observaes da amostra, relativamente sua mdia, e dividindo pelo nmero de observaes da amostra menos um: Medidas de Disperso Varincia

Pode ser obtida atravs da amostra ou da populao.Medidas de Disperso VarinciaVarincia populacionalVarincia amostral

Medidas de Disperso Desvio PadroUma vez que a varincia envolve a soma de quadrados, a unidade em que se exprime no a mesma que a dos dados. Assim, para obter uma medida da variabilidade ou disperso com as mesmas unidades que os dados, tomamos a raiz quadrada da varincia e obtemos o desvio padro: Desvio padro amostralDesvio padro populacional

Medidas de Disperso Desvio Padro

56,25757,457,758,558,959,559,559,56060,360,560,560,560,860,860,860,96161,261,561,661,862,562,96363,864,765,466,3

Grf2

56.263.196605952258.470060714460.8333333333

5763.196605952258.470060714460.8333333333

57.463.196605952258.470060714460.8333333333

57.763.196605952258.470060714460.8333333333

58.563.196605952258.470060714460.8333333333

58.963.196605952258.470060714460.8333333333

59.563.196605952258.470060714460.8333333333

59.563.196605952258.470060714460.8333333333

59.563.196605952258.470060714460.8333333333

6063.196605952258.470060714460.8333333333

60.363.196605952258.470060714460.8333333333

60.563.196605952258.470060714460.8333333333

60.563.196605952258.470060714460.8333333333

60.563.196605952258.470060714460.8333333333

60.863.196605952258.470060714460.8333333333

60.863.196605952258.470060714460.8333333333

60.863.196605952258.470060714460.8333333333

60.963.196605952258.470060714460.8333333333

6163.196605952258.470060714460.8333333333

61.263.196605952258.470060714460.8333333333

61.563.196605952258.470060714460.8333333333

61.663.196605952258.470060714460.8333333333

61.863.196605952258.470060714460.8333333333

62.563.196605952258.470060714460.8333333333

62.963.196605952258.470060714460.8333333333

6363.196605952258.470060714460.8333333333

63.863.196605952258.470060714460.8333333333

64.763.196605952258.470060714460.8333333333

65.463.196605952258.470060714460.8333333333

66.363.196605952258.470060714460.8333333333

normal centro

Distribuio normal

ZpZp

0.10.07973.10.998061-0.68269

0.20.15853.20.99863

0.30.23583.30.99903

0.40.31083.40.99933

0.50.38293.50.99953

0.60.45153.60.99968

0.70.51613.70.99978

0.80.57633.80.99986

0.90.63193.90.99990

1.00.68274.00.99994

1.10.72874.10.99996

1.20.76994.20.999973

1.30.80644.30.999983

1.40.83854.40.999989

1.50.86644.50.999993

1.60.89044.60.999996

1.70.91094.70.999997

1.80.92814.80.9999984

1.90.94264.90.9999990

2.00.95455.00.9999994

2.10.96435.10.9999997

2.20.97225.20.99999980

2.30.97865.30.99999988

2.40.98365.40.99999993

2.50.98765.50.99999996

2.60.99075.60.999999979

2.70.99315.70.999999988

2.80.99495.80.999999993

2.90.99635.90.999999996

3.00.99736.00.999999998

exercicio estatstica

5762.557.756.262.966.3

64.765.460.559.561.660.8

60.360.859.56061.863.8

57.461.260.9616359.5

60.560.861.558.558.960.5

mdia60.8333333333

amplitude10.1

variancia populacional5.3988888889

variancia amostral5.5850574713

desvio padro pop2.3235509224

desvio padro amostral2.3632726189

coef variao3.8848317023

somatoria x1825

somatria quadrtica111182.8

grfico s

56.263.196605952258.470060714460.8333333333

5763.196605952258.470060714460.8333333333

57.463.196605952258.470060714460.8333333333

57.763.196605952258.470060714460.8333333333

58.563.196605952258.470060714460.8333333333

58.963.196605952258.470060714460.8333333333

59.563.196605952258.470060714460.8333333333

59.563.196605952258.470060714460.8333333333

59.563.196605952258.470060714460.8333333333

6063.196605952258.470060714460.8333333333

60.363.196605952258.470060714460.8333333333

60.563.196605952258.470060714460.8333333333

60.563.196605952258.470060714460.8333333333

60.563.196605952258.470060714460.8333333333

60.863.196605952258.470060714460.8333333333

60.863.196605952258.470060714460.8333333333

60.863.196605952258.470060714460.833333333313.333333333366.6666666667

60.963.196605952258.470060714460.833333333386.6666666667

6163.196605952258.470060714460.8333333333

61.263.196605952258.470060714460.8333333333

61.563.196605952258.470060714460.8333333333

61.663.196605952258.470060714460.8333333333

61.863.196605952258.470060714460.8333333333

62.563.196605952258.470060714460.8333333333

62.963.196605952258.470060714460.8333333333

6363.196605952258.470060714460.8333333333

63.863.196605952258.470060714460.8333333333

64.763.196605952258.470060714460.8333333333

65.463.196605952258.470060714460.8333333333

66.363.196605952258.470060714460.8333333333

60.833333333356.25757.457.758.558.9

59.559.559.56060.360.5

60.560.560.860.860.860.9

6161.261.561.661.862.5

62.96363.864.765.466.3

grfico s

histograma

eixofifi

0.302020

0.458080

0.60113.3113.3

0.75186.7186.7

0.90173.3173.3

1.00140140

1.201010

1.4055

15

20200

252010

302020

358035

408057

458080

50113.391

55113.3103

60113.3113.3

65186.7134

70186.7158

75186.7186.7

80173.3182

85173.3178

90173.3173.3

95140157

100140140

1051095

1101052

1151010

120108.7

12557.5

13056.3

13555

14002.5

15000

histograma

00

00

00

00

00

00

00

00

&A

Page &P

fi

fi

grafico normal

0

0

0

0

0

0

0

0

normal

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

rea da normal

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

rea da normal

&A

Page &P

Plan1

student

Tabela da distribuio Normal

Z =-1.74p =0.04093

Z0.000.010.020.030.040.050.060.070.080.09

-3.00.00130.00130.00130.00120.00120.00110.00110.00110.00100.0010

-2.90.00190.00180.00180.00170.00160.00160.00150.00150.00140.0014

-2.80.00260.00250.00240.00230.00230.00220.00210.00210.00200.0019

-2.70.00350.00340.00330.00320.00310.00300.00290.00280.00270.0026

-2.60.00470.00450.00440.00430.00410.00400.00390.00380.00370.0036

-2.50.00620.00600.00590.00570.00550.00540.00520.00510.00490.0048

-2.40.00820.00800.00780.00750.00730.00710.00690.00680.00660.0064

-2.30.01070.01040.01020.00990.00960.00940.00910.00890.00870.0084

-2.20.01390.01360.01320.01290.01250.01220.01190.01160.01130.0110

-2.10.01790.01740.01700.01660.01620.01580.01540.01500.01460.0143

-2.00.02280.02220.02170.02120.02070.02020.01970.01920.01880.0183

-1.90.02870.02810.02740.02680.02620.02560.02500.02440.02390.0233

-1.80.03590.03510.03440.03360.03290.03220.03140.03070.03010.0294

-1.70.04460.04360.04270.04180.04090.04010.03920.03840.03750.0367

-1.60.05480.05370.05260.05160.05050.04950.04850.04750.04650.0455

-1.50.06680.06550.06430.06300.06180.06060.05940.05820.05710.0559

-1.40.08080.07930.07780.07640.07490.07350.07210.07080.06940.0681

-1.30.09680.09510.09340.09180.09010.08850.08690.08530.08380.0823

-1.20.11510.11310.11120.10930.10750.10560.10380.10200.10030.0985

-1.10.13570.13350.13140.12920.12710.12510.12300.12100.11900.1170

-1.00.15870.15620.15390.15150.14920.14690.14460.14230.14010.1379

-0.90.18410.18140.17880.17620.17360.17110.16850.16600.16350.1611

-0.80.21190.20900.20610.20330.20050.19770.19490.19220.18940.1867

-0.70.24200.23890.23580.23270.22960.22660.22360.22060.21770.2148

-0.60.27430.27090.26760.26430.26110.25780.25460.25140.24830.2451

-0.50.30850.30500.30150.29810.29460.29120.28770.28430.28100.2776

