Wine Chemistry and Biochemistry

680 P.J. Mart ́ın-Alvarez ́

13.1.2 Statistical Treatment to Compare Two Independent Samples

Let

{

x 1 , 1 ,x 2 , 1 ,x 3 , 1 ,...,xn 1 , 1

}

and

{

x 1 , 2 ,x 2 , 2 ,x 3 , 2 ,...,xn 2 , 2

}

be two independent

random samples ofn 1 andn 2 observations of a continuous random variableX, from

two populationsW 1 andW 2 whereXhas mean valuesμ 1 ,μ 2 and standard deviation

valuesσ 1 ,σ 2 , respectively. We have the descriptive values ̄xj,sj j= 1 ,2. For the

graphic processing we can use the histograms or the box plots. Accepting normal

distributions (X ∼ N(μj,σj) in Wj), and for asignificance levelα, we can

calculate the corresponding confidence intervals for the parametersμj,σj,ortotest

hypotheses about them.

13.1.2.1 Hypothesis Test for Two Means or Two-Sample T Test

To test the null hypothesisH 0 ≡μ 1 =μ 2 against the alternative hypothesisH 1 ≡

μ 1 =μ 2 , assuming normal distributions and equality of variances (σ 12 =σ 22 ), the

test statistic istcal=

x ̄ 1 −x ̄ 2

sp

√

1

n 1 +

1

n 2

which, underH 0 , has a t-Student distribution with

ν=n 1 +n 2 −2df,andwheres^2 p=

(n 1 −1)s 12 +(n 2 −1)s 22

n 1 +n 2 − 2 is the pooled sample variance.

When the variances are not equal, the test statistic is:tcal=

x ̄ 1 −x ̄ 2

√

s^21

n 1 +

s^22

n 2

which, under

H 0 , has a t-Student distribution withν=

(s^21 /n 1 +s 22 /n 2 )^2
(s^21 /n 1 )^2 /(n 1 +1)+(s 22 /n 2 )^2 /(n 2 +1)
−2df.Forafixedvalueofα,if|tcal|>t 1 −α/ 2 ,νthenH 0 ≡μ 1 =μ 2 is rejected

andH 1 ≡μ 1 =μ 2 is accepted, otherwise (|tcal|≤t 1 −α/ 2 ,ν),H 0 should not be

rejected. With the associated probability (P= 2 pr ob(tν>|tcal|)), ifP<αthen

H 0 ≡μ 1 =μ 2 is rejected, otherwise (P>α)H 0 ≡μ 1 =μ 2 is not rejected.

The nonparametric Mann-Whitney U-test for the comparison of two indepen-

dent sets of data, without accepting normal distributions, uses the ranks of all

the observations, previously arranged from the lowest to the highest value, to test

H 0 ≡median 1 =median 2 (Massart et al. 1990; O’Mahony 1986). IfP<α,H 0

is rejected.

13.1.2.2 Hypothesis Test for Two Variances

To test the null hypothesisH 0 ≡σ 12 =σ 22 againstH 1 ≡σ 12 >σ 22 (one-tail),

and assuming normal distributions ands 12 >s^22 , the test statistic isFcal=s 12 /s 22

which, underH 0 , has an F-distribution withn 1 −1andn 2 −1df.Forafixedvalue

ofα,ifFcal<F 1 −α,n 1 − 1 ,n 2 − 1 ,H 0 ≡σ 12 =σ 22 should not be rejected, otherwise

(Fcal>F 1 −α,n 1 − 1 ,n 2 − 1 )H 1 ≡σ 12 >σ 22 is accepted. With the associated probability

(P= pr ob(Fn 1 − 1 ,n 2 − 1 >Fcal)),ifP<αthen the variancesare significantly

different. This test can be used to compare the precision of two analytical methods

for the same sample.