exerc (1).pdf

UNIFEI - Universidade Federal de Itajubá Instituto de Engenharia de Produção & Gestão

EXERCÍCIOS E TABELAS

PEDRO PAULO BALESTRASSI

ANDERSON PAULO DE PAIVA

Itajubá/2009

EPR503 – Exercícios e Tabelas.

2

Capítulo 1 - ESTATÍSTICA DESCRITIVA 1) Um projeto é necessário avaliar-se a folga que separa dois componentes, já que se desconfia que em algumas vezes a montagem do dispositivo não será possível. O componente A (eixo) tem sua dimensão com distribuição normal, com média 9,90 mm e desvio-padrão de 0,02mm. O componente B (Furo) tem também distribuição normal, mas com média 10,05 mm e desvio-padrão de 0,04 mm, conforme mostra a figura ao lado. Determine a média, desvio padrão e a tolerância (T) para a folga entre os componentes.

2) Um eixo é formado por 3 componentes x1, x2 e x3. Sabe-se que estas dimensões (em mm) se distribuem normalmente, tal que x1~N(40; 2), x2~N(35; 3) e x3~N(45; 5). Determine o (i) comprimento médio, (ii) o desvio padrão e a (iii) tolerância do sistema após a montagem.

3) The demand for bottled water increases during the hurricane season in Florida. The operations manager at a plant that bottles drinking water wants to be sure that the filling process for one-gallon bottles is operating properly. Currently, the company is testing the weights of one-gallon (3,78 l) bottles. A random sample of 75 bottles is tested and the weights are described in the following table. 3,57 3,63 3,64 3,65 3,67 3,67 3,67 3,69 3,69 3,71 3,71 3,71 3,71 3,72 3,723,73 3,74 3,74 3,74 3,74 3,74 3,75 3,75 3,75 3,76 3,76 3,77 3,77 3,77 3,773,77 3,77 3,78 3,78 3,79 3,79 3,79 3,79 3,80 3,81 3,81 3,81 3,81 3,81 3,823,82 3,82 3,82 3,82 3,84 3,84 3,85 3,85 3,86 3,86 3,87 3,87 3,88 3,89 3,893,90 3,91 3,93 3,93 3,94 3,94 3,94 3,94 3,95 3,96 3,96 3,98 3,99 4,06 4,11

a) Find the mean, range, variance and standard deviation of the bottle weights; b) Find the five-number summary of the weights and the interquartile range for this

data; c) What is the value of the coefficient of variation? Construct a box-and-whisker plot.

4) The accompanying table shows test scores of 40 students. Write a complete report according to the aforementioned items of Question 3.

54 62 68 73 79 83 89 93 56 62 70 75 81 85 89 93 56 66 70 77 81 86 90 94 59 67 73 78 82 86 90 95 60 68 73 79 83 88 91 98

AB

.


3

5) The following data shows the percentage returns for the 25 largest U.S. common stock mutual funds for a particular day. Write a summary report as cited in Question 3.

133 187 194 199 203 203 215 240 245 246 247 256 280 296 303 307 308 308 312 323 329 368 380 395 509 -

6) Solve Questions 3, 4 and 5 using the grouped data techniques. 7) Mostre que:

( a ) ( )x xii

n

− ==∑ 0

1

( b ) ( ) ( )∑ ∑∑ ∑== =

−=−=−n

i

ii

n

i

n

iii n

xxxnxxx

1

222

1 1

22

( c ) ( )n x x n x nxii

k

i i ii

k

= =∑ ∑− = −

1

2 2 2

1

( d ) ( )f x x f x xii

k

i i ii

k

= =∑ ∑− = −

1

2 2 2

1

8) O conjunto abaixo representa as notas do exame final de uma determinada turma:

10 45 51 60 62 65 67 71 75 81 15 45 53 60 62 66 67 71 75 81 23 47 54 60 63 66 68 71 75 83 33 48 54 60 63 66 69 73 76 85 34 48 55 60 64 66 70 73 76 85 35 50 57 61 64 66 70 74 77 86 37 50 58 61 64 66 70 74 77 88 42 50 58 61 65 67 71 75 79 92

Construir uma distribuição de freqüência, adotando um intervalo de classe conveniente, o histograma e o polígono de freqüência. Calcular a média, o desvio padrão, a mediana, o 1o quartil e o 65o percentil.

9) Por engano, um professor omitiu uma nota no conjunto de notas de 10 alunos. Se as nove notas restantes são 48, 71, 79, 95, 45, 57, 75, 83, 97 e a média das 10 notas é 72, qual o valor da nota omitida.

10) São os seguintes os números de alunos e respectivos QI médios em três estabelecimentos de ensino: Qual é o QI médio global dos três estabelecimentos? Que tipo de média foi utilizada?

Estabelecimento No de Alunos QI Médio A 790 104 B 155 110 C 530 106


4

11) Em certo ano, uma empresa pagou a cada um de seus 45 estagiários um salário médio mensal de R$ 800, a cada um de seus 67 funcionários juniores R$ 2.000, a cada um de seus 58 plenos R$ 2.600 e a cada um de seus 32 seniores R$ 3.100. Qual o salário médio mensal dos 202 empregados?

12) Dado o histograma abaixo, calcular a média, a variância, a mediana e o 3o quartil. Sabe-se que o número total de observações é 90.

10 20 30 40 50 60 70 80 90

23 3

4

13

20

25

15

5

13) Dado o conjunto de observações 120 107 95 118 150 130 132 109 136 ( a ) Determinar os quartis. ( b ) Calcular a média aparada com m = 1. ( c ) A média aparada praticamente coincide com a média aritmética simples. Por

quê?

14) Calcule x , S2 e S para os dados abaixo, onde os valores de x são pontos médios de intervalos. Determine o escore padronizado para x = 80:

xi 15 25 35 45 55 65 75 85 95 105 ni 1 5 12 18 21 19 10 7 6 1

15) A série a seguir representa os tempos, em minutos, de espera numa fila de ônibus, durante 13 dias, de um cidadão que se dirige diariamente ao seu emprego:

15 10 2 17 6 8 3 10 2 9 5 9 13 Calcular a média e a mediana.

16) Dada a seguinte distribuição de idades dos membros de uma sociedade, Calcular: a) x , s, ~x , Q1, Q3;

b) o intervalo interquartílico Q3 - Q1;

Idade ni Idade ni 15 ├── 20 16 35 ├── 40 17 20 ├── 25 35 40 ├── 45 8 25 ├── 30 44 45 ├── 50 2 30 ├── 35 27 50 ├── 55 1


5

17) Para comparar a precisão de dois micrômetros, um técnico estuda medidas tomadas com ambos os aparelhos. Com um mediu repetidamente o diâmetro de uma pequena esfera de rolamento; as mensurações acusam média de 5,32 mm e d.p. de 0,019 mm. Com outro mediu o comprimento natural de uma mola, tendo as mensurações acusado média de 6,4 cm e d.p. de 0,03 cm. Qual dos dois aparelhos é relativamente mais preciso?

18) O que acontece com a mediana, a média e o desvio padrão de uma série de dados quando: a) Cada observação é multiplicada por 2; b) Soma-se 10 a cada observação; c) Subtrai-se a média geral x de cada observação; d) De cada observação subtrai-se x e divide-se pelo desvio padrão.

19) A tabela abaixo mostra o tempo gasto por empregados numa determinada operação em uma fábrica: a) Esboce o histograma correspondente; b) Calcule a média, o desvio padrão; c) O presidente decide separar os funcionários mais rápidos (com tempo inferior a um desvio padrão abaixo da média) para receberem promoção. Qual a porcentagem desses funcionários?

20) Em testes de resistência à tração de um tipo de aço, obteve-se um coeficiente de variação de 8%. O escore padronizado obtido para uma resistência de 28 kg/mm2 foi de -0,4. a) Qual a resistência média? b) Qual o desvio padrão? c) O que significa o resultado negativo do escore padronizado?

21) Os números de lugares vagos em vôos entre duas cidades foram agrupados nas classes 0├──5 5├──10 10├──20 20├──25 25├──30 30 ou mais.

Com esta distribuição é possível determinar o número de vôos em que há: (a) menos de 20 assentos vagos; (b) mais de 20; (c) ao menos 9; (d) no máximo 9; (e) exatamente 5; (f) entre 10 e 25?

22) Os pontos médios de uma distribuição de leituras de temperaturas são 16, 25, 34, 43, 52, 61. Determinar os limites de classe e o intervalo de classe.

Tempo (min) Freqüência 10 ├── 15 5 15 ├── 20 57 20 ├── 25 42 25 ├── 30 28 30 ├── 40 18


6

NOTA: Os exercícios seguintes foram extraídos do capítulo 2 do livro Statistics for Business and Economics (Newbold et al., 2003).