-0.40.34460.34090.33720.33360.33000.32640.32280.31920.31560.3121

-0.30.38210.37830.37450.37070.36690.36320.35940.35570.35200.3483

-0.20.42070.41680.41290.40900.40520.40130.39740.39360.38970.3859

-0.10.46020.45620.45220.44830.44430.44040.43640.43250.42860.4247

0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359

0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753

0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.6141

0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.6517

0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.6879

0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.7224

0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.7549

0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.7852

0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.8133

0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.8389

1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.8621

1.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.8830

1.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.9015

1.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.9177

1.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.9319

1.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.9441

1.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.9545

1.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.9633

1.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.9706

1.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.9767

2.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.9817

2.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.9857

2.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890

2.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916

2.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936

2.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952

2.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964

2.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974

2.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981

2.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986

3.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990

student

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

Relogio comparador

x40.0077x60.000

media40.007860.6250.61012

dp0.0002s2.235

n10

t1.581distribuio normal0.0000000019

rea a esquerda

confiana7.2599.9999999981

38.99%

distribuio normal

rea a direita

61.01%

AnoInscriesEstado CivilInscries

20052753Solteiro952

20062875Casado1537

20072803Vivo684

20082920Divorciado651

total11351Desquitado730

Outros382

Total4936

EstadoInscries

AM203

PE322Grau de escolaridadesexo

MT219MasculinoFeminino

ES2351 grau incompleto82116

SC3071 grau completo235128

Total12862 grau324204

3 grau9282

total733530

0

0

0

0

Inscries

Inscries

0

0

0

0

0

0

Inscries

sexo Masculino

sexo Feminino

sexo Masculino

sexo Feminino

90%95%99%

0.050.0250.0050.04

16.31412.70663.6560.03

22.9204.3039.9255

32.3533.1825.841%95

42.1322.7764.604

52.0152.5714.032

61.9432.4473.70715

71.8952.3653.499

81.8602.3063.355K =2.13

91.8332.2623.250

101.8122.2283.169

111.7962.2013.106

121.7822.1793.055

131.7712.1603.012

141.7612.1452.977

151.7532.1312.947

161.7462.1202.921

171.7402.1102.898

181.7342.1012.878

191.7292.0932.861

201.7252.0862.845

211.7212.0802.831

221.7172.0742.819

231.7142.0692.807

241.7112.0642.797

251.7082.0602.787

261.7062.0562.779

271.7032.0522.771

281.7012.0482.763

291.6992.0452.756

301.6972.0422.750

401.6842.0212.704

601.6712.0002.660

1201.6581.9802.617

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

mmmm

0.00-3

0.120

0.231

0.352

0.441

0.565

0.667

0.755

0.834

0.934

1.013

1.5-2-1

2.0-2-1

2.5-4-3

3.0-5-5

3.5-3-2

4.0-2-1

4.5-10

5.011

6.021

7.022

8.047

9.056

10.078

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

entrada

sada

(mm)

(m)

Erros do Relgio comparador

Medidas de Disperso Coeficiente de VariaoCoeficiente de Variao (Cv) Medida relativa de disperso utilizada para comparar grandezas diferentes. Observe as duas distribuies a seguir e avalie qual das duas apresenta maior disperso.Avaliando pelo desvio padro, conclumos que ambas apresentam a mesma disperso. Porm, percebemos claramente que a disperso do processo B muito mais significativa que a do processo A.

ProcessoMdiaDesvio padroA20 00050B20050

Cv = S(x) x 100 XAssim, no adequado comparar disperses de duas ou mais distribuies pelo desvio padro, sendo necessria uma outra medida que leve em considerao a magnitude da varivel. Assim: Medidas de Disperso Coeficiente de Variao

Exerccio:Sejam os dados abaixo um conjunto de valores resultados de 40 medies independentes e consecutivas:Calcule: a) mdia ; amplitude ; desvio padro amostral e populacional ; somatria de x ; somatria quadrtica ; varincia amostral e populacional.

61,059,257,062,557,756,262,966,3 58,256,564,765,4 60,559,561,660,862,259,060,360,859,560,061,863,861,159,757,461,260,961,063,0 59,560,262,960,560,861,558,558,960,5

Distribuio RetangularTambm conhecida como distribuio uniforme.No lanamento de um dado de seis lados, a probabilidade de se tirar um nmero de uma em seis

Grf2

0.1666666667

0.1666666667

0.1666666667

0.1666666667

0.1666666667

0.1666666667

06 seg

dia 06/07 - segunda feiratempo

minutos

apresentaoinstrutor10ok

alunos20

curso10408:10 s 8:50

Estatsticaestatstica descritiva10ok

estatstica inferencial10

Amostra10

Populao10408:50 s 9:30

Intervalo para lanche9:30 s 9:50

Tabulao de resultadoselementos essenciais10ok

elementos esclarecedores10

Apresentao10

exerccios30609:50 s 10:50

Sries estatsticassrie cronolgica, geogrfica e especificativa10ok

1010:50 s 11:00

Distribuio de frequnciadados no agrupados15ok

dados agrupados15

definio das classes205011:00 s 11:50

Intervalo para almoo12:00 s 13:00

Distribuio de frequnciahistograma20ok

frequencias acumuladas20

exerccios20

frequencia corrigida1013:10 s 14:40

exerccios2090

medidas de tendencia centralmdia, moda e mediana20ok

mdia atpica103014:40 s 15:10

Lanche15:10 s 15:30

medidas de tendencia centralseparatrizes20ok

medidas de tendencia na distribuio de frequencia40

6015:30 s 16:303806.3333333333

07 ter

dia 07/07 - tera feiratempo

minutos

exerccios2020ok8:10 s 8:30

medidas de dispersoAmplitude e desvio mdio15ok

varincia e desvio padro15

coeficiente de variao5


9:20 s 9:40

Estatstica na calculadora e no excelFunes estatsticas nas calculadoras e excel30ok

9:40 s 10:40

exerccios3060

Lanche

Representa-o grficagrfico de coluna e barra, linhas e setor15ok

grficos mistos10

Regras para elaborao1510:40 s 11:50

exerccios3070

Almoo12:00 s 13:00

Distribuiesretangular10

triangular1013:10 s 14:50

normal20

normal reduzida15

exerccios30

student15100

Lanche14:50 s 15:10

compatibiliza-o de valoresregras de arredondamento10

algarismo significativo10

compatibilizao de valores10

exerccios154515:10 s 15:553455.75

08 qua

dia 08/07 - quarta feiratempo

minutos

Testessnedecor15

dixon15

15

erro normalizado15


intervalo9:30 s 9:50

Definio de Incerteza de medioprtica de medio (disperso e valor mdio)1209:50 s 11:50

