Design of Engineering Experiments
Chapter 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
1
Portland Cement Formulation (page 24)
Chapter 2
2
Graphical View of the Data
Dot Diagram, Fig. 2.1, pp. 24
Chapter 2
3
If you have a large sample, a
histogram may be useful
Chapter 2
4
Box Plots, Fig. 2.3, pp. 26
Chapter 2
5
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 twosample t-test
Chapter 2
6
The Hypothesis Testing Framework
• Sampling from a normal distribution
• Statistical hypotheses: H :   
0
1
2
H1 : 1  2
Chapter 2
7
Estimation of Parameters
1 n
y   yi estimates the population mean 
n i 1
n
1
2
2
2
S 
( yi  y ) estimates the variance 

n  1 i 1
Chapter 2
8
Summary Statistics (pg. 36)
Formulation 1
Formulation 2
“New recipe”
“Original recipe”
y1  16.76
y2  17.04
S  0.100
S 22  0.061
S1  0.316
S 2  0.248
n1  10
n2  10
2
1
Chapter 2
9
How the Two-Sample t-Test Works:
Use the sample means to draw inferences about the population means
y1  y2  16.76  17.04  0.28
Difference in sample means
Standard deviation of the difference in sample means
 
2
y
2
n
This suggests a statistic:
Z0 
y1  y2
 12
n1
Chapter 2

 22
n2
10
How the Two-Sample t-Test Works:
Use S and S to estimate  and 
2
1
2
2
The previous ratio becomes
2
1
2
2
y1  y2
2
1
2
2
S
S

n1 n2
However, we have the case where  12   22   2
Pool the individual sample variances:
2
2
(
n

1)
S

(
n

1)
S
2
1
2
2
Sp  1
n1  n2  2
Chapter 2
11
How the Two-Sample t-Test Works:
The test statistic is
y1  y2
t0 
1 1
Sp

n1 n2
• 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
Chapter 2
12
The Two-Sample (Pooled) t-Test
(n1  1) S12  (n2  1) S 22 9(0.100)  9(0.061)
S 

 0.081
n1  n2  2
10  10  2
2
p
S p  0.284
y1  y2
16.76  17.04
t0 

 2.20
1 1
1 1
Sp

0.284

n1 n2
10 10
The two sample means are a little over two standard deviations apart
Is this a "large" difference?
Chapter 2
13
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
14
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
Chapter 2
t0 = -2.20
15
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
Chapter 2
t0 = -2.20
16
The Two-Sample (Pooled) t-Test
t0 = -2.20
•
•
•
•
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
Chapter 2
17
Computer Two-Sample t-Test Results
Chapter 2
18
Checking Assumptions –
The Normal Probability Plot
Chapter 2
19
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
20
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
L    U where P( L    U )  1  
• The 100(1- α)% confidence interval on the
difference in two means:
y1  y2  t / 2,n1  n2 2 S p (1/ n1 )  (1/ n2 )  1  2 
y1  y2  t / 2,n1  n2 2 S p (1/ n1 )  (1/ n2 )
Chapter 2
21
Chapter 2
22
A função t.test no R
t.test(stats)
Student's t-Test
Description
Performs one and two sample t-tests on vectors of data.
Usage
t.test(x, y = NULL,
alternative = c("two.sided", "less", "greater"), mu = 0,
paired = FALSE, var.equal = FALSE, conf.level = 0.95, ...)
Chapter 2
23
Argumentos da função t.test
x-
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
24
Argumentos da função t.test
conf.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
25
Exemplo dos dados sobre cimento
• Arquivo em cimento.txt com nome das
variáveis.
• Ler e realizar o teste t no R.
Chapter 2
26
Usando o R
• dados=read.table(“m://aulas//flavia//cimento.txt”,header=T)
• stripchart(dados,at=c(1,1.1))
• boxplot(dados)
Chapter 2
27
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 = 0.0422
alternative 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
28
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
29
• 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
30
Resumindo...
(ni  1)
S

2
i
2
i
ind
 n2 1 , i  1,2
~
i
t al que
 22 S12
 2 ~ Fn 1,n 1
2
 1 S2
1
Chapter 2
2
31
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:
S12
S12
 F / 2,n1 1,n2 1 ou 2  F1 / 2,n1 1,n2 1
2
S2
S2
em que
P (X  F ,n, m )   , com X ~ Fn ,m .
Chapter 2
32
• No R está disponível a função var.test
var.test(dados$m,dados$u,ratio=1,alternative="two.sided",conf.level=0.95)
F test to compare two variances
data: dados$m and dados$u
F = 1.6293, num df = 9, denom df = 9, p-value = 0.4785
alternative 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
Chapter 2
33