23) The demand for bottled water increases during the hurricane season in Florida. The operations manger at a plant that bottles drinking water wants to be sure that the filling process for one-gallon bottles is operating properly. Currently, the company is testing the weights of one-gallon (3,78 l) bottles. A random sample of 75 bottles is tested and the weights are in the following table. Write a report of your findings. 3,57 3,63 3,64 3,65 3,67 3,67 3,67 3,69 3,69 3,71 3,71 3,71 3,71 3,72 3,723,73 3,74 3,74 3,74 3,74 3,74 3,75 3,75 3,75 3,76 3,76 3,77 3,77 3,77 3,773,82 3,82 3,82 3,82 3,84 3,84 3,85 3,85 3,86 3,86 3,87 3,87 3,88 3,89 3,893,90 3,91 3,93 3,93 3,94 3,94 3,94 3,94 3,95 3,96 3,96 3,98 3,99 4,06 4,11

24) The following table shows test scores of 40 students.

54 67 75 83 90 56 68 77 83 90 56 68 78 85 91 59 70 79 86 93 60 70 79 86 93 62 73 81 88 94 62 73 81 89 95 66 73 82 89 98

a) Find the 65th percentile, the median and the first and third quartiles. b) Considering the grouped data, find the coefficient of variation (C.V. %) of the distribution. 25) A sample of 40 accounting students recorded the time spent studying the course material in the week before the final exam. Find all the descriptive statistics for the data.

1,0 2,3 2,4 2,6 2,8 2,8 2,9 3,0 3,5 3,6 3,6 3,9 4,4 4,5 4,8 5,0 5,2 5,5 6,2 6,7

26) A sample of 20 financial analysts was asked to provide forecasts of earnings per share of a corporation for next year. The results are summarized in the following table.

Forecast ($ per Share) Number of Analysts 9,95----|10,45 2 10,45---|10,95 8 10,95---|11,45 6 11,45---|11,95 3 11,95---|12,45 1

Total 20 Estimate the sample mean and standard deviation of the forecasts.


7

Capítulo 2 - DISTRIBUIÇÕES DISCRETAS 1) Uma empresa envia 5 propostas para seus clientes e, historicamente, sabe-se que a probabilidade da proposta se converter em um contrato é de 40%. Determine:

a) a probabilidade da empresa fechar ao menos um contrato; b) a probabilidade da empresa fechar entre 2 e 4 contratos (inclusive); c) a probabilidade da empresa não fechar nenhum contrato.

2) Suponha que um novo sistema de controle computadorizado de reclamações tenha sido instalado em uma seguradora de saúde. Apenas 40% das reclamações requerem interferência humana para serem resolvidas. Em um dia particular, 100 reclamações chegam ao processamento. Supondo que o número de reclamações que requerem intervenção humana segue distribuição binomial, determine:

a) a probabilidade de haver entre 37 e 43 reclamações (inclusive) que necessitem de intervenção humana;

b) a probabilidade de haver exatamente 38 reclamações que necessitam deste tipo de interferência;

c) a probabilidade haver mais de 42 reclamações que requerem atenção humana. 3) As linhas telefônicas de um sistema de reservas de uma Cia. Aérea estão ocupadas 35% do tempo. Considere que 10 chamadas aconteçam.

a) Qual é a probabilidade de que, para exatamente 3 chamadas, as linhas estejam ocupadas?

b) Qual é a probabilidade de que, para no mínimo uma chamada, as linhas estejam ocupadas?

c) Qual é a probabilidade de haver mais de cinco chamadas não atendidas em função das linhas estarem todas ocupadas?

4) Por que nem todos os passageiros de aviões aparecem na hora do embarque, uma Cia. Aérea vende 125 passagens para um vôo que só comporta 120 pessoas. É o que se denomina Overbooking. Se a probabilidade de um passageiro não comparecer para o embarque é de 10%, determinar:

a) a probabilidade de que cada passageiro que compareça possa embarcar. b) a probabilidade do vôo decolar com assentos vazios.

5) Um teste de múltipla escolha contém 25 questões, cada qual com 4 alternativas. Suponha que um estudante “chute” todas as questões.

a) Qual a probabilidade dele acertar mais de 20 questões? b) Qual a probabilidade dele acertar menos de 5 questões?

6) Uma empresa recebe um carregamento com 20 itens. Como a inspeção individual de cada item é muita cara, há uma orientação de se verificar aleatoriamente uma amostra de apenas 6 itens, aceitando-se o lote se não houver mais do que um item defeituoso. Qual é a probabilidade de que um lote com cinco itens defeituosos seja aceito?


8

7) Um lote de peças contém 100 unidades de um fornecedor A e 200 de um fornecedor B. Se quatro peças forem selecionadas, ao acaso e sem reposição, qual será a probabilidade de que elas sejam todas provenientes do fornecedor local? 9) Um dia de produção de 850 peças fabricadas contém 50 peças que não satisfazem os requerimentos dos consumidores. Duas peças são selecionadas ao acaso, sem reposição, da produção do dia. Qual é a probabilidade de que haja duas peças não-conformes na amostra? 10) O número médio de chamadas que chegam a uma central é de 10/hora. Determine: a) P(x = 5) b) P(3 ou menos) c) P(exatamente 15) d) P(8 ou menos) 11) O número de falhas em um fio delgado de cobre segue distribuição de Poisson, com média igual a 2,3 falhas/mm. Determine:

a) a probabilidade de existir exatamente 2 falhas em um milímetro de fio; b) a probabilidade de haver 10 falhas em cada 5 milímetros; c) a probabilidade de haver, no mínimo, uma falha em 2 milímetros.

12) Em uma seção de uma auto-estrada, o número de buracos, que é bastante significante para requerer reparo, segue distribuição de Poisson com média de 2 buracos por km.

a) Qual é a probabilidade de que não haja buracos que requeiram reparo em 5 km de auto-estrada?

b) Qual é a probabilidade de que no mínimo um buraco requeira reparo em 0,5 quilômetro?

13) O número de erros em um livro-texto segue distribuição de Poisson, com média igual a 0,01 erros por página. Qual é a probabilidade de que haja três erros ou menos em 100 páginas? 14) Suponha que X seja uma variável aleatória que segue distribuição de Poisson, com média igual a 0,4. Determine:

a) P (X = 0) b) P (X ≤ 2) c) P (X = 4) d) P (X = 8) e) P (X ≤ 4)

15) Suponha que o número de consumidores que entrem em um banco em uma hora seja variável aleatória de Poisson. Suponha também que P (X = 0) = 0,05. Determine a média e a variância de X.


9

Capítulo 2 - APROXIMAÇÃO DA BINOMIAL E POISSON PELA NORMAL Quando o tamanho da amostra utilizada n é grande (ou extremamente grande), o cálculo das distribuições Binomial e Poisson torna-se difícil e impraticável. Nesses casos, duas aproximações podem ser propostas, a saber:

• Se X for uma variável aleatória binomial, então Z é aproximadamente uma variável aleatória normal padronizada, se:

( )pnpnpXZ−

−=

1

A aproximação será boa para np > 5 e n(1-p) > 5.

• Se X for uma variável aleatória de Poisson, então Z é aproximadamente uma variável aleatória normal padronizada, se:

λλ−

=XZ

A aproximação será boa para λ> 5. EXERCÍCIOS: 1) Suponha que x seja uma variável aleatória binomial com n = 200 e p = 0,4. Usando a aproximação pela Normal, determine:

a) ( )70≤xP b) ( )9070 ≤≤ xP 2) Suponha que x seja uma variável aleatória binomial com n = 100 e p = 0,1. Usando a aproximação pela Normal, determine: a) ( )4≤xP - método exato b) ( )4≤xP - aproximado c) ( )128 ≤≤ xP - aproximado 3) A fabricação de chips semicondutores produz 2% de chips defeituosos. Considere que os chips sejam independentes e que um lote contenha 1.000 chips. a) Aproxime a probabilidade de que mais de 25 chips sejam defeituosos; b) Aproxime a probabilidade de 20 a 30 chips serem defeituosos. 4) Um fornecedor embarca um lote de 1.000 conectores elétricos. Uma amostra de 25 é selecionada ao acaso e sem reposição. Considere que o lote contenha 100 conectores defeituosos. a) Usando uma aproximação binomial, qual é a probabilidade de que não haja conectores defeituosos na amostra? b) Usando a aproximação normal, analise novamente o item (4.a). c) Refaça os itens 4.a e 4.b, considerando que o lote tenha tamanho igual a 500. As aproximações continuam satisfatórias? 5) Suponha que o número de partículas de asbesto em uma amostra de 1 cm2 seja uma v.a. de Poisson com média igual a 1.000. Qual é a probabilidade de que 10 cm2 contenham mais de 10.000 partículas?