120medio de temperatura, medio de uma pea e medio de rugosidade medio de presso

12:00 s 13:00

Almoo

Diferena entre tolerncia e incerteza de medio1513:00 s 13:45

Diferena entre erro e incerteza de medio15

adequao ao uso1545

Incerteza de medioAplicao da ISO 14253-115

definio de incerteza15

Guia para expresso da incerteza10

13:45 s 14:55

Incerteza de mediotipos de incerteza e fontes de incerteza15

incerteza combinada e incerteza expandida15

70

grau de liberdade da uc15

fator de abrangncia k153015:20 s 15:50

09 qui

dia 09/07 - quinta feiratempo

minutos

modelo de planilha de incerteza158:00 s 9:30

campos da planilha de incerteza15

coeficiente de sensibilidade15

apresentao dos resultados15

roteiro p/ o clculo da incerteza3090

exemplos de calculo de incertezaincerteza de um voltmetro609:50 s 11:50

incerteza de um micrometro60120

60120

6060

dia 10/07 - sexta feiratempo

minutos

Visita aos laboratrios de metrologia e elastmeros908:00 s 9:30

incerteza de um manometro60

incerteza de uma balana609:50 s 12:00

Plan2

PlanoInscries

Alfa65321

Beta57960

Gama23502

mega29030

total175813

Plan2

Inscries

Plan3

Inscries

Inscries

10.1666666667

20.1666666667

30.1666666667

40.1666666667

50.1666666667

60.1666666667

Distribuio TriangularNo lanamento de dois dados, a probabilidade de se tirar um 7 maior que a probabilidade de se tirar um 2.

21311411151111611111711111181111191111101111111121

Grf3

1

11

111

1111

11111

111111

11111

1111

111

11

1

06 seg

dia 06/07 - segunda feiratempo

minutos

apresentaoinstrutor10ok

alunos20

curso10408:10 s 8:50

Estatsticaestatstica descritiva10ok

estatstica inferencial10

Amostra10

Populao10408:50 s 9:30

Intervalo para lanche9:30 s 9:50

Tabulao de resultadoselementos essenciais10ok

elementos esclarecedores10

Apresentao10


Sries estatsticassrie cronolgica, geogrfica e especificativa10ok

1010:50 s 11:00

Distribuio de frequnciadados no agrupados15ok

dados agrupados15

definio das classes205011:00 s 11:50

Intervalo para almoo12:00 s 13:00

Distribuio de frequnciahistograma20ok

frequencias acumuladas20

exerccios20

frequencia corrigida1013:10 s 14:40

exerccios2090

medidas de tendencia centralmdia, moda e mediana20ok

mdia atpica103014:40 s 15:10

Lanche15:10 s 15:30

medidas de tendencia centralseparatrizes20ok

medidas de tendencia na distribuio de frequencia40

6015:30 s 16:303806.3333333333

07 ter

dia 07/07 - tera feiratempo

minutos

exerccios2020ok8:10 s 8:30

medidas de dispersoAmplitude e desvio mdio15ok

varincia e desvio padro15

coeficiente de variao5


9:20 s 9:40

Estatstica na calculadora e no excelFunes estatsticas nas calculadoras e excel30ok

9:40 s 10:40

exerccios3060

Lanche

Representa-o grficagrfico de coluna e barra, linhas e setor15ok

grficos mistos10

Regras para elaborao1510:40 s 11:50

exerccios3070

Almoo12:00 s 13:00

Distribuiesretangular10

triangular1013:10 s 14:50

normal20

normal reduzida15

exerccios30

student15100

Lanche14:50 s 15:10

compatibiliza-o de valoresregras de arredondamento10

algarismo significativo10

compatibilizao de valores10

exerccios154515:10 s 15:553455.75

08 qua

dia 08/07 - quarta feiratempo

minutos

Testessnedecor15

dixon15

15

erro normalizado15


intervalo9:30 s 9:50

Definio de Incerteza de medioprtica de medio (disperso e valor mdio)1209:50 s 11:50

