Transcript t-Test
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
Chapter 2
Graphical View of the Data
Dot Diagram, Fig. 2.1, pp. 24
3
Chapter 2
If you have a large sample, a histogram may be useful
4
Chapter 2
Box Plots, Fig. 2.3, pp. 26
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
two sample t-test
Chapter 2 6
The Hypothesis Testing Framework
• Sampling from a
normal
distribution • Statistical hypotheses:
H H
1 0 : : 1 1 2 2 Chapter 2 7
Estimation of Parameters
y
1
n i n
1
y i
estimates the population mean
S
2
n
1 1
i n
1 (
y i
y
2 ) estimates the variance 2 Chapter 2 8
Chapter 2
Summary Statistics (pg. 36)
Formulation 1 “New recipe”
y
1 16.76
S
1 2
S
1 0.100
0.316
n
1 10
Formulation 2 “Original recipe”
y
2 17.04
S
2 2
S n
2 2 0.061
0.248
10 9
How the Two-Sample t-Test Works:
Use the sample means to draw inferences about the population means
y
1
y
2 0.28
Difference in sample means Standard deviation of the difference in sample means y 2 2
n
This suggests a statistic: Z 0
y
1 1 2
n
1
y
2
n
2 2 2 Chapter 2 10
How the Two-Sample t-Test Works:
Use
S
1 2 and
S
2 2 to estimate 1 2 The previous ratio becomes and 2 2
y
1
y
2
S
1 2
S
2 2
n
1
n
2 However, we have the case where 1 2 2 2 2 Pool the individual sample variances:
S p
2 (
n
1 1)
n
1
S
1 2
n
2 (
n
2 2 1)
S
2 2 Chapter 2 11
How the Two-Sample t-Test Works:
The test statistic is
t
0
S p y
1
y
2 1
n
1 1
n
2 • Values of
t
0 hypothesis that are near zero are consistent with the null • • Values of
t
0 that are very different from zero are consistent with the alternative hypothesis
t
0 is a “distance” measure-how far apart the averages are expressed in standard deviation units • Notice the interpretation of
t
0 as a
signal-to-noise
ratio Chapter 2 12
The Two-Sample (Pooled) t-Test
S
2
p S p
(
n
1 1)
n
1
S
1 2
n
2 (
n
2 2 1)
S
2 2 0.284
0.081
t
0
S p y
1
y
2 1
n
1 1
n
2 0.284
1 10 1 10 2.20
The two sample means are a little over two standard deviations apart Is t his 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
t
0 really is • In 1908, W. S. Gosset derived the
reference distribution
for
t
0 … called the
t
distribution • Tables of the
t
distribution – see textbook appendix Chapter 2
t
0 = -2.20
15
The Two-Sample (Pooled) t-Test
• A value of
t
0 between –2.101 and 2.101 is consistent with equality of means • It is possible for the means to be equal and
t
0 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
t
0 = -2.20
Chapter 2 16
The Two-Sample (Pooled) t-Test
t
0 = -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 rareness of the event)
wrongly rejecting
the null hypothesis of equal means (it measures The
P-
value in our problem is
P
= 0.042
Chapter 2 17
Computer Two-Sample t-Test Results
Chapter 2 18
Chapter 2
Checking Assumptions – The Normal Probability Plot
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
L U
of a confidence interval
U
) • The 100(1- α)%
confidence interval
on the difference in two means:
y
1
y
2
t
/ 2,
n
1 2
S p
(1/
n
1
n
2 )
y
1
y
2
t
/ 2,
n
1 1 2 2
S p
(1/
n
1
n
2 ) 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 y alternative a (non-empty) numeric vector of data values.
an optional (non-empty) numeric vector of data values.
mu a character string specifying the alternative hypothesis, must be one of “ two.sided" (default), "greater" or "less" . You can specify just the initial letter.
a number indicating the true value of the mean (or difference in means if you are performing a two sample test).
paired var.equal a logical indicating whether you want a paired t-test.
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.
24 Chapter 2
Argumentos da função
t.test
conf.level formula data confidence level of the interval.
a formula of the form lhs ~ rhs where lhs variable giving the data values and rhs is a numeric a factor with two levels giving the corresponding groups.
an optional matrix or data frame containing the variables in the formula.
subset na.action an optional vector specifying a subset of observations to be used.
a function which indicates what should happen when the data contain NA s. 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
Chapter 2
Resumindo...
(
n i
1 )
S i
2
i
2
ind
~ 2
n i
1 ,
i
1 , 2 tal que 1 2 2 2
S
1 2
S
2 2 ~
F n
1 1 ,
n
2 1 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:
S
1 2
S
2 2
F
/ 2 ,
n
1 1 ,
n
2 1 ou
S
1 2
S
2 2
F
1 / 2 ,
n
1 1 ,
n
2 1 em que P(X F , n, m ) , com
X
~
F n
,
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