10

Capítulo 2 - CORREÇÃO DE CONTINUIDADE A aproximação normal de uma distribuição de probabilidade discreta é, algumas vezes, modificada por um fator de correção de 0,5. O valor deste fator está relacionado ao fato de que variáveis discretas só podem assumir valores inteiros. De maneira genérica, a correção da continuidade poderá ser realizada substituindo-se o limite inferior de um intervalo, a, por (a-0,5) e o limite superior, b, por (b+0,5) (Newbold et al., 2003). Assim, para o caso binominal, pode-se escrever que:

( ) ( )( )

( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛

−−+

≤≤−−−

≅≤≤pnpnpbZ

pnpnpaPbxaP

15,0

15,0

O fator de correção pode ser usado quando 9)1(5 <−< pnp . 6) Suponha que X seja uma v.a. binomial com n = 50 e p = 0,1. Pelo fato de X ser uma v.a. discreta, ( ) ( )5,22 ≤=≤ xPxP . No entanto, a aproximação normal para ( )2≤xP pode ser melhorada aplicando-se a aproximação para ( )5,2≤xP . Compare os resultados nos dois casos. Use a correção de continuidade para ( )10<xP . 7) Suponha que X seja uma v.a. binomial com n = 50 e p = 0,1. Pelo fato de X ser uma v.a. discreta, ( ) ( )5,12 ≥=≥ xPxP . No entanto, a aproximação normal para ( )2≥xP pode ser melhorada aplicando-se a aproximação para ( )5,2≥xP . Compare os resultados usando Z = 1,5 e Z = 2. Compare estes resultados com o valor exato de ( )2≥xP . 8) Suponha que X seja uma v.a. binomial com n = 50 e p = 0,1. Pelo fato de X ser uma v.a. discreta, ( ) ( )5,55,152 ≤≤=≤≤ xPxP . No entanto, a aproximação normal para ( )52 ≤≤ xP pode ser melhorada aplicando-se a aproximação para ( )5,55,1 ≤≤ xP .

Compare os resultados usando Z11 = 1,5 e Z12 = 5,5 e Z21 = 2 e Z22 = 5. Compare estes resultados com o valor exato de ( )52 ≤≤ xP . 9) (Newbold, Example 6.8): A salesman makes initial telephone contact with potential customers in an effort to assess whether a follow-up visit to their homes is likely to be worthwhile. His experience suggests that 40% of the initial contacts lead to follow-up visits. If he contacts 100 people by telephone, what is the probability that between 45 and 50 home visits will result? 10) (Newbold, 6.28): A car rental company has determined that the probability a car will need service work in any given month is 20%. The company has 900 cars. a) What is the probability that more than 200 cars will require service work in a particular day? b) What is the probability that fewer than 175 cars will need service work in a given month? 11) (Newbold, 6.31): Suppose that half of all students of a university are dissatisfied with the food service. Taking a random sample of 40 students and using the normal approximation (with and without correction), assess the probability of ( )2218 ≤≤ xP .


11

Capítulos 3 e 4 - ESTIMAÇÃO E INTERVALOS DE CONFIANÇA 1) Suponha que o tempo de permanência dos clientes em uma loja seja normalmente distribuído. Uma amostra aleatória de 16 consumidores revelou um tempo médio de 25 min. Assuma que 6=σ min. Encontre os intervalos de confiança de 90%, 95% e 99% para o tempo médio populacional, µ . 2) Um processo de empacotamento de açúcar produz unidades cujos pesos são normalmente distribuídos, com desvio padrão (populacional) de 1,2 Kg. Uma amostra de 25 pacotes revelou média de 19,8 Kg. Encontre intervalos de 95% e 99% para o valor real do peso médio de todos os pacotes produzidos. 3) Em um processo de fabricação de tijolos, o desvio padrão para o peso médio unitário é de 0,12 lb. Uma amostra aleatória com 16 tijolos revelou uma média de 4,07 lb. Encontre um intervalo de confiança de 95% para o valor real do peso médio dos tijolos produzidos. 4) 470430 << µ é um intervalo de confiança de 95% para as vidas (em minutos) de um produto. Suponha que este resultado tenha sido obtido com uma amostra de tamanho 100. Determine o valor da média amostral e construa um intervalo de confiança de 99%. 5) Determine o tamanho da amostra necessária para se estimar, com 95% de confiança, o salário médio de uma categoria, de modo que a média amostral esteja a menos de $500,00 do valor real. Suponha 6250=σ . 6) O preço típico dos livros utilizados em uma escola varia entre $10 e $90. Quantos exemplares devem ser selecionados para se estimar, com confiança de 95%, o preço médio real destes itens. Suponha que a média amostral esteja a menos de $2 do valor verdadeiro. (Dica: use R25,0=σ ). 7) Uma empresa de limpeza urbana deseja estimar o peso médio do lixo doméstico descartado pela população. Determine o tamanho da amostra necessária para estimar essa média, para que se tenha 95% de confiança em que a média amostral esteja a menos de 2 kg do valor populacional real, cujo desvio padrão é 12,46 Kg. 8) Quando uma população N é relativamente pequena (ou finita) e a amostragem se processar sem reposição, a margem de erro E utilizada na determinação da amostra deve ser corrigida, tal que:

12/ −−

=N

nNn

ZE σα

Determine a fórmula corrigida para a determinação do tamanho da amostra. 9) Uma clínica está oferecendo um tratamento de redução de peso. Uma amostra formada pelos registros históricos de 10 pacientes revelou uma média de 16,37 lb e desvio padrão de 5,38. Encontre intervalos de confiança de 90%, 95% e 99% para o peso médio verdadeiro perdido por todos os pacientes. 10) Uma universidade deseja estimar o salário anual médio que um aluno graduado em administração recebe 5 anos após sua formatura. Uma amostra de 25 administradores revelou =x US$ 42.270 e s = US$ 4.780. Construa um intervalo de confiança de 95% para o valor médio real.


12

Capítulo 5 – Testes de Hipótese 1) A company which receives shipments of batteries tests a random sample of nine of them before agreeing to take a shipment. The company is concerned that the true mean lifetime for all batteries in the shipment should be at least 50 hours. From past experience, it is safe to conclude that the population distribution of lifetimes is normal, with standard deviation of 3 hours. For one particular shipment, the mean lifetime for a sample of nine batteries was 48.2 hours. Test at 5% level the null hypothesis that the population mean lifetime is at least 50 hours. 2) An engineering research center claims that through the use of a new computer control system, automobiles should achieve on average an additional 3 miles per gallon of gas. A random sample of 100 automobiles was used to evaluate this product. The sample mean increase in miles per gallon achieved was 2.4 and the sample standard deviation was 1.8 miles per gallon. Test the hypothesis that the population mean is at least 3 miles per gallon using 5% significance level. Find the P-value of this test, and interpret your findings. 3) A beer distributor claims that a new display, featuring a life-size picture of a well-known rock singer, will increase product sales in supermarkets by an average of 50 cases in a week. For a random sample of 20 liquor weekly sales, the average sales increase was 41.3 cases and the sample standard deviation was 12.2 cases. Test at the 5% level the hypothesis that the population mean sales increase is at least 50 cases. 4) In contract negotiations, a company claims that a new incentive scheme has resulted in average weekly earning of at least $400 for all customer service workers. A union representative takes a random sample of 15 workers and finds that their weekly earnings have an average of $381.25 and a standard deviation of $48.60. Assume a normal distribution.

a) Test the company’s claim; b) If the same sample results had been obtained from a random sample of 50

employees, could the company’s claim be rejected at a lower significance level than in part (a)?

5) A bearing used in an automotive application is supposed to have a nominal inside diameter of 1.5 inches. A random sample of 25 bearings is selected and the average inside diameter of these bearing is 1.4975 inches. Bearing diameter is known to be normally distributed with standard deviation 0.01 inch. Test the null hypothesis using a two-sided approach and considering %1=α .

6) The production manager of a company has asked you to evaluate a proposed new procedure for producing its double-hung windows. The present process has a mean production of 80 units per hour with a population standard deviation of 8=σ . The manager indicates that she does not want to change to a new procedure unless there is strong


13

evidence that the mean production level is higher with the new process. Taking a random sample of n = 25 production hours the obtained sample mean was 83. Test with a significance level of %5=α if there is strong evidence that the production hour increased. Find the respective P-value.

7) A manufacturing process involves drilling holes whose diameter are normally distributed with population mean of 2 inches and population standard deviation of 0.06 inches. A random sample of 9 measurements had a sample mean of 1.95 inches. Use a significance level of %5=α to determine if the observed sample mean is unusual and suggests that the drilling machine should be adjusted. (Hint: use two-sided hypothesis test).

8) A manufacturer of detergent claims that the contents of boxes sold weigh on average at least 16 ounces. The distribution of weight is known to be normal, with 4.0=σ ounce. A random sample of 16 boxes yielded a sample mean weight of 15.84 ounces. Test at 5% and 10% significance levels the null hypothesis that the population mean weight is at least 16 ounces.

