Chapter 2 1

Design of Engineering ExperimentsChapter 2 – Some Basic Statistical Concepts• Describing sample data

– Random samples– Sample mean, variance, standard deviation– Populations versus samples– Population mean, variance, standard deviation– Estimating parameters

• Simple comparative experiments– The hypothesis testing framework– The two-sample t-test– Checking assumptions, validity

Chapter 2 2

Portland Cement Formulation (page 24)

Chapter 2 3

Graphical View of the DataDot Diagram, Fig. 2.1, pp. 24

Chapter 2 4

If you have a large sample, a histogram may be useful

Chapter 2 5

Box Plots, Fig. 2.3, pp. 26

Chapter 2 6

The Hypothesis Testing Framework

• Statistical hypothesis testing is a useful framework for many experimental situations

• Origins of the methodology date from the early 1900s

• We will use a procedure known as the two-sample t-test

Chapter 2 7

The Hypothesis Testing Framework

• Sampling from a normal distribution• Statistical hypotheses:

0 1 2

1 1 2

:

:

H

H

Chapter 2 8

Estimation of Parameters

1

2 2 2

1

1 estimates the population mean

1( ) estimates the variance

1

n

ii

n

ii

y yn

S y yn

Chapter 2 9

Summary Statistics (pg. 36)

1

21

1

1

16.76

0.100

0.316

10

y

S

S

n

Formulation 1

“New recipe”

Formulation 2

“Original recipe”

2

22

2

2

17.04

0.061

0.248

10

y

S

S

n

Chapter 2 10

How the Two-Sample t-Test Works:

1 2

22y

Use the sample means to draw inferences about the population means

16.76 17.04 0.28

Difference in sample means

Standard deviation of the difference in sample means

This suggests a statistic:

y y

n

1 20 2 2

1 2

1 2

Zy y

n n

Chapter 2 11

How the Two-Sample t-Test Works:2 2 2 2

1 2 1 2

1 2

2 21 2

1 2

2 2 21 2

2 22 1 1 2 2

1 2

Use and to estimate and

The previous ratio becomes

However, we have the case where

Pool the individual sample variances:

( 1) ( 1)

2p

S S

y y

S Sn n

n S n SS

n n

Chapter 2 12

How the Two-Sample t-Test Works:

• Values of t0 that are near zero are consistent with the null hypothesis

• Values of t0 that are very different from zero are consistent with the alternative hypothesis

• t0 is a “distance” measure-how far apart the averages are expressed in standard deviation units

• Notice the interpretation of t0 as a signal-to-noise ratio

1 20

1 2

The test statistic is

1 1

p

y yt

Sn n

Chapter 2 13

The Two-Sample (Pooled) t-Test2 2

2 1 1 2 2

1 2

1 20

1 2

( 1) ( 1) 9(0.100) 9(0.061)0.081

2 10 10 2

0.284

16.76 17.04 2.20

1 1 1 10.284

10 10

The two sample means are a little over two standard deviations apart

Is t

p

p

p

n S n SS

n n

S

y yt

Sn n

his a "large" difference?

Chapter 2 14

William Sealy Gosset (1876, 1937)

Gosset's interest in barley cultivation led him to speculate that design of experiments should aim, not only at improving the average yield, but also at breeding varieties whose yield was insensitive (robust) to variation in soil and climate.

Developed the t-test (1908)

Gosset was a friend of both Karl Pearson and R.A. Fisher, an achievement, for each had a monumental ego and a loathing for the other.

Gosset was a modest man who cut short an admirer with the comment that “Fisher would have discovered it all anyway.”

Chapter 2 15

The Two-Sample (Pooled) t-Test

• So far, we haven’t really done any “statistics”

• We need an objective basis for deciding how large the test statistic t0

really is• In 1908, W. S. Gosset

derived the reference distribution for t0 … called the t distribution

• Tables of the t distribution – see textbook appendix

t0 = -2.20

Chapter 2 16

The Two-Sample (Pooled) t-Test• A value of t0 between

–2.101 and 2.101 is consistent with equality of means

• It is possible for the means to be equal and t0 to exceed either 2.101 or –2.101, but it would be a “rare event” … leads to the conclusion that the means are different

• Could also use the P-value approach

t0 = -2.20

Chapter 2 17

The Two-Sample (Pooled) t-Test

• The P-value is the area (probability) in the tails of the t-distribution beyond -2.20 + the probability beyond +2.20 (it’s a two-sided test)

• The P-value is a measure of how unusual the value of the test statistic is given that the null hypothesis is true

• The P-value the risk of wrongly rejecting the null hypothesis of equal means (it measures rareness of the event)