120medio de temperatura, medio de uma pea e medio de rugosidade medio de presso

12:00 s 13:00

Almoo

Diferena entre tolerncia e incerteza de medio1513:00 s 13:45

Diferena entre erro e incerteza de medio15

adequao ao uso1545

Incerteza de medioAplicao da ISO 14253-115

definio de incerteza15

Guia para expresso da incerteza10

13:45 s 14:55

Incerteza de mediotipos de incerteza e fontes de incerteza15

incerteza combinada e incerteza expandida15

70

grau de liberdade da uc15

fator de abrangncia k153015:20 s 15:50

09 qui

dia 09/07 - quinta feiratempo

minutos

modelo de planilha de incerteza158:00 s 9:30

campos da planilha de incerteza15

coeficiente de sensibilidade15

apresentao dos resultados15

roteiro p/ o clculo da incerteza3090

exemplos de calculo de incertezaincerteza de um voltmetro609:50 s 11:50

incerteza de um micrometro60120

60120

6060

dia 10/07 - sexta feiratempo

minutos

Visita aos laboratrios de metrologia e elastmeros908:00 s 9:30

incerteza de um manometro60

incerteza de uma balana609:50 s 12:00

Plan2

PlanoInscries

Alfa65321

Beta57960

Gama23502

mega29030

total175813

Plan2

Inscries

Plan3

Inscries

Inscries

123456

1234567

2345678

3456789

45678910

567891011

6789101112

21

311

4111

51111

611111

7111111

81111100000100000

9111100001110000

1011100011111000

111100111111100

12101111111110

11111111111

23456789101112

Histograma8 classes

Grf1

22

44

88

1515

1414

88

44

22

fi

fi

normal centro

Distribuio normal

ZpZp

0.10.07973.10.998061-0.68269

0.20.15853.20.99863

0.30.23583.30.99903

0.40.31083.40.99933

0.50.38293.50.99953

0.60.45153.60.99968

0.70.51613.70.99978

0.80.57633.80.99986

0.90.63193.90.99990

1.00.68274.00.99994

1.10.72874.10.99996

1.20.76994.20.999973

1.30.80644.30.999983

1.40.83854.40.999989

1.50.86644.50.999993

1.60.89044.60.999996

1.70.91094.70.999997

1.80.92814.80.9999984

1.90.94264.90.9999990

2.00.95455.00.9999994

2.10.96435.10.9999997

2.20.97225.20.99999980

2.30.97865.30.99999988

2.40.98365.40.99999993

2.50.98765.50.99999996

2.60.99075.60.999999979

2.70.99315.70.999999988

2.80.99495.80.999999993

2.90.99635.90.999999996

3.00.99736.00.999999998


5762.557.756.262.966.3

64.765.460.559.561.660.8

60.360.859.56061.863.8

57.461.260.9616359.5

60.560.861.558.558.960.5

mdia60.8333333333

amplitude10.1






somatoria x1825


grfico s

5763.163272618958.436727381160.8

56.263.163272618958.436727381160.8

57.463.163272618958.436727381160.8

57.763.163272618958.436727381160.8

58.563.163272618958.436727381160.8

58.963.163272618958.436727381160.8

59.563.163272618958.436727381160.8

59.563.163272618958.436727381160.8

59.563.163272618958.436727381160.8

6063.163272618958.436727381160.8

60.363.163272618958.436727381160.8

60.563.163272618958.436727381160.8

60.563.163272618958.436727381160.8

60.563.163272618958.436727381160.8

60.863.163272618958.436727381160.8

60.863.163272618958.436727381160.8

60.863.163272618958.436727381160.8

60.963.163272618958.436727381160.8

6163.163272618958.436727381160.8

61.263.163272618958.436727381160.8

61.563.163272618958.436727381160.8

61.663.163272618958.436727381160.8

61.863.163272618958.436727381160.8

62.563.163272618958.436727381160.8

62.963.163272618958.436727381160.8

6363.163272618958.436727381160.8

63.863.163272618958.436727381160.8

64.763.163272618958.436727381160.8

65.463.163272618958.436727381160.8

66.363.163272618958.436727381160.8

grfico s

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

histograma

eixofifi

1022

2044

3088

401515

501414

6088

7044

8022

histograma

00

00

00

00

00

00

00

00

&A

Page &P

fi

fi

grafico normal

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

grafico normal

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&A

Page &P

normal

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

rea da normal


Z =-1.74p =0.04093

Z0.000.010.020.030.040.050.060.070.080.09

-3.00.00130.00130.00130.00120.00120.00110.00110.00110.00100.0010

-2.90.00190.00180.00180.00170.00160.00160.00150.00150.00140.0014

-2.80.00260.00250.00240.00230.00230.00220.00210.00210.00200.0019

-2.70.00350.00340.00330.00320.00310.00300.00290.00280.00270.0026

-2.60.00470.00450.00440.00430.00410.00400.00390.00380.00370.0036

-2.50.00620.00600.00590.00570.00550.00540.00520.00510.00490.0048

-2.40.00820.00800.00780.00750.00730.00710.00690.00680.00660.0064

-2.30.01070.01040.01020.00990.00960.00940.00910.00890.00870.0084

-2.20.01390.01360.01320.01290.01250.01220.01190.01160.01130.0110

-2.10.01790.01740.01700.01660.01620.01580.01540.01500.01460.0143

-2.00.02280.02220.02170.02120.02070.02020.01970.01920.01880.0183

-1.90.02870.02810.02740.02680.02620.02560.02500.02440.02390.0233

-1.80.03590.03510.03440.03360.03290.03220.03140.03070.03010.0294

-1.70.04460.04360.04270.04180.04090.04010.03920.03840.03750.0367

-1.60.05480.05370.05260.05160.05050.04950.04850.04750.04650.0455

-1.50.06680.06550.06430.06300.06180.06060.05940.05820.05710.0559

-1.40.08080.07930.07780.07640.07490.07350.07210.07080.06940.0681

-1.30.09680.09510.09340.09180.09010.08850.08690.08530.08380.0823

-1.20.11510.11310.11120.10930.10750.10560.10380.10200.10030.0985

-1.10.13570.13350.13140.12920.12710.12510.12300.12100.11900.1170

-1.00.15870.15620.15390.15150.14920.14690.14460.14230.14010.1379

-0.90.18410.18140.17880.17620.17360.17110.16850.16600.16350.1611

-0.80.21190.20900.20610.20330.20050.19770.19490.19220.18940.1867

-0.70.24200.23890.23580.23270.22960.22660.22360.22060.21770.2148

-0.60.27430.27090.26760.26430.26110.25780.25460.25140.24830.2451

-0.50.30850.30500.30150.29810.29460.29120.28770.28430.28100.2776

-0.40.34460.34090.33720.33360.33000.32640.32280.31920.31560.3121

-0.30.38210.37830.37450.37070.36690.36320.35940.35570.35200.3483

-0.20.42070.41680.41290.40900.40520.40130.39740.39360.38970.3859

-0.10.46020.45620.45220.44830.44430.44040.43640.43250.42860.4247

0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359

0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753

0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.6141

0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.6517

0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.6879

0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.7224

0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.7549

0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.7852

0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.8133

0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.8389

1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.8621

1.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.8830

1.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.9015

1.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.9177

1.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.9319

1.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.9441

1.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.9545

1.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.9633

1.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.9706

1.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.9767

2.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.9817

2.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.9857

2.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890

2.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916

2.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936

2.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952

2.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964

2.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974

2.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981

2.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986

3.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990

rea da normal

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

student

x40.0077x60.000

media40.007860.6250.61012

dp0.0002s2.235

n10


rea a esquerda

confiana7.2599.9999999981

38.99%

distribuio normal

rea a direita

61.01%

Relogio comparador

90%95%99%

0.050.0250.005uc0.04

16.31412.70663.656ua0.03

22.9204.3039.925na5

32.3533.1825.841%95

42.1322.7764.604

52.0152.5714.032

61.9432.4473.707neff15

71.8952.3653.499

81.8602.3063.355K =2.13

91.8332.2623.250

101.8122.2283.169

111.7962.2013.106

121.7822.1793.055

131.7712.1603.012

141.7612.1452.977

151.7532.1312.947

161.7462.1202.921

171.7402.1102.898

181.7342.1012.878

191.7292.0932.861

201.7252.0862.845

211.7212.0802.831

221.7172.0742.819

231.7142.0692.807

241.7112.0642.797

251.7082.0602.787

261.7062.0562.779

271.7032.0522.771

281.7012.0482.763

291.6992.0452.756

301.6972.0422.750

401.6842.0212.704

601.6712.0002.660

1201.6581.9802.617

Relogio comparador

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

mmmm

0.00-3

0.120

0.231

0.352

0.441

0.565

0.667

0.755

0.834

0.934

1.013

1.5-2-1

2.0-2-1

2.5-4-3

3.0-5-5

3.5-3-2

4.0-2-1

4.5-10

5.011

6.021

7.022

8.047

9.056

10.078

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

entrada

sada

(mm)

(m)