9) A company which receives shipments of batteries tests a random sample of 9 of them before agreeing to take a shipment. The company is concerned that the true mean lifetime for all batteries in the shipment should be at least 50 hours. From past experience, it is safe to conclude that lifetime is normally distributed with 3=σ hours. For one particular shipment, the mean lifetime for a sample of nine batteries was 48.2 hours. Test at 10% significance level the null hypothesis that the population mean lifetime is at least 50 hours.

10) A pharmaceutical manufacturer is concerned about the impurity concentration in pills, and it is anxious that this does not exceed 3%. It is known that from a particular production run, impurity concentrations follow a normal distribution with standard deviation of 0.4 %. A random sample of 64 pills was checked, and the sample mean impurity concentration was found to be 3.07%. Test at the 5% level the null hypothesis that the population mean impurity concentration is 3% against the alternative that it is more than 3%. Find the respective P-value in this case.

11) The accounts of a corporation show that, on average, accounts payable are $125.32. An auditor checked a random sample of 16 of these accounts. The sample mean was $131.78 and the sample standard deviation was $25.41. Assume that the population distribution is normal. Test as the 5% significance level against a two-sided alternative the null hypothesis that the population mean is $125.32. Find the P-value of this test.

12) A process that produces bottles of shampoo, when operating correctly, produces bottles whose contents weigh, on average, 20 ounces. A random sample of nine bottles from a single production run yielded the following content weights (in ounces):

21,4 19,7 19,7 20,6 20,8 20,1 19,7 20,3 20,9

Assuming that the population distribution is normal, test at the 5% level against a two-sided alternative the null hypothesis that the process is operating correctly.

13) A statistics instructor is interested in the ability of student to asses the difficulty of a test they have taken. This test was taken by a large group of students, and the average score was 78.5. A random sample of eight students was asked to predict this average score. Their predictions were:

72 83 78 65 69 77 81 71


14

Assuming a normal distribution, test the null hypothesis that the population mean prediction would be 78.5. Use a tow-sided alternative and a 10% significance level.

14) A random sample of 172 marketing students was asked to rate on a scale from one (not important) to five (extremely important) health benefits as a job characteristic. The sample rating was 3.31 and the sample standard deviation was 0.70. Test at 1% significance level the null hypothesis that the population mean rating is at most 3.0 against the alternative that it is bigger than 3.0. Capítulo 5 – Testes de Hipótese para duas Médias 1) Two catalysts may be used in a batch chemical process. Twelve batches were prepared using catalyst 1, resulting in an average yield of 86 and a sample standard deviation of 3. Fifteen batches were prepared using catalyst 2, and they resulted in an average yield of 89 with a standard deviation of 2. Assume that yield measurements are approximately normally distributed with the same standard deviation. Is there evidence to support a claim that catalyst 2 produces a higher mean yield than catalyst 1? Use =α 1.0%. 2) The melting points of two alloys used in formulating solder were investigated by melting 21 samples of each material. The sample mean and standard deviation for alloy 1 was =1x 420 ºF and =1s 4 ºF, while for alloy 2 they were =2x 426 ºF and =2s 3 ºF. Do the sample data support the claim that both alloys have the same melting point? Use a =α 0.05 in your conclusions and assume that both populations are normally distributed and have the same standard deviation. 3) A new filtering device is installed in a chemical unit. Before its installation, a random sample yielded the following information about the percentage of impurity: n1 = 8 and n2 = 9, =1x 12.5, =2

1s 101.17, =2x 10.2, and =22s 94.73. Adopting =α 5%, has the filtering

device reduced the percentage of impurity significantly? Assume that both populations have the same standard deviation. 4) A product developer is interested in reducing the drying time of primer paint. Two formulations of the paint are tested; formulation 1 is the standard chemistry, and formulation 2 has a new drying ingredient that should reduce the drying time. From experience, it is known that the standard deviation of drying time is 8 minutes, and this inherent variability should be unaffected by the addition of the new ingredient. Ten specimens are painted with formulation 1, and another 10 specimens are painted with formulation 2; the 20 specimens are painted in random order. The two sample average drying times are 1211 =x minutes and 1122 =x minutes, respectively. What conclusions can the product developer draw about the effectiveness of the new ingredient, using =α 0.05? 5) Two machines are used for filling plastic bottles with a net volume of 16.0 oz. The fill volume is normal, with standard deviation =1σ 0.020 and =2σ 0.025 ounces. A member of the quality engineering staff suspects that both machines fill to the same mean net volume, whether or not this volume is 16.0 oz. A random sample of 10 bottles is taken from the output of each machine.

Machine 1 16,03 16,04 16,05 16,05 16,02 16,01 15,96 15,98 16,02 15,99Machine 2 16,02 15,97 15,96 16,01 15,99 16,03 16,04 16,02 16,01 16,00


15

Using a =α 0.05, do you think the engineer is correct? 6) Two types of plastic are suitable for use by an electronics component manufacturer. The breaking strength of this plastic is important. It is known that 0,121 ==σσ psi. From a random sample of size n1 = 10 and n2 = 12, we obtain 5.1621 =x and 0.1552 =x psi . The company will not adopt plastic 1 unless its mean breaking strength exceeds that of plastic 2 by at least 10 psi. Based on the sample information, should it use plastic 1? Use a 0.05 significance level in your findings. 7) Two machines are used to fill plastic bottles with dishwashing detergent. The standard deviations of fill volume are known to be =1σ 0.10 fluid ounces and =1σ 0.15 fluid ounces for the two machines, respectively. Two random samples of n1 = 12 bottles from machine 1 and n2 = 10 bottles from machine 2 are selected, and the sample mean fill volumes are 87.301 =x fluid ounces and 68.302 =x fluid ounces. Assume normality. Using a significance level of 1% and 5%, is it possible to conclude that the machines have the same performance? 8) Ten individuals have participated in a diet-modification program to stimulate weight loss. Their weight both before and after participation in the program are shown in the following list. Is there evidence to support the claim that this particular diet-modification program is effective in producing a mean weight reduction? Use =α 0.05.

Before 195 213 247 201 187 210 215 246 294 310 After 187 195 221 190 175 197 199 221 278 285

Otherwise, is there evidence to support the claim that this particular diet modification program will result in a mean weight loss of at least 10 pounds? Use =α 0.05. 9) In a study comparing banks in Germany and Great Britain, a sample of 145 matched pairs of banks was formed. Each pair contained one German and one Great Britain bank. The pairings were made in such a way that the two members were as similar as possible in regard to such factors as sizes and age. The ratio of total loans outstanding to total assets was calculated for each of the banks. For this ratio, the sample mean difference (German – Great Britain) was 0.0518, and the sample standard deviation of the differences was 0.3055. Test the hypothesis that the two population means are equal. 10) Fifteen adult males between the ages of 35 and 50 participated in a study to evaluate the effect of diet and exercise on blood cholesterol levels. The total cholesterol was measured in each subject initially and then three months after participating in an aerobic exercise program and switching to a low-fat diet. The data are shown in the accompanying table. Do the data support the claim that low-fat diet and aerobic exercise are of value in producing a mean reduction in blood cholesterol levels? Use α=0.05.

Blood Cholesterol Level Subject 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Before 265 240 258 295 251 245 287 314 260 279 283 240 238 225 247After 229 231 227 240 238 241 234 256 247 239 246 218 219 226 233


16

Capítulo 5 – TESTES DE HIPÓTESE PARA UMA PROPORÇÃO

1) Of a random sample of 361 owners of small businesses that had gone into bankruptcy, 105 reported conducting no marketing studies prior to opening the business. Test the null hypothesis that at most 25% of all member of this population conducted no marketing studies before opening the business.

2) A semiconductor manufacturer produces controller used in automobile engine applications. The customer requires that the process fallout or fraction of defectives at a critical manufacturing step not exceed 5% and that the manufacturer demonstrate process capability at this level of quality using 5% of significance level. The semiconductor manufacturer takes a random sample of 200 devices and finds that four of them are defective. Can the manufacturer demonstrate process capability for the customer?

3) A researcher claims that at least 10% of all helmets have manufacturing flaws that could potentially cause injury to the wearer. A sample of 200 helmets revealed that 16 helmets contained such defects. Does this finding support the researcher’s claim? (Use 1% significance level). Find the P-value for this test.

Capítulo 5 – TESTES DE HIPÓTESE PARA UMA VARIÂNCIA

1) At the insistence of a government inspector, a new safety device is installed in an assembly-line operation. After the installation of this device, a random sample of eight days output gave the following results for number of finished components produced: 618, 660, 638, 625, 571, 598, 639 e 582. Management is concerned about the variability of daily output and views as undesirable any variance above 500. Test at the 10% significance level the null hypothesis that the population variance for daily output does not exceed 500.

2) Plastic sheets produced by a machine are periodically monitored for possible fluctuations in thickness. If the true variance in thicknesses exceeds 2.25 square millimeters, there is cause for concern about product quality. Thickness measurements for a random sample of ten sheets produced in a particular shift were taken, giving the following results (in mm): 226, 226, 232, 227, 225, 228, 225, 228, 229 e 230. Test the null hypothesis that the population variance is at most 2.25.