• The P-value in our problem is P = 0.042

t0 = -2.20

Chapter 2 18

Computer Two-Sample t-Test Results

Chapter 2 19

Checking Assumptions – The Normal Probability Plot

Chapter 2 20

Importance of the t-Test

• Provides an objective framework for simple comparative experiments

• Could be used to test all relevant hypotheses in a two-level factorial design, because all of these hypotheses involve the mean response at one “side” of the cube versus the mean response at the opposite “side” of the cube

Chapter 2 21

Confidence Intervals (See pg. 44)• Hypothesis testing gives an objective statement

concerning the difference in means, but it doesn’t specify “how different” they are

• General form of a confidence interval

• The 100(1- α)% confidence interval on the difference in two means:

where ( ) 1 L U P L U

1 2

1 2

1 2 / 2, 2 1 2 1 2

1 2 / 2, 2 1 2

(1/ ) (1/ )

(1/ ) (1/ )

n n p

n n p

y y t S n n

y y t S n n

Chapter 2 22

Chapter 2 23

A função t.test no R

t.test(stats)Student's t-TestDescriptionPerforms one and two sample t-tests on vectors of data. Usaget.test(x, y = NULL, alternative = c("two.sided", "less", "greater"), mu = 0,paired = FALSE, var.equal = FALSE, conf.level = 0.95, ...)

Chapter 2 24

Argumentos da função t.testx - a (non-empty) numeric vector of data values.

y - an optional (non-empty) numeric vector of data values.

alternative - a character string specifying the alternative hypothesis, must be one of “two.sided" (default), "greater" or "less". You can specify just the initial letter.

mu - a number indicating the true value of the mean (or difference in means if you are performing a two sample test).

paired - a logical indicating whether you want a paired t-test.

var.equal - a logical variable indicating whether to treat the two variances as being equal. If TRUE then the pooled variance is used to estimate the variance otherwise the Welch (or Satterthwaite) approximation to the degrees of freedom is used.

Chapter 2 25

Argumentos da função t.testconf.level

- confidence level of the interval.

formula - a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs a factor with two levels giving the corresponding groups.

data - an optional matrix or data frame containing the variables in the formula.

subset - an optional vector specifying a subset of observations to be used.

na.action -

a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Chapter 2 26

Exemplo dos dados sobre cimento

• Arquivo em cimento.txt com nome das variáveis.

• Ler e realizar o teste t no R.

Chapter 2 27

Usando o R

• dados=read.table(“m://aulas//flavia//cimento.txt”,header=T)

• stripchart(dados,at=c(1,1.1))

• boxplot(dados)

Chapter 2 28

t.test(dados$m,dados$u,alternative="two.sided",var.equal=T,paired=F,conf.level=.95)

Two Sample t-test

data: dados$m and dados$u t = -2.1869, df = 18, p-value = 0.0422alternative hypothesis: true difference in means is not equal to 0 95 percent confidence interval: -0.54507339 -0.01092661 sample estimates:mean of x mean of y 16.764 17.042

Chapter 2 29

Comparando as variâncias

• Dadas duas amostras independentes de duas distribuições normais, antes de realizar o teste t, para comparar as médias, é necessário verificar se é razoável ou não considerar variâncias iguais ou não, para saber se adotaremos o teste t “pooled” (combinado) ou se adotaremos uma aproximação para o número de graus de liberdade da distribuição amostral da estatística de teste, adotando uma aproximação e não a distribuição exata.

Chapter 2 30

• Se as amostras provêm de fato de populações normais temos que a variância amostral a menos de constante tem distribuição de qui-quadrado com número de graus de liberdade n-1, em que n é o tamanho da amostra.

• Como as amostras são independentes, segue que a menos da constante, as duas variâncias amostrais são independentemente distribuídas segundo uma distribuição de qui-quadrado.

Chapter 2 31

Resumindo...

1,122

21

21

22

212

2

21~

que tal

2,1,~)1(

nn

ni

ii

FS

S

i

indS

ni

Chapter 2 32

Teste de igualdade das variâncias

• Sob a hipótese de que as variâncias são iguais, segue que a estatística de teste é dada pela razão das variâncias amostrais e, num teste bilateral de nível de significância α, rejeitaremos a hipótese nula se:

.~ com,)FP(X

que em

ou

,mn,,

1,1,2/122

21

1,1,2/22

21

2121

mn

nnnn

FX

FS

SF

S

S

Chapter 2 33

• No R está disponível a função var.test

F test to compare two variances

data: dados$m and dados$u F = 1.6293, num df = 9, denom df = 9, p-value = 0.4785alternative hypothesis: true ratio of variances is not equal to 1 95 percent confidence interval: 0.4046845 6.5593806 sample estimates:ratio of variances 1.629257

var.test(dados$m,dados$u,ratio=1,alternative="two.sided",conf.level=0.95)