Histograma16 classes

Grf3

11

22

33

55

66

88

99

1111

1414

1111

99

88

66

44

22

11

fi

fi

normal centro

Distribuio normal

ZpZp

0.10.07973.10.998061-0.68269

0.20.15853.20.99863

0.30.23583.30.99903

0.40.31083.40.99933

0.50.38293.50.99953

0.60.45153.60.99968

0.70.51613.70.99978

0.80.57633.80.99986

0.90.63193.90.99990

1.00.68274.00.99994

1.10.72874.10.99996

1.20.76994.20.999973

1.30.80644.30.999983

1.40.83854.40.999989

1.50.86644.50.999993

1.60.89044.60.999996

1.70.91094.70.999997

1.80.92814.80.9999984

1.90.94264.90.9999990

2.00.95455.00.9999994

2.10.96435.10.9999997

2.20.97225.20.99999980

2.30.97865.30.99999988

2.40.98365.40.99999993

2.50.98765.50.99999996

2.60.99075.60.999999979

2.70.99315.70.999999988

2.80.99495.80.999999993

2.90.99635.90.999999996

3.00.99736.00.999999998


5762.557.756.262.966.3

64.765.460.559.561.660.8

60.360.859.56061.863.8

57.461.260.9616359.5

60.560.861.558.558.960.5

mdia60.8333333333

amplitude10.1






somatoria x1825


grfico s

5763.163272618958.436727381160.8

56.263.163272618958.436727381160.8

57.463.163272618958.436727381160.8

57.763.163272618958.436727381160.8

58.563.163272618958.436727381160.8

58.963.163272618958.436727381160.8

59.563.163272618958.436727381160.8

59.563.163272618958.436727381160.8

59.563.163272618958.436727381160.8

6063.163272618958.436727381160.8

60.363.163272618958.436727381160.8

60.563.163272618958.436727381160.8

60.563.163272618958.436727381160.8

60.563.163272618958.436727381160.8

60.863.163272618958.436727381160.8

60.863.163272618958.436727381160.8

60.863.163272618958.436727381160.8

60.963.163272618958.436727381160.8

6163.163272618958.436727381160.8

61.263.163272618958.436727381160.8

61.563.163272618958.436727381160.8

61.663.163272618958.436727381160.8

61.863.163272618958.436727381160.8

62.563.163272618958.436727381160.8

62.963.163272618958.436727381160.8

6363.163272618958.436727381160.8

63.863.163272618958.436727381160.8

64.763.163272618958.436727381160.8

65.463.163272618958.436727381160.8

66.363.163272618958.436727381160.8

grfico s

histograma

eixofifi

511

1022

1533

2055

2566

3088

3599

401111

451414

501111

5599

6088

6566

7044

7522

8011

100

histograma

&A

Page &P

fi

fi

grafico normal

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

grafico normal

&A

Page &P

normal

rea da normal


Z =-1.74p =0.04093

Z0.000.010.020.030.040.050.060.070.080.09

-3.00.00130.00130.00130.00120.00120.00110.00110.00110.00100.0010

-2.90.00190.00180.00180.00170.00160.00160.00150.00150.00140.0014

-2.80.00260.00250.00240.00230.00230.00220.00210.00210.00200.0019

-2.70.00350.00340.00330.00320.00310.00300.00290.00280.00270.0026

-2.60.00470.00450.00440.00430.00410.00400.00390.00380.00370.0036

-2.50.00620.00600.00590.00570.00550.00540.00520.00510.00490.0048

-2.40.00820.00800.00780.00750.00730.00710.00690.00680.00660.0064

-2.30.01070.01040.01020.00990.00960.00940.00910.00890.00870.0084

-2.20.01390.01360.01320.01290.01250.01220.01190.01160.01130.0110

-2.10.01790.01740.01700.01660.01620.01580.01540.01500.01460.0143

-2.00.02280.02220.02170.02120.02070.02020.01970.01920.01880.0183

-1.90.02870.02810.02740.02680.02620.02560.02500.02440.02390.0233

-1.80.03590.03510.03440.03360.03290.03220.03140.03070.03010.0294

-1.70.04460.04360.04270.04180.04090.04010.03920.03840.03750.0367

-1.60.05480.05370.05260.05160.05050.04950.04850.04750.04650.0455

-1.50.06680.06550.06430.06300.06180.06060.05940.05820.05710.0559

-1.40.08080.07930.07780.07640.07490.07350.07210.07080.06940.0681

-1.30.09680.09510.09340.09180.09010.08850.08690.08530.08380.0823

-1.20.11510.11310.11120.10930.10750.10560.10380.10200.10030.0985

-1.10.13570.13350.13140.12920.12710.12510.12300.12100.11900.1170

-1.00.15870.15620.15390.15150.14920.14690.14460.14230.14010.1379

-0.90.18410.18140.17880.17620.17360.17110.16850.16600.16350.1611

-0.80.21190.20900.20610.20330.20050.19770.19490.19220.18940.1867

-0.70.24200.23890.23580.23270.22960.22660.22360.22060.21770.2148

-0.60.27430.27090.26760.26430.26110.25780.25460.25140.24830.2451

-0.50.30850.30500.30150.29810.29460.29120.28770.28430.28100.2776

-0.40.34460.34090.33720.33360.33000.32640.32280.31920.31560.3121

-0.30.38210.37830.37450.37070.36690.36320.35940.35570.35200.3483

-0.20.42070.41680.41290.40900.40520.40130.39740.39360.38970.3859

-0.10.46020.45620.45220.44830.44430.44040.43640.43250.42860.4247

0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359

0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753

0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.6141

0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.6517

0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.6879

0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.7224

0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.7549

0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.7852

0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.8133

0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.8389

1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.8621

1.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.8830

1.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.9015

1.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.9177

1.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.9319

1.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.9441

1.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.9545

1.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.9633

1.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.9706

1.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.9767

2.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.9817

2.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.9857

2.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890

2.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916

2.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936

2.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952

2.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964

2.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974

2.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981

2.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986

3.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990

rea da normal

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

student

x40.0077x60.000

media40.007860.6250.61012

dp0.0002s2.235

n10


rea a esquerda

confiana7.2599.9999999981

38.99%

distribuio normal

rea a direita

61.01%

Relogio comparador

90%95%99%

0.050.0250.0050.04

16.31412.70663.6560.03

22.9204.3039.9255

32.3533.1825.841%95

42.1322.7764.604

52.0152.5714.032

61.9432.4473.70715

71.8952.3653.499

81.8602.3063.355K =2.13

91.8332.2623.250

101.8122.2283.169

111.7962.2013.106

121.7822.1793.055

131.7712.1603.012

141.7612.1452.977

151.7532.1312.947

161.7462.1202.921

171.7402.1102.898

181.7342.1012.878

191.7292.0932.861

201.7252.0862.845

211.7212.0802.831

221.7172.0742.819

231.7142.0692.807

241.7112.0642.797

251.7082.0602.787

261.7062.0562.779

271.7032.0522.771

281.7012.0482.763

291.6992.0452.756

301.6972.0422.750

401.6842.0212.704

601.6712.0002.660

1201.6581.9802.617

Relogio comparador

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

mmmm

0.00-3

0.120

0.231

0.352

0.441

0.565

0.667

0.755

0.834

0.934

1.013

1.5-2-1

2.0-2-1

2.5-4-3

3.0-5-5

3.5-3-2

4.0-2-1

4.5-10

5.011

6.021

7.022

8.047

9.056

10.078

entrada

sada

(mm)

(m)