3) A company produces electric devices operated by a thermostatic control. The standard deviation of the temperature at which these controls actually operate should not exceed 2.0 degrees Fahrenheit. For a random sample of 20 of these controls, the sample standard deviation of operating temperatures was 2.36 degrees Fahrenheit. Stating any assumptions you need to make, test at the 5% level the null hypothesis that the population standard deviation is 2.0 against the alternative that it is bigger.

4) One way to evaluate the effectiveness of a teaching assistant is to examine the scores achieved by his or her students in an examination at the end of the course. Obviously, the mean score is of interest. However, the variance also contains useful information – some teachers have a style that works very well with more able students but is unsuccessful with less able or poorly motivated students. A professor sets a standard examination at end of each semester for all sections of a course. The variance of the scores on this test is typically very close to 300. A new teaching assistant has a class of 30 students, whose test scores had a variance of 480. Regarding these students’ test scores as a


17

random sample from a normal population, test against a two-sided alternative the null hypothesis that the population variance of their scores is 300.

5) A company produces electric devices operated by a thermostatic control. The standard deviation of the temperature at which these controls actually operate should not exceed 2.0 degrees Fahrenheit. For a random sample of 20 of these controls, the sample standard deviation of operating temperatures was 2.36 degrees Fahrenheit. Stating any assumptions you need to make, test at the 5% level the null hypothesis that the population standard deviation is 2.0 against the alternative that it is bigger.

6) The sugar content of the syrup in canned peaches is normally distributed. A random sample of 10 cans yields a sample standard deviation of 4.8 milligrams. a) Find a 95% two-sided confidence interval for population standard deviation; b) Test the hypothesis: Ho: s2 = 18 versus H1: s2≠ 18, using 5% significance level; c) What is the P-value for this test?

Capítulo 5 - TESTE DE HIPÓTESES PARA DUAS PROPORÇÕES 1) Em uma amostral aleatória de 200 motoristas de carros nacionais em uma cidade, 165 afirmaram usar regularmente o cinto de segurança, enquanto, em uma outra amostral de 250 motoristas de carros estrangeiros na mesma cidade, 198 afirmam usar regularmente este dispositivo. Usando nível de significância de 5%, determine se existe diferença significativa quanto ao uso do cinto. 2) De uma amostral aleatória de 500 adultos residentes em um bairro, 385 foram favoráveis ao aumento do limite de velocidade em uma auto-estrada, enquanto em outra amostra de 400 adultos, residentes em outro bairro, 267 foram favoráveis a essa mudança. Usando nível de significância de 5%, pode-se afirmar que existe uma diferença significativa entre as opiniões dos moradores dos dois bairros? 3) Duas máquinas injetoras de plástico diferentes estão sendo comparadas quanto às proporções de produtos defeituosos que produzem. Duas amostras aleatórias de 300 peças fabricadas são selecionadas de ambas as máquinas. Foram encontradas 15 peças defeituosas na máquina 1 e 8, na máquina 2. Considerando-se um nível de significância de 5%, é razoável concluir que ambas as máquinas produzam a mesma fração de peças defeituosas? Capítulo 5 - TESTE DE HIPÓTESES PARA DUAS VARIÂNCIAS 4) Duas companhias químicas podem fornecer uma matéria-prima cuja concentração é um CTQ de cliente. A concentração média é a mesma, porém, suspeita-se que suas variâncias difiram. O desvio padrão de uma amostra aleatória de 10 bateladas produzidas pela companhia 1 é s1=4,7 g/l, enquanto que uma amostra de 16 bateladas da companhia 2 é s2=5,8 g/l. Considerando o nível de significância de 5%, há evidência suficiente para concluir que as variâncias das duas populações difiram? 5) The melting points of two alloys used in formulating solder were investigated by melting 21 samples of each material. The sample standard deviation for alloy 1 was =1s 4 ºF, while for alloy 2 they were =2s 3 ºF. Do the sample data support the claim that both


18

alloys have the same variance? Use a 5% significance level in your conclusions and assume that both populations are normally distributed. 6) A new filtering device is installed in a chemical unit. Before its installation, a random sample yielded the following information about the percentage of impurity: n1 = 8 and n2 = 9, =2

1s 101.17 and =22s 94.73. Adopting =α 5%, do the sample data support the claim

that both alloys have the same variance? 7) The diameter of steel rods manufactured on two different extrusion machines is being investigated. Two random samples of sizes n1 =15 and n2 =17 are selected, and sample variances are =2

1s 0.35 and =22s 0.40, respectively. Using =α 5%, test the null hypothesis

that 22

21 σσ = .


19

Capítulo 6 - ANOVA 1) A corporation is trying to decide which of three types of automobile to order for its fleet – domestic, Japanese or European. Five cars of each type were ordered, and after 10.000 miles of driving, the operating cost per mile of each was assessed. The following tables in cents per mile were obtained.

Domestic 18,0 17,6 15,4 19,1 16,9 Japanese 20,1 15,6 16,1 15,3 15,4 European 19,3 17,4 15,1 18,6 16,1

Using =α 5%, test the null hypothesis that the population mean operating costs per mile are the same for these three types of cars. 2) A researcher is interested in investigating the tensile strength (lb/in2) of a new synthetic fiber that will be used to make cloth for men’s shirts. Suspecting that the strength is affected by the percent of cotton used in the blend of materials for the fiber, she decides to conduct an experiment varying the percent of cotton and measuring the tensile strength. According to the following data, is there any evidence that the amount of cotton used in the fiber influences the tensile strength?

15% 7 7 15 11 9 20% 12 17 12 18 18 25% 14 18 18 19 19 30% 19 25 22 19 23 35% 7 10 11 15 11

3) A manufacturer of television sets is interested in the effect on tube conductivity of four different types of coating for color picture tubes. The following conductivity data are obtained. Is there a difference in conductivity due to coating type? Use 05,0=α .

Coating Type Conductivity A 143 141 150 146 B 152 149 137 143 C 134 136 132 127 D 129 127 132 129

4) An experiment was run to determine whether four specific firing temperatures affect the density of a certain type of brick. The experiment led to the data in the following table. Using =α 5%, does the firing temperature affect the density of the bricks?

Temperature Density 100 21,8 21,9 21,7 21,6 125 21,7 21,4 21,5 21,4 150 21,9 21,8 21,8 21,6 175 21,9 21,7 21,8 21,4

5) An agricultural experiment designed to assess differences in yields of corn for four different varieties, using three different fertilizers, produced the results (in bushels per


20

acre) shown in the table below. Test at a 5% significance level, the null hypothesis of equality among the population means.

Variety Fertilizer A B C D Type 1 86 88 77 84 Type 2 92 91 81 93 Type 3 75 80 83 79

6) The objective of a study is to verify whether three marks of automobile have the same fuel consumption. An experiment is then conducted choosing five people to randomly drive each car three times. The data are available on the following table. Test at a 5% significance level, the null hypothesis of equality among the population means. Is there evidence of difference among the drivers? Is there strong evidence of interaction between the driver and the car?

Automobile Driver Car A Car B Car C Driver 1 25,0 25,4 25,2 24,0 24,4 23,9 25,9 25,8 25,4 Driver 2 24,8 24,8 24,5 23,5 23,8 23,8 25,2 25,0 25,4 Driver 3 26,1 26,3 26,2 24,6 24,9 24,9 25,7 25,9 25,5 Driver 4 24,1 24,4 24,4 23,9 24,0 23,8 24,0 23,6 23,5 Driver 5 24,0 23,6 24,1 24,4 24,4 24,1 25,1 25,2 25,3

7) An engineer suspects that the surface finish of metal parts is influenced by the type of paint used and the drying time. He selected, according to the table below, three drying times and two types of paint. Three parts are tested with each combination of paint type and drying time. State and test the appropriate hypotheses using Anova and %.5=α 8) An experiment describes an investigation about the effect of two factors (glass type and phosphor type) on the brightness of a television tube. The response variable measured is the current (in microamps) necessary to obtain a specified brightness level. Considering a

%5=α , carry out the Anova calculations and discuss your findings.

Question 7 Question 8


21

Capítulo 6 - ANOVA/ANCOVA (MÚLTIPLOS FATORES) 1) A soft drink distributor is studying the effectiveness of delivery methods. Three different types of hand trucks have been developed, and an experiment is performed in the company’s methods engineering laboratory. The variable of interest is the delivery time in minutes (y); however, this response is also strongly related to the case volume delivered (x). Each hand truck is used four times and the data are stored in the following table. Using =α 5%, analyze these data and draw appropriate conclusions.

