11-12 Statistical tests anova

57 Pages·2016·1.04 MB·English

by  Makra László

SSTTAATTIISSTTIICCSS
F-test
two-sample t-test
Cochran-test
Variance analysis (ANOVA)
One- and two-sided tests
• Two-sided tests • One-side test
– H : no change – H : the average did not
0 0
increase
– H : there is change
a
(any direction) – H : the average increased
a
t-distribution with 7 t-distribution with 7
degrees of freedom degrees of freedom
In case of p-values: p(one-sided)=p(two-sided)/2
Interpretation of the significance
• Significant difference: p<α , p<0.05. It is stated that the
compared populations are different. The probability error
of the decision is small (maximum α − this is the error of
the first kind or Type I error).
• Non-significant difference: p>α , p>0.05. In this case, all
you can say is that there is not sufficient information to
detect the difference. Maybe
– actually there is no difference;
– there is a difference, however only a few number of elements
were available;
– there was a standard deviation;
– it was wrong of the test method;
– 1
• Statistical significance should always be thought through
whether for example it is significant in agricultural point
of view;
• When providing statistical significance, indication of the
p-value is also advisable;
FF--tteesstt ffoorr ddeetteeccttiinngg iiddeennttiittyy ooff vvaarriiaanncceess ooff ttwwoo
nnoorrmmaallllyy ddiissttrriibbuutteedd rraannddoomm vvaarriiaabblleess
(cid:1) Our hypothesis for the identity of the variances of two independent
random variables of normal distribution with unknown expectation and
variance is checked by the so-called F-test.
2 2
(cid:1) H : σ = σ
0 1 2 TToo bbee ppeeffoorrmmeedd bbeeffoorree tt--tteesstt!!
2 2
(cid:1) H : σ > σ
1 2
1
(cid:1) The test is always carried out as a one-sided test (it could be carried out
otherwise as well)
2
*
s
. F = 1 s∗2 > s∗2
(cid:1) Test statistics: s z ,2 where 1 2
*
s
2
numerator: DF = n -1 denominator: DF = n -1
1 1 2 2
(cid:1) If H fulfils, then F is of F-distribution with degrees of freedom n -1, n -1
0 sz 1 2
(cid:1) Decision principle: for F ≤ F 0-hypothesis is accepted, otherwise not.
sz α
Two-sample t-test
Y −Y
1 2
s2 s2
1 + 2
n n
1 2
If the samples are independent,
normally distributed and their
standard deviations do not differ
significantly, they be seen as two
parts of a single sample.
Y −Y
1 2
( ) ( )
2 2
n + n n −1 s + n −1 s
1 2 ⋅ 1 1 2 2
n n n + n − 2
1 2 1 2
The test statistics received is of t-distribution with degrees of freedom n + n - 2
1 2
Conditions of the t-test:
• For one-sample t-test:
• the random variable is normally distributed;
• the sample elements are independent;
• For two-sample t-test, in addition:
• standard deviations of the same two random variables
are identical;
degree of
freedom critical values belongong to p
Table of the t-distribution
and the test statistics of the
one-sample t-test
FF--tteesstt ffoorr ddeetteeccttiinngg iiddeennttiittyy ooff vvaarriiaanncceess ooff ttwwoo nnoorrmmaallllyy ddiissttrriibbuutteedd rraannddoomm vvaarriiaabblleess
Hypothesistesting
Nonparametrictests Parametrictests
Multiple-sampletests
Fittingtest withχ2-test
H: F=F
0 0 For theexpectanceof For thevarianceof more
more thanonenormally thanonenormally
distributedrandom distributedrandom
Homogeneitytest with
χ2-test variables variables
H: F(ξ)=G(η)
0
Varianceanalysis Cochran test
Independencetest with H0: µ1=µ2=…=µn H0: σ1=σ2=…=σr
χ2-test σ1=σ2=…=σn n1=n2=…=nr=n
H: ξandηindependent
0
One-sampletests Two-sampletests
For theexpectanceof For thevarianceof For theexpectanceof For thevarianceof two
normallydistributed normallydistributed twonormallydistributed normallydistributed
random variable random variable random variables random variables
One-samplez-test χ2-test for the
F-test
H0: µ=µ0 variance For independentsamples For pairedsamples H: σ2 =σ2
σknown, orn>30 H: σ2=σ2 0 1 2
0 0
Two-samplez-test
One-samplet-test
H: µ=µ H0: µ1=µ2 Pairedt-test
σu0nknow0n σ1, σ2known, or H0: µ1-µ2=d0
n,n>30
1 2
Two-samplet-test
H: µ =µ
0 1 2
σ,σ unknown, σ = σ
1 2 1 2
TTaasskk ((FF--tteesstt))
CO emissions of cigarettes from two different brands were
tested. The data were as follows. May we assume that the
standard deviation of the CO emission of the two brands are
the same?
„A” „B”
n 11 10
Mean 16,4 mg 15,6 mg
s* 1,2 mg 1,1 mg
SSoolluuttiioonn ooff tthhee ttaasskk ((FF--tteesstt))
H : σ = σ
0 1 2
H : σ > σ
1 1 2
α = 0,05, DF = 10, DF = 9
1 2
F = 3,13
0,05
2
1,2
F = = 1,19
sz 2
1,1
(cid:1) Since F occurs in the acceptance range, there is no
sz
reason to reject H in the 5% significance level.
0