HistogramaTendendo a infinitas classes

Grf2

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

normal centro

Distribuio normal

ZpZp

0.10.07973.10.998061-0.68269

0.20.15853.20.99863

0.30.23583.30.99903

0.40.31083.40.99933

0.50.38293.50.99953

0.60.45153.60.99968

0.70.51613.70.99978

0.80.57633.80.99986

0.90.63193.90.99990

1.00.68274.00.99994

1.10.72874.10.99996

1.20.76994.20.999973

1.30.80644.30.999983

1.40.83854.40.999989

1.50.86644.50.999993

1.60.89044.60.999996

1.70.91094.70.999997

1.80.92814.80.9999984

1.90.94264.90.9999990

2.00.95455.00.9999994

2.10.96435.10.9999997

2.20.97225.20.99999980

2.30.97865.30.99999988

2.40.98365.40.99999993

2.50.98765.50.99999996

2.60.99075.60.999999979

2.70.99315.70.999999988

2.80.99495.80.999999993

2.90.99635.90.999999996

3.00.99736.00.999999998


5762.557.756.262.966.3

64.765.460.559.561.660.8

60.360.859.56061.863.8

57.461.260.9616359.5

60.560.861.558.558.960.5

mdia60.8333333333

amplitude10.1






somatoria x1825


grfico s

5763.163272618958.436727381160.8

56.263.163272618958.436727381160.8

57.463.163272618958.436727381160.8

57.763.163272618958.436727381160.8

58.563.163272618958.436727381160.8

58.963.163272618958.436727381160.8

59.563.163272618958.436727381160.8

59.563.163272618958.436727381160.8

59.563.163272618958.436727381160.8

6063.163272618958.436727381160.8

60.363.163272618958.436727381160.8

60.563.163272618958.436727381160.8

60.563.163272618958.436727381160.8

60.563.163272618958.436727381160.8

60.863.163272618958.436727381160.8

60.863.163272618958.436727381160.8

60.863.163272618958.436727381160.8

60.963.163272618958.436727381160.8

6163.163272618958.436727381160.8

61.263.163272618958.436727381160.8

61.563.163272618958.436727381160.8

61.663.163272618958.436727381160.8

61.863.163272618958.436727381160.8

62.563.163272618958.436727381160.8

62.963.163272618958.436727381160.8

6363.163272618958.436727381160.8

63.863.163272618958.436727381160.8

64.763.163272618958.436727381160.8

65.463.163272618958.436727381160.8

66.363.163272618958.436727381160.8

grfico s

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

0000

histograma

eixofifi

1022

2044

3088

401515

501414

6088

7044

8022

histograma

00

00

00

00

00

00

00

00

&A

Page &P

fi

fi

grafico normal

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

grafico normal

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

&A

Page &P

normal

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

rea da normal


Z =-1.74p =0.04093

Z0.000.010.020.030.040.050.060.070.080.09

-3.00.00130.00130.00130.00120.00120.00110.00110.00110.00100.0010

-2.90.00190.00180.00180.00170.00160.00160.00150.00150.00140.0014

-2.80.00260.00250.00240.00230.00230.00220.00210.00210.00200.0019

-2.70.00350.00340.00330.00320.00310.00300.00290.00280.00270.0026

-2.60.00470.00450.00440.00430.00410.00400.00390.00380.00370.0036

-2.50.00620.00600.00590.00570.00550.00540.00520.00510.00490.0048

-2.40.00820.00800.00780.00750.00730.00710.00690.00680.00660.0064

-2.30.01070.01040.01020.00990.00960.00940.00910.00890.00870.0084

-2.20.01390.01360.01320.01290.01250.01220.01190.01160.01130.0110

-2.10.01790.01740.01700.01660.01620.01580.01540.01500.01460.0143

-2.00.02280.02220.02170.02120.02070.02020.01970.01920.01880.0183

-1.90.02870.02810.02740.02680.02620.02560.02500.02440.02390.0233

-1.80.03590.03510.03440.03360.03290.03220.03140.03070.03010.0294

-1.70.04460.04360.04270.04180.04090.04010.03920.03840.03750.0367

-1.60.05480.05370.05260.05160.05050.04950.04850.04750.04650.0455

-1.50.06680.06550.06430.06300.06180.06060.05940.05820.05710.0559

-1.40.08080.07930.07780.07640.07490.07350.07210.07080.06940.0681

-1.30.09680.09510.09340.09180.09010.08850.08690.08530.08380.0823

-1.20.11510.11310.11120.10930.10750.10560.10380.10200.10030.0985

-1.10.13570.13350.13140.12920.12710.12510.12300.12100.11900.1170

-1.00.15870.15620.15390.15150.14920.14690.14460.14230.14010.1379

-0.90.18410.18140.17880.17620.17360.17110.16850.16600.16350.1611

-0.80.21190.20900.20610.20330.20050.19770.19490.19220.18940.1867

-0.70.24200.23890.23580.23270.22960.22660.22360.22060.21770.2148

-0.60.27430.27090.26760.26430.26110.25780.25460.25140.24830.2451

-0.50.30850.30500.30150.29810.29460.29120.28770.28430.28100.2776

-0.40.34460.34090.33720.33360.33000.32640.32280.31920.31560.3121

-0.30.38210.37830.37450.37070.36690.36320.35940.35570.35200.3483

-0.20.42070.41680.41290.40900.40520.40130.39740.39360.38970.3859

-0.10.46020.45620.45220.44830.44430.44040.43640.43250.42860.4247

0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359

0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753

0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.6141

0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.6517

0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.6879

0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.7224

0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.7549

0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.7852

0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.8133

0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.8389

1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.8621

1.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.8830

1.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.9015

1.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.9177

1.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.9319

1.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.9441

1.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.9545

1.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.9633

1.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.9706

1.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.9767

2.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.9817

2.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.9857

2.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890

2.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916

2.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936

2.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952

2.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964

2.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974

2.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981

2.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986

3.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990

rea da normal

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

student

x40.0077x60.000

media40.007860.6250.61012

dp0.0002s2.235

n10


rea a esquerda

confiana7.2599.9999999981

38.99%

distribuio normal

rea a direita

61.01%

Relogio comparador

90%95%99%

0.050.0250.005uc0.04

16.31412.70663.656ua0.03

22.9204.3039.925na5

32.3533.1825.841%95

42.1322.7764.604

52.0152.5714.032

61.9432.4473.707neff15

71.8952.3653.499

81.8602.3063.355K =2.13

91.8332.2623.250

101.8122.2283.169

111.7962.2013.106

121.7822.1793.055

131.7712.1603.012

141.7612.1452.977

151.7532.1312.947

161.7462.1202.921

171.7402.1102.898

181.7342.1012.878

191.7292.0932.861

201.7252.0862.845

211.7212.0802.831

221.7172.0742.819

231.7142.0692.807

241.7112.0642.797

251.7082.0602.787

261.7062.0562.779

271.7032.0522.771

281.7012.0482.763

291.6992.0452.756

301.6972.0422.750

401.6842.0212.704

601.6712.0002.660

1201.6581.9802.617

Relogio comparador

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

mmmm

0.00-3

0.120

0.231

0.352

0.441

0.565

0.667

0.755

0.834

0.934

1.013

1.5-2-1

2.0-2-1

2.5-4-3

3.0-5-5

3.5-3-2

4.0-2-1

4.5-10

5.011

6.021

7.022

8.047

9.056

10.078

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

00

entrada

sada

(mm)

(m)


Pode-se afirmar que esta distribuio a mais importante dentre as demais, tendo em vista a sua grande aplicabilidade.O histograma constitudo de uma srie de retngulos.

medida que o nmero de dados aumenta, os intervalos de classe podem ser reduzidos e essa sucesso de retngulos vai-se aproximando de uma curva contnua.

necessrio que se lembre, esta curva contnua uma abstrao matemtica, uma vez que para se chegar mesma necessrio um nmero infinito e medies.Distribuio Normal

A distribuio normal envolve apenas dois parmetros estatsticos, a mdia e o desvio padro da populao.O desvio padro o ponto em que h inverso da curvatura.A rea total sob a curva f(x) igual unidade. Em virtude da simetria, as reas esquerda da mdia so iguais s da direita.A rea abaixo da curva normal equivale a 100 % ou 1, e significa que todos os valores possveis se encontram representados pela curva. Distribuio Normal

Grfico2

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

Plan1

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

Plan1

&A

Page &P

Plan2

&A

Page &P

Plan3

Distribuio Normal

Z0,000,010,020,030,040,050,060,070,080,09-1,00,15870,15620,15390,15150,14920,14690,14460,14230,14010,1379-0,90,18410,18140,17880,17620,17360,17110,16850,16600,16350,1611-0,80,21190,20900,20610,20330,20050,19770,19490,19220,18940,1867-0,70,24200,23890,23580,23270,22960,22660,22360,22060,21770,2148-0,60,27430,27090,26760,26430,26110,25780,25460,25140,24830,2451-0,50,30850,30500,30150,29810,29460,29120,28770,28430,28100,2776-0,40,34460,34090,33720,33360,33000,32640,32280,31920,31560,3121-0,30,38210,37830,37450,37070,36690,36320,35940,35570,35200,3483-0,20,42070,41680,41290,40900,40520,40130,39740,39360,38970,3859-0,10,46020,45620,45220,44830,44430,44040,43640,43250,42860,42470,00,50000,50400,50800,51200,51600,51990,52390,52790,53190,53590,10,53980,54380,54780,55170,55570,55960,56360,56750,57140,57530,20,57930,58320,58710,59100,59480,59870,60260,60640,61030,61410,30,61790,62170,62550,62930,63310,63680,64060,64430,64800,65170,40,65540,65910,66280,66640,67000,67360,67720,68080,68440,68790,50,69150,69500,69850,70190,70540,70880,71230,71570,71900,72240,60,72570,72910,73240,73570,73890,74220,74540,74860,75170,75490,70,75800,76110,76420,76730,77040,77340,77640,77940,78230,78520,80,78810,79100,79390,79670,79950,80230,80510,80780,81060,81330,90,81590,81860,82120,82380,82640,82890,83150,83400,83650,83891,00,84130,84380,84610,84850,85080,85310,85540,85770,85990,8621