Hand Truck Type

Type I Type II Type III y x y x y x 27 24 25 26 40 38 44 40 35 32 22 26 33 35 46 42 53 50 41 40 26 25 18 20

2) Three different machines produce a monofilament fiber for a textile company. The process engineer is interested in determining if there is a difference in the breaking strength of the fiber produced by the three machines. However, the strength of a fiber is related to its diameter, with thicker fibers being generally stronger than thinner ones. A random sample of five fiber specimens is selected from each machine. The fiber strength (y) and the corresponding diameter (x) for each specimen are shown in the following table. Using =α 5%, analyze these data and draw your conclusions.

Breaking Strength Data (y = strength in pounds and x = diameter in 10-4 inches).

Machine I Machine II Machine III y x y x y x 36 20 40 22 35 21 41 25 48 28 37 23 39 24 39 22 42 26 42 25 45 30 34 21 49 32 44 28 32 15

3) An engineer is studying the effect of cutting speed on the rate of metal removal in a machining operation. However, the rate of metal removal is also related to the hardness of the test specimen. Five observations are taken at each cutting speed. The amount of metal removed (y) and the hardness of the specimen (x) are shown in the following table. Analyze the data using an analysis of covariance and %.5=α

Cutting Speed (rpm) 1000 1200 1400

y x y x y x 68 120 112 165 118 175 90 140 94 140 82 132 98 150 65 120 73 124 77 125 74 125 92 141 88 136 85 133 80 130


22

4) The quality control department of a fabric finishing plant is studying the effect of several factors on the dyeing of cotton-synthetic cloth used to manufacture men’s shirts. Three operators, three cycle times, and two temperatures were selected, and three small specimens of cloth were dyed each set of conditions. The finished cloth was compared to a standard, and a numerical score was assigned. Analyze the following data and draw conclusions.

5) The percentage of hardwood concentration in raw pulp, the vat pressure, and the cooking time of the pulp are being investigated for their effects on the strength of paper. Three levels of hardwood concentration, three levels of pressure, and two cooking times are selected. A factorial experiment with two replicates is conducted, and the following data are obtained. Write your findings.


23

Capítulo 7 – CORRELAÇÃO E REGRESSÃO 1) The electric power consumed each month by a chemical plant is thought to be related to the average ambient temperature (x1), the number of days in the month (x2), the average product purity (x3), and the tons of product produced (x4). The past historical data are available and are presented in the table 1:

Table 1 Table 2 2) A study was performed on wear of a bearing y and its relationship to the oil viscosity (x1) and the load (x2). The data in table 2 were obtained. Fit a multiple linear regression model and an interaction model to these data and predict the wear when x1 = 25 and x2 = 1000. 3) The production manager of a company was interested in estimating the mathematical relationship between the number of electronic assemblies produced during an eight-hour shift and the average cost per assembly. This function would then be used to estimate cost for various production order bids and to determine the production level that would minimize average cost. According to the data below, build a linear and a quadratic model to mean cost per unit.

Number of Units (x)

Mean Cost per Unit (y)

100 5,11 210 4,42 290 4,07 415 3,52 509 3,33 613 3,44 697 3,77 806 4,07 908 4,28 x

y

9008007006005004003002001000

5,0

4,5

4,0

3,5

S 0,126875R-Sq 96,2%R-Sq(adj) 94,9%


24

4) (Cobb-Douglas): A company began producing small fishing boats in the early 1970’s. The production method uses a workstation with a set of jigs and power tools that can be operated by a varying number of workers. Over the years the number of workstations (units of capital) and the workforce has grown to meet the demand for boats. To expand the sales the company needs to know how much the number of workstations and employees must be increased to achieve a higher level of production. Use an exponential Cobb-Douglas model to solve the problem, considering the data below.

Year K L Y Year K L Y Year K L Y 1976 1,0 2,0 40 1984 4,5 7,1 100 1992 13,2 15,6 300 1977 1,2 2,1 45 1985 4,5 8,5 130 1993 14,0 17,0 340 1978 1,2 2,7 52 1986 4,1 8,9 161 1994 14,8 18,0 370 1979 1,1 3,0 57 1987 6,0 10,0 215 1995 15,0 20,9 405 1980 2,0 3,1 65 1988 8,1 13,9 260 1996 16,0 21,0 430 1981 3,0 3,6 75 1989 7,9 16,1 265 1997 16,0 21,4 440 1982 4,0 4,0 86 1990 11,0 14,0 275 1998 17,0 21,8 460 1983 4,5 6,0 95 1991 12,0 14,0 282 1999 17,0 22,1 472

5) In economics the demand function has an exponential form 1

0ββ PQ = , where Q is the

quantity demanded and P is the price per unit of the goods produced. According to the historical data below, adjust an exponential model to the demand. Price 5,5 6,0 6,5 6,0 5,0 6,5 4,5 5,0 Demand 420,0 380,0 350,0 400,0 440,0 380,0 450,0 420,0 6) The following data shows the cooling temperatures (y) of a freshly brewed cup of coffee after it is poured from the brewing pot into a serving cup. The brewing pot temperature is approximately 180º F. Determine an exponential regression model equation to represent this data in function of cooling time (x). x 0 5 8 11 15 18 22 25 30 34 38 42 45 50 y 179,5 168,7 158,1 149,2 141,7 134,6 125,4 123,5 116,3 113,2 109,1 105,7 102,2 100,5 7) (Lagged Values): The following data shows the private consumption (y) and the disposable income (x) in a country. Using these data, estimate the above mentioned models.

a) ( ) ( ) ( )1210 −++= ttt xxy γββ ; b) ( ) ( ) ( )1210 logloglog −++= ttt xxy γββ .

y 170 181 196 213 233 252 267 278 333 336 x 179 195 204 216 247 264 285 307 343 359 y 360 394 404 427 462 491 527 554 557 564 x 377 421 434 453 495 541 596 623 656 695 y 580 618 652 663 693 761 831 919 1003 x 731 773 827 856 901 978 1093 1206 1296


25

TABELAS


26

Φ( )x e dttx= −

−∞∫1

22 2

π

X

x Φ(x) x Φ(x) x Φ(x) x Φ(x) x Φ(x)-3,10 -3,09 -3,08 -3,07 -3,06 -3,05 -3,04 -3,03 -3,02 -3,01 -3,00 -2,99 -2,98 -2,97 -2,96 -2,95 -2,94 -2,93 -2,92 -2,91 -2,90 -2,89 -2,88 -2,87 -2,86 -2,85 -2,84 -2,83 -2,82 -2,81 -2,80 -2,79 -2,78 -2,77 -2,76 -2,75 -2,74 -2,73 -2,72 -2,71 -2,70 -2,69 -2,68 -2,67 -2,66 -2,65 -2,64 -2,63 -2,62 -2,61 -2,60 -2,59 -2,58 -2,57 -2,56 -2,55

0,000968 0,001001 0,001035 0,001070 0,001107 0,001144 0,001183 0,001223 0,001264 0,001306 0,001350 0,001395 0,001441 0,001489 0,001538 0,001589 0,001641 0,001695 0,001750 0,001807 0,001866 0,001926 0,001988 0,002052 0,002118 0,002186 0,002256 0,002327 0,002401 0,002477 0,002555 0,002635 0,002718 0,002803 0,002890 0,002980 0,003072 0,003167 0,003264 0,003364 0,003467 0,003573 0,003681 0,003793 0,003907 0,004025 0,004145 0,004269 0,004396 0,004527 0,004661 0,004799 0,004940 0,005085 0,005234 0,005386

-2,54 -2,53 -2,52 -2,51 -2,50 -2,49 -2,48 -2,47 -2,46 -2,45 -2,44 -2,43 -2,42 -2,41 -2,40 -2,39 -2,38 -2,37 -2,36 -2,35 -2,34 -2,33 -2,32 -2,31 -2,30 -2,29 -2,28 -2,27 -2,26 -2,25 -2,24 -2,23 -2,22 -2,21 -2,20 -2,19 -2,18 -2,17 -2,16 -2,15 -2,14 -2,13 -2,12 -2,11 -2,10 -2,09 -2,08 -2,07 -2,06 -2,05 -2,04 -2,03 -2,02 -2,01 -2,00 -1,99

0,005543 0,005703 0,005868 0,006037 0,006210 0,006387 0,006569 0,006756 0,006947 0,007143 0,007344 0,007549 0,007760 0,007976 0,008198 0,008424 0,008656 0,008894 0,009137 0,009387 0,009642 0,009903 0,010170 0,010444 0,010724 0,011011 0,011304 0,011604 0,011911 0,012224 0,012545 0,012874 0,013209 0,013553 0,013903 0,014262 0,014629 0,015003 0,015386 0,015778 0,016177 0,016586 0,017003 0,017429 0,017864 0,018309 0,018763 0,019226 0,019699 0,020182 0,020675 0,021178 0,021692 0,022216 0,022750 0,023295

