View
14
Download
1
Embed Size (px)
DESCRIPTION
Estatística básica, voltada para área de metrologia.
Citation preview
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
exerccios15508:30 s 9:20
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
exerccios15758:15 s 9:30
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
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
exerccios15508:30 s 9:20
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
exerccios15758:15 s 9:30
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
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
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
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
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
t1.581distribuio normal0.0000000019
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)
Erros do Relgio comparador
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
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
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
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
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
t1.581distribuio normal0.0000000019
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)
Erros do Relgio comparador
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
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
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
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
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
t1.581distribuio normal0.0000000019
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)
Erros do Relgio comparador
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)
Tabela da distribuio Normal
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
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
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
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
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)
Erros do Relgio comparador
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)
Tabela da distribuio Normal
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
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
ampli