Grf1

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

normal (2)


Z =-1.74p =0.04093

Z0.000.010.020.030.040.050.060.070.080.09

-1.00.15870.15620.15390.15150.14920.14690.14460.14230.14010.1379

-0.90.18410.18140.17880.17620.17360.17110.16850.16600.16350.1611

-0.80.21190.20900.20610.20330.20050.19770.19490.19220.18940.1867

-0.70.24200.23890.23580.23270.22960.22660.22360.22060.21770.2148

-0.60.27430.27090.26760.26430.26110.25780.25460.25140.24830.2451

-0.50.30850.30500.30150.29810.29460.29120.28770.28430.28100.2776

-0.40.34460.34090.33720.33360.33000.32640.32280.31920.31560.3121

-0.30.38210.37830.37450.37070.36690.36320.35940.35570.35200.3483

-0.20.42070.41680.41290.40900.40520.40130.39740.39360.38970.3859

-0.10.46020.45620.45220.44830.44430.44040.43640.43250.42860.4247

0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359

0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753

0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.6141

0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.6517

0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.6879

0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.7224

0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.7549

0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.7852

0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.8133

0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.8389

1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.8621

normal (2)

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

normal centro

Distribuio normal

ZpZp

0.10.07973.10.998061-0.68269

0.20.15853.20.99863

0.30.23583.30.99903

0.40.31083.40.99933

0.50.38293.50.99953

0.60.45153.60.99968

0.70.51613.70.99978

0.80.57633.80.99986

0.90.63193.90.99990

1.00.68274.00.99994

1.10.72874.10.99996

1.20.76994.20.999973

1.30.80644.30.999983

1.40.83854.40.999989

1.50.86644.50.999993

1.60.89044.60.999996

1.70.91094.70.999997

1.80.92814.80.9999984

1.90.94264.90.9999990

2.00.95455.00.9999994

2.10.96435.10.9999997

2.20.97225.20.99999980

2.30.97865.30.99999988

2.40.98365.40.99999993

2.50.98765.50.99999996

2.60.99075.60.999999979

2.70.99315.70.999999988

2.80.99495.80.999999993

2.90.99635.90.999999996

3.00.99736.00.999999998


5762.557.756.262.966.3

64.765.460.559.561.660.8

60.360.859.56061.863.8

57.461.260.9616359.5

60.560.861.558.558.960.5

mdia60.8333333333

amplitude10.1






somatoria x1825


grfico s

5763.163272618958.436727381160.8

56.263.163272618958.436727381160.8

57.463.163272618958.436727381160.8

57.763.163272618958.436727381160.8

58.563.163272618958.436727381160.8

58.963.163272618958.436727381160.8

59.563.163272618958.436727381160.8

59.563.163272618958.436727381160.8

59.563.163272618958.436727381160.8

6063.163272618958.436727381160.8

60.363.163272618958.436727381160.8

60.563.163272618958.436727381160.8

60.563.163272618958.436727381160.8

60.563.163272618958.436727381160.8

60.863.163272618958.436727381160.8

60.863.163272618958.436727381160.8

60.863.163272618958.436727381160.8

60.963.163272618958.436727381160.8

6163.163272618958.436727381160.8

61.263.163272618958.436727381160.8

61.563.163272618958.436727381160.8

61.663.163272618958.436727381160.8

61.863.163272618958.436727381160.8

62.563.163272618958.436727381160.8

62.963.163272618958.436727381160.8

6363.163272618958.436727381160.8

63.863.163272618958.436727381160.8

64.763.163272618958.436727381160.8

65.463.163272618958.436727381160.8

66.363.163272618958.436727381160.8

grfico s

histograma

eixofifi

0.302020

0.458080

0.60113.3113.3

0.75186.7186.7

0.90173.3173.3

1.00140140

1.201010

1.4055

15

20200

252010

302020

358035

408057

458080

50113.391

55113.3103

60113.3113.3

65186.7134

70186.7158

75186.7186.7

80173.3182

85173.3178

90173.3173.3

95140157

100140140

1051095

1101052

1151010

120108.7

12557.5

13056.3

13555

14002.5

15000

histograma

&A

Page &P

fi

fi

grafico normal

normal

rea da normal

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

rea da normal

&A

Page &P

Plan1

student


Z =-1.74p =0.04093

Z0.000.010.020.030.040.050.060.070.080.09

-3.00.00130.00130.00130.00120.00120.00110.00110.00110.00100.0010

-2.90.00190.00180.00180.00170.00160.00160.00150.00150.00140.0014

-2.80.00260.00250.00240.00230.00230.00220.00210.00210.00200.0019

-2.70.00350.00340.00330.00320.00310.00300.00290.00280.00270.0026

-2.60.00470.00450.00440.00430.00410.00400.00390.00380.00370.0036

-2.50.00620.00600.00590.00570.00550.00540.00520.00510.00490.0048

-2.40.00820.00800.00780.00750.00730.00710.00690.00680.00660.0064

-2.30.01070.01040.01020.00990.00960.00940.00910.00890.00870.0084

-2.20.01390.01360.01320.01290.01250.01220.01190.01160.01130.0110

-2.10.01790.01740.01700.01660.01620.01580.01540.01500.01460.0143

-2.00.02280.02220.02170.02120.02070.02020.01970.01920.01880.0183

-1.90.02870.02810.02740.02680.02620.02560.02500.02440.02390.0233

-1.80.03590.03510.03440.03360.03290.03220.03140.03070.03010.0294

-1.70.04460.04360.04270.04180.04090.04010.03920.03840.03750.0367

-1.60.05480.05370.05260.05160.05050.04950.04850.04750.04650.0455

-1.50.06680.06550.06430.06300.06180.06060.05940.05820.05710.0559

-1.40.08080.07930.07780.07640.07490.07350.07210.07080.06940.0681

-1.30.09680.09510.09340.09180.09010.08850.08690.08530.08380.0823

-1.20.11510.11310.11120.10930.10750.10560.10380.10200.10030.0985

-1.10.13570.13350.13140.12920.12710.12510.12300.12100.11900.1170

-1.00.15870.15620.15390.15150.14920.14690.14460.14230.14010.1379

-0.90.18410.18140.17880.17620.17360.17110.16850.16600.16350.1611

-0.80.21190.20900.20610.20330.20050.19770.19490.19220.18940.1867

-0.70.24200.23890.23580.23270.22960.22660.22360.22060.21770.2148

-0.60.27430.27090.26760.26430.26110.25780.25460.25140.24830.2451

-0.50.30850.30500.30150.29810.29460.29120.28770.28430.28100.2776

-0.40.34460.34090.33720.33360.33000.32640.32280.31920.31560.3121

-0.30.38210.37830.37450.37070.36690.36320.35940.35570.35200.3483

-0.20.42070.41680.41290.40900.40520.40130.39740.39360.38970.3859

-0.10.46020.45620.45220.44830.44430.44040.43640.43250.42860.4247

0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359

0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753

0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.6141

0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.6517

0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.6879

0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.7224

0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.7549

0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.7852

0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.8133

0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.8389

1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.8621

1.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.8830

1.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.9015

1.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.9177

1.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.9319

1.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.9441

1.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.9545

1.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.9633