-1,98 -1,97 -1,96 -1,95 -1,94 -1,93 -1,92 -1,91 -1,90 -1,89 -1,88 -1,87 -1,86 -1,85 -1,84 -1,83 -1,82 -1,81 -1,80 -1,79 -1,78 -1,77 -1,76 -1,75 -1,74 -1,73 -1,72 -1,71 -1,70 -1,69 -1,68 -1,67 -1,66 -1,65 -1,64 -1,63 -1,62 -1,61 -1,60 -1,59 -1,58 -1,57 -1,56 -1,55 -1,54 -1,53 -1,52 -1,51 -1,50 -1,49 -1,48 -1,47 -1,46 -1,45 -1,44 -1,43

0,023852 0,024419 0,024998 0,025588 0,026190 0,026803 0,027429 0,028067 0,028717 0,029379 0,030054 0,030742 0,031443 0,032157 0,032884 0,033625 0,034379 0,035148 0,035930 0,036727 0,037538 0,038364 0,039204 0,040059 0,040929 0,041815 0,042716 0,043633 0,044565 0,045514 0,046479 0,047460 0,048457 0,049471 0,050503 0,051551 0,052616 0,053699 0,054799 0,055917 0,057053 0,058208 0,059380 0,060571 0,061780 0,063008 0,064256 0,065522 0,066807 0,068112 0,069437 0,070781 0,072145 0,073529 0,074934 0,076359

-1,42 -1,41 -1,40 -1,39 -1,38 -1,37 -1,36 -1,35 -1,34 -1,33 -1,32 -1,31 -1,30 -1,29 -1,28 -1,27 -1,26 -1,25 -1,24 -1,23 -1,22 -1,21 -1,20 -1,19 -1,18 -1,17 -1,16 -1,15 -1,14 -1,13 -1,12 -1,11 -1,10 -1,09 -1,08 -1,07 -1,06 -1,05 -1,04 -1,03 -1,02 -1,01 -1,00 -0,99 -0,98 -0,97 -0,96 -0,95 -0,94 -0,93 -0,92 -0,91 -0,90 -0,89 -0,88 -0,87

0,077804 0,079270 0,080757 0,082264 0,083793 0,085344 0,086915 0,088508 0,090123 0,091759 0,093418 0,095098 0,096801 0,098525 0,100273 0,102042 0,103835 0,105650 0,107488 0,109349 0,111233 0,113140 0,115070 0,117023 0,119000 0,121001 0,123024 0,125072 0,127143 0,129238 0,131357 0,133500 0,135666 0,137857 0,140071 0,142310 0,144572 0,146859 0,149170 0,151505 0,153864 0,156248 0,158655 0,161087 0,163543 0,166023 0,168528 0,171056 0,173609 0,176186 0,178786 0,181411 0,184060 0,186733 0,189430 0,192150

-0,86 -0,85 -0,84 -0,83 -0,82 -0,81 -0,80 -0,79 -0,78 -0,77 -0,76 -0,75 -0,74 -0,73 -0,72 -0,71 -0,70 -0,69 -0,68 -0,67 -0,66 -0,65 -0,64 -0,63 -0,62 -0,61 -0,60 -0,59 -0,58 -0,57 -0,56 -0,55 -0,54 -0,53 -0,52 -0,51 -0,50 -0,49 -0,48 -0,47 -0,46 -0,45 -0,44 -0,43 -0,42 -0,41 -0,40 -0,39 -0,38 -0,37 -0,36 -0,35 -0,34 -0,33 -0,32 -0,31

0,194894 0,197663 0,200454 0,203269 0,206108 0,208970 0,211855 0,214764 0,217695 0,220650 0,223627 0,226627 0,229650 0,232695 0,235762 0,238852 0,241964 0,245097 0,248252 0,251429 0,254627 0,257846 0,261086 0,264347 0,267629 0,270931 0,274253 0,277595 0,280957 0,284339 0,287740 0,291160 0,294599 0,298056 0,301532 0,305026 0,308538 0,312067 0,315614 0,319178 0,322758 0,326335 0,329969 0,333598 0,337243 0,340903 0,344578 0,348268 0,351973 0,355691 0,359424 0,363169 0,366928 0,370700 0,374484 0,378281

x Φ(x) x Φ(x) x Φ(x) x Φ(x) x Φ(x) -0,30 -0,29

0,382089 0,385908

0,38 0,39

0,648027 0,651732

1,06 1,07

0,855428 0,857690

1,74 1,75

0,959071 0,959941

2,42 2,43

0,992240 0,992451


27

x Φ(x) x Φ(x) x Φ(x) x Φ(x) x Φ(x) -0,28 -0,27 -0,26 -0,25 -0,24 -0,23 -0,22 -0,21 -0,20 -0,19 -0,18 -0,17 -0,16 -0,15 -0,14 -0,13 -0,12 -0,11 -0,10 -0,09 -0,08 -0,07 -0,06 -0,05 -0,04 -0,03 -0,02 -0,01 0,00 0,01 0,02 0,03 0,04 0,05 0,06 0,07 0,08 0,09 0,10 0,11 0,12 0,13 0,14 0,15 0,16 0,17 0,18 0,19 0,20 0,21 0,22 0,23 0,24 0,25 0,26 0,27 0,28 0,29 0,30 0,31 0,32 0,33 0,34 0,35 0,36 0,37

0,389739 0,393580 0,397432 0,401294 0,405165 0,409046 0,412936 0,416834 0,420740 0,424655 0,428576 0,432505 0,436441 0,440382 0,444330 0,448283 0,452242 0,456205 0,460172 0,464144 0,468119 0,472097 0,476078 0,480061 0,484047 0,488034 0,492022 0,496011 0,500000 0,503989 0,507978 0,511967 0,515953 0,519939 0,523922 0,527903 0,531881 0,535856 0,539828 0,543795 0,547758 0,551717 0,555670 0,559618 0,563559 0,567495 0,571424 0,575345 0,579260 0,583166 0,587064 0,590954 0,594835 0,598706 0,602568 0,606420 0,610261 0,614092 0,617911 0,621719 0,625516 0,629300 0,633072 0,636831 0,640576 0,644309

0,40 0,41 0,42 0,43 0,44 0,45 0,46 0,47 0,48 0,49 0,50 0,51 0,52 0,53 0,54 0,55 0,56 0,57 0,58 0,59 0,60 0,61 0,62 0,63 0,64 0,65 0,66 0,67 0,68 0,69 0,70 0,71 0,72 0,73 0,74 0,75 0,76 0,77 0,78 0,79 0,80 0,81 0,82 0,83 0,84 0,85 0,86 0,87 0,88 0,89 0,90 0,91 0,92 0,93 0,94 0,95 0,96 0,97 0,98 0,99 1,00 1,01 1,02 1,03 1,04 1,05

0,655422 0,659097 0,662757 0,666402 0,670031 0,673645 0,677242 0,680822 0,684386 0,687933 0,691462 0,694974 0,698468 0,701944 0,705401 0,708840 0,712260 0,715661 0,719043 0,722405 0,725747 0,729069 0,732371 0,735653 0,738914 0,742154 0,745373 0,748571 0,751748 0,754903 0,758036 0,761148 0,764238 0,767305 0,770350 0,773373 0,776373 0,779350 0,782305 0,785236 0,788145 0,791030 0,793892 0,796731 0,799546 0,802337 0,805106 0,807850 0,810570 0,813267 0,815940 0,818589 0,821214 0,823814 0,826391 0,828944 0,831472 0,833977 0,836457 0,838913 0,841345 0,843752 0,846136 0,848495 0,850830 0,853141

1,08 1,09 1,10 1,11 1,12 1,13 1,14 1,15 1,16 1,17 1,18 1,19 1,20 1,21 1,22 1,23 1,24 1,25 1,26 1,27 1,28 1,29 1,30 1,31 1,32 1,33 1,34 1,35 1,36 1,37 1,38 1,39 1,40 1,41 1,42 1,43 1,44 1,45 1,46 1,47 1,48 1,49 1,50 1,51 1,52 1,53 1,54 1,55 1,56 1,57 1,58 1,59 1,60 1,61 1,62 1,63 1,64 1,65 1,66 1,67 1,68 1,69 1,70 1,71 1,72 1,73

0,859929 0,862143 0,868334 0,866500 0,868643 0,870762 0,872857 0,874928 0,876976 0,878999 0,881000 0,882977 0,884930 0,886860 0,888767 0,890651 0,892512 0,894350 0,896165 0,897958 0,899727 0,901475 0,903199 0,904902 0,906582 0,908241 0,909877 0,911492 0,913085 0,914657 0,916207 0,917736 0,919243 0,920730 0,922196 0,923641 0,925066 0,926471 0,927855 0,929219 0,930563 0,931888 0,933193 0,934478 0,935744 0,936992 0,938220 0,939429 0,940620 0,941792 0,942947 0,944083 0,945201 0,946301 0,947384 0,948449 0,949497 0,950529 0,951543 0,952540 0,953521 0,954486 0,955435 0,956367 0,957284 0,958185

1,76 1,77 1,78 1,79 1,80 1,81 1,82 1,83 1,84 1,85 1,86 1,87 1,88 1,89 1,90 1,91 1,92 1,93 1,94 1,95 1,96 1,97 1,98 1,99 2,00 2,01 2,02 2,03 2,04 2,05 2,06 2,07 2,08 2,09 2,10 2,11 2,12 2,13 2,14 2,15 2,16 2,17 2,18 2,19 2,20 2,21 2,22 2,23 2,24 2,25 2,26 2,27 2,28 2,29 2,30 2,31 2,32 2,33 2,34 2,35 2,36 2,37 2,38 2,39 2,40 2,41

0,960796 0,961636 0,962462 0,963273 0,964070 0,964852 0,965621 0,966375 0,967116 0,967843 0,968557 0,969258 0,969946 0,970621 0,971283 0,971933 0,972571 0,973197 0,973810 0,974412 0,975002 0,975581 0,976148 0,976705 0,977250 0,977784 0,978308 0,978822 0,979325 0,979818 0,980301 0,980774 0,981237 0,981691 0,982136 0,982571 0,982997 0,983414 0,983823 0,984222 0,984614 0,984997 0,985371 0,985738 0,986097 0,986447 0,986791 0,987126 0,987455 0,987776 0,988089 0,988396 0,988696 0,988989 0,989276 0,989556 0,989830 0,990097 0,990358 0,990613 0,990863 0,991106 0,991344 0,991576 0,991802 0,992024

2,44 2,45 2,46 2,47 2,48 2,49 2,50 2,51 2,52 2,53 2,54 2,55 2,56 2,57 2,58 2,59 2,60 2,61 2,62 2,63 2,64 2,65 2,66 2,67 2,68 2,69 2,70 2,71 2,72 2,73 2,74 2,75 2,76 2,77 2,78 2,79 2,80 2,81 2,82 2,83 2,84 2,85 2,86 2,87 2,88 2,89 2,90 2,91 2,92 2,93 2,94 2,95 2,96 2,97 2,98 2,99 3,00 3,01 3,02 3,03 3,04 3,05 3,06 3,07 3,08 3,09

0,992656 0,993857 0,993053 0,993244 0,993431 0,993613 0,993790 0,994963 0,994132 0,994297 0,994457 0,994614 0,994766 0,995915 0,995060 0,995201 0,995339 0,995473 0,995604 0,995731 0,995855 0,996975 0,996093 0,996207 0,996319 0,996427 0,996533 0,996636 0,996736 0,996833 0,996928 0,997020 0,997110 0,997197 0,997282 0,997365 0,997445 0,997523 0,997599 0,997673 0,997744 0,997814 0,997882 0,997948 0,998012 0,998074 0,998134 0,998193 0,998250 0,998305 0,998359 0,998411 0,998462 0,998511 0,998559 0,998605 0,998650 0,998694 0,998736 0,998777 0,998817 0,998856 0,998893 0,998930 0,998965 0,998999


28

tα0

v α g. l. 0,25 0,10 0,05 0,025 0,01 0,005 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 60 120 ∞

1,000 0,816 0,765 0,741 0,727

0,718 0,711 0,706 0,703 0,700

0,697 0,695 0,694 0,692 0,691

0,690 0,689 0,688 0,688 0,687

0,686 0,686 0,685 0,685 0,684

0,684 0,684 0,683 0,683 0,683

0,681 0,679 0,677 0,674

3,078 1,886 1,638 1,533 1,476

1,440 1,415 1,397 1,383 1,372

1,363 1,356 1,350 1,345 1,341

1,337 1,333 1,330 1,328 1,325

1,323 1,321 1,319 1,318 1,316

1,315 1,314 1,313 1,311 1,310

1,303 1,296 1,289 1,282

6,314 2,920 2,353 2,132 2,015

1,943 1,895 1,860 1,833 1,812

1,796 1,782 1,771 1,761 1,753

1,746 1,740 1,734 1,729 1,725

1,721 1,717 1,714 1,711 1,708

1,706 1,703 1,701 1,699 1,697

1,684 1,671 1,658 1,645

12,706 4,303 3,182 2,776 2,571

2,447 2,365 2,306 2,262 2,228

2,201 2,179 2,160 2,145 2,131

2,120 2,110 2,101 2,093 2,086

2,080 2,074 2,069 2,064 2,060

2,056 2,052 2,048 2,045 2,042

2,021 2,000 1,980 1,960

31,821 6,965 4,541 3,747 3,365

3,143 2,998 2,896 2,821 2,764

2,718 2,681 2,650 2,624 2,602

2,583 2,567 2,552 2,539 2,528

2,518 2,508 2,500 2,492 2,485

2,479 2,473 2,467 2,462 2,457

2,423 2,390 2,358 2,326

63,657 9,925 5,841 4,604 4,032

3,707 3,499 3,355 3,250 3,169

3,106 3,055 3,012 2,977 2,947

2,921 2,898 2,878 2,861 2,845

2,831 2,819 2,807 2,797 2,787

2,779 2,771 2,763 2,756 2,750

2,704 2,660 2,617 2,576


29

χα2

α

α g. l. 0,995 0,990 0,975 0,950 0,050 0,025 0,010 0,005 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 40 50 60 70 80 90 100

0,00004 0,01003 0,07172 0,20699 0,41174

0,67573 0,98927 1,34419 1,73493 2,15585

2,60321 3,07382 3,56503 4,07468 4,60094

5,14224 5,69724 6,26481 6,84398 7,43386

8,03386 8,64272 9,26042 9,88623

10,5197

11,1603 11,8076 12,4613 13,1211 13,7867

20,7065 27,9907 35,5346 43,2752 51,1720

59,1963 67,3276

0,00016 0,02010 0,11483 0,29711 0,55430

0,87209 1,23904 1,64648 2,08791 2,55821

3,05347 3,57056 4,10691 4,66043 5,22935

5,81221 6,40776 7,01491 7,63273 8,26040

8,89720 9,54249

10,1957 10,8564 11,5240

12,1981 12,8786 13,5648 14,2565 14,9535

22,1643 29,7067 37,4848 45,4418 53,5400

61,7541 70,0648

0,00098 0,05064 0,21580 0,48442 0,83121

1,23735 1,68987 2,17973 2,70039 2,42697

3,81575 4,40379 5,00874 5,62872 6,26214

6,90766 7,56418 8,23075 8,90655 9,59083

10,2829 10,9823 11,6885 12,4011 13,1197

13,8439 14,5733 15,3079 16,0471 16,7908

24,4331 32,3574 40,4817 48,7576 57,1532

65,6466 74,2219

0,00393 0,10259 0,35185 0,71072 1,14548

1,63539 2,16735 2,73264 3,32511 3,94030

4,57481 5,22603 5,89186 6,57063 7,26094

7,96164 8,67176 9,39046

10,1170 10,8508

11,5913 12,3380 13,0905 13,8484 14,6114

15,3791 16,1513 16,9279 17,7083 18,4926

26,5093 34,7642 43,1879 51,7393 60,3915

69,1260 77,9295

3,84146 5,99147 7,81473 9,48773

11,0705

12,5916 14,0671 15,5073 16,9190 18,3070

19,6751 21,0261 22,3621 23,6848 24,9958

26,2962 27,5871 28,8693 30,1435 31,4104

32,6705 33,9244 35,1725 36,4151 37,6525

38,8852 40,1133 41,3372 42,5569 43,7729

55,7585 67,5048 79,0819 90,5312

101,879

113,145 124,342

5,02389 7,37776 9,34840

11,1433 12,8325

14,4494 16,0128 17,5346 19,0228 20,4831

21,9200 23,3367 24,7356 26,1190 27,4884

28,8454 30,1910 31,5264 32,8523 34,1696

35,4789 36,7807 38,0757 39,3641 40,6465

41,9232 43,1944 44,4607 45,7222 46,9792

59,3417 71,4202 83,2976 95,0231

106,629

118,136 129,561

6,63490 9,21034

11,3449 13,2767 15,0863

16,8119 18,4753 20,0902 21,6660 23,2093

24,7250 26,2170 27,6883 29,1413 30,5779

31,9999 33,4087 34,8053 36,1908 37,5662

38,9321 40,2894 41,6384 42,9798 44,3141

45,6417 46,9630 48,2782 49,5879 50,8922

63,6907 76,1539 88,3794

100,425 112,329

124,116 135,807

7,87944 10,5966 12,8381 14,8602 16,7496

18,5476 20,2777 21,9550 23,5893 25,1882

26,7569 28,2995 29,8194 31,3193 32,8013

34,2672 35,7185 37,1564 38,5822 39,9968

41,4010 42,7956 44,1813 45,5585 46,9278

48,2899 49,6449 50,9933 52,3356 53,6720

66,7659 74,4900 91,9517

104,215 116,321

128,299 140,169


30

exerc (1).pdf

Documents