1.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.9706

1.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.9767

2.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.9817

2.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.9857

2.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890

2.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916

2.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936

2.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952

2.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964

2.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974

2.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981

2.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986

3.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990

student

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

Relogio comparador

x40.0077x60.000

media40.007860.6250.61012

dp0.0002s2.235

n10


rea a esquerda

confiana7.2599.9999999981

38.99%

distribuio normal

rea a direita

61.01%

AnoInscriesEstado CivilInscries

20052753Solteiro952

20062875Casado1537

20072803Vivo684

20082920Divorciado651

total11351Desquitado730

Outros382

Total4936

EstadoInscries

AM203

PE322Grau de escolaridadesexo

MT219MasculinoFeminino

ES2351 grau incompleto82116

SC3071 grau completo235128

Total12862 grau324204

3 grau9282

total733530

Inscries

Inscries

Inscries

sexo Masculino

sexo Feminino

sexo Masculino

sexo Feminino

90%95%99%

0.050.0250.0050.04

16.31412.70663.6560.03

22.9204.3039.9255

32.3533.1825.841%95

42.1322.7764.604

52.0152.5714.032

61.9432.4473.70715

71.8952.3653.499

81.8602.3063.355K =2.13

91.8332.2623.250

101.8122.2283.169

111.7962.2013.106

121.7822.1793.055

131.7712.1603.012

141.7612.1452.977

151.7532.1312.947

161.7462.1202.921

171.7402.1102.898

181.7342.1012.878

191.7292.0932.861

201.7252.0862.845

211.7212.0802.831

221.7172.0742.819

231.7142.0692.807

241.7112.0642.797

251.7082.0602.787

261.7062.0562.779

271.7032.0522.771

281.7012.0482.763

291.6992.0452.756

301.6972.0422.750

401.6842.0212.704

601.6712.0002.660

1201.6581.9802.617

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

mmmm

0.00-3

0.120

0.231

0.352

0.441

0.565

0.667

0.755

0.834

0.934

1.013

1.5-2-1

2.0-2-1

2.5-4-3

3.0-5-5

3.5-3-2

4.0-2-1

4.5-10

5.011

6.021

7.022

8.047

9.056

10.078

entrada

sada

(mm)

(m)


Distribuio Normal

Algumas reas da curva normalIntervalo% da rea dentro do intervalo 0,67 50,00 1,0 68,26 1,96 95,00 2,0 95,46 3,0 99,73 4,0 99,99

Distribuio Normal

ZpZpZp0,10,07972,10,96434,10,999960,20,15852,20,97224,20,9999730,30,23582,30,97864,30,9999830,40,31082,40,98364,40,9999890,50,38292,50,98764,50,9999930,60,45152,60,99074,60,9999960,70,51612,70,99314,70,9999970,80,57632,80,99494,80,99999840,90,63192,90,99634,90,99999901,00,68273,00,99735,00,99999941,10,72873,10,998065,10,99999971,20,76993,20,998635,20,999999801,30,80643,30,999035,30,999999881,40,83853,40,999335,40,999999931,50,86643,50,999535,50,999999961,60,89043,60,999685,60,9999999791,70,91093,70,999785,70,9999999881,80,92813,80,999865,80,9999999931,90,94263,90,999905,90,9999999962,00,95454,00,999946,00,999999998

Distribuio t de student

Quanto mais vamos diminuindo o nmero de medies estamos nos afastando de uma distribuio normal com o desvio da populao () e a mdia da populao ().

Isto a amostra vai perdendo ou deixando de representar com fidelidade as caractersticas da populao.

E como geralmente trabalhamos com um pequeno nmero de medies ( 3 a 10), utilizamos a distribuio t para amostras pequenas (em geral < 30 elementos).Distribuio t de student

O valor de t tabelado e determinado atravs do conhecimento da tamanho da amostra (n) e do nvel de confiana que se deseja determinar o intervalo em que esta a mdia da populao (). Distribuio t de studentDistribuio normalDistribuio t

O nvel utilizado na metrologia de 95 % de confiana.

Para identificar o valor t na tabela necessrio conhecer o grau de liberdade (v) associado a um nvel de confiana.

O grau de liberdade de um conjunto de n repeties igual n-1.

( graus de liberdade ) = n - 1 Uma derivao matemtica de graus de liberdade esta alem do escopo deste curso, mas o conceito bsico pode ser estabelecido: Graus de liberdade uma medida de segurana envolvida quando um desvio padro amostral usado para estimar o desvio padro verdadeiro de um universo.Distribuio t de student

Distribuio t de student

95%95%2,5%2,5%112,706172,11024,303182,10133,182192,09342,776202,08652,571212,08062,447222,07472,365232,06982,306242,06492,262252,060102,228262,056112,201272,052122,179282,048132,160292,045142,145302,042152,131602,000162,1201201,980

Grf2

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

normal (2)


Z =-1.74p =0.04093

Z0.000.010.020.030.040.050.060.070.080.09

-1.00.15870.15620.15390.15150.14920.14690.14460.14230.14010.1379

-0.90.18410.18140.17880.17620.17360.17110.16850.16600.16350.1611

-0.80.21190.20900.20610.20330.20050.19770.19490.19220.18940.1867

-0.70.24200.23890.23580.23270.22960.22660.22360.22060.21770.2148

-0.60.27430.27090.26760.26430.26110.25780.25460.25140.24830.2451

-0.50.30850.30500.30150.29810.29460.29120.28770.28430.28100.2776

-0.40.34460.34090.33720.33360.33000.32640.32280.31920.31560.3121

-0.30.38210.37830.37450.37070.36690.36320.35940.35570.35200.3483

-0.20.42070.41680.41290.40900.40520.40130.39740.39360.38970.3859

-0.10.46020.45620.45220.44830.44430.44040.43640.43250.42860.4247

0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359

0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753

0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.6141

0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.6517

0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.6879

0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.7224

0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.7549

0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.7852

0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.8133

0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.8389

1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.8621

normal (2)

0.6

1

1.4

1.7

2.3

2.7

3.2

3.7

4.2

5

5.8

6.5

7.8

10

12

15

19

24

27.5

30.5

32.5

34

35

35.4

35.4

35

34

32.5

30.5

27.5

24

19

15

12

10

7.8

6.5

5.8

5

4.2

3.7

3.2

2.7

2.3

1.7

1.4

1

0.6

normal centro

Distribuio normal

ZpZpZp

0.10.07972.10.96434.10.99996

0.20.15852.20.97224.20.999973

0.30.23582.30.97864.30.999983

0.40.31082.40.98364.40.999989

0.50.38292.50.98764.50.999993

0.60.45152.60.99074.60.999996

0.70.51612.70.99314.70.999997

0.80.57632.80.99494.80.9999984

0.90.63192.90.99634.90.9999990

1.00.68273.00.99735.00.9999994

1.10.72873.10.998065.10.9999997

1.20.76993.20.998635.20.99999980

1.30.80643.30.999035.30.99999988

1.40.83853.40.999335.40.99999993

1.50.86643.50.999535.50.99999996

1.60.89043.60.999685.60.999999979

1.70.91093.70.999785.70.999999988

1.80.92813.80.999865.80.999999993

1.90.94263.90.999905.90.999999996

2.00.95454.00.999946.00.999999998


5762.557.756.262.966.3

64.765.460.559.561.660.8

60.360.859.56061.863.8

57.461.260.9616359.5

60.560.861.558.558.960.5

mdia60.8333333333

ampli

Documents

Estatística