Wine Chemistry and Biochemistry

678 P.J. Mart ́ın-Alvarez ́

13.1.1 Statistical Treatment for Only One Sample

Let{x 1 ,x 2 ,x 3 ,...,xn}be a random sample ofnobservations of a continuous

random variableX, from a population whereX has a meanμand a standard

deviationσ.Wehavethesample mean( ̄x =

∑

xi/n)andthesample standard

deviation

(

s=

√∑

(xi− ̄x)^2 /(n−1)

)

. The median, the lower (Q 1 ) and upper

(Q 3 ) quartiles, the relative standard deviation

(

RSD(%)=CV(%)= 100 s ̄x

)

,and

the standard error (s/

√

n) can also be calculated. For the graphical processing we

can use the histogram or box plot (withmin, Q 1 , median, Q 3 ,andmaxvalues, or with

Q 1 − 1 .5(Q 3 −Q 1 ),Q 1 , median, Q 3 ,Q 3 + 1 .5(Q 3 −Q 1 ) values to show potential

outliers). Accepting X is a normally distributed random variable with meanμand

standard deviationσ(X∼N(μ, σ)), that can be verified with thenormal probabil-

ity plotor with thenormality tests(Shapiro-Wilks, Kolmogoroff-Smirnov-Lilliefors,

etc.), and since ̄xandsare estimators ofμandσ, respectively, with a fixedsig-

nificance levelα(e.g.α= 0 .05), we can calculate the correspondingconfidence

intervalsat 100(1−α)% forμ:

[

x ̄−t 1 −α/ 2 ,n− 1 s/

√

n,x ̄+t 1 −α/ 2 ,n− 1 s/

√

n

]

and

forσ^2 :

[

(n−1)s^2 /χ 1 −α/ 2 ,n− 1 ,(n−1)s^2 /χα/ 2 ,n− 1

]

,wheret 1 −α/ 2 ,n− 1 is the critical

value of the t-Student distribution withn−1 degrees of freedom (df) such that

pr ob(tn− 1 ≤t 1 −α/ 2 ,n− 1 )= 1 −α/ 2 ,χα/^22 ,n− 1 andχ 12 −α/ 2 ,n− 1 are the critical values

of theχ^2 -distribution withn−1 df, such thatpr ob(χn^2 − 1 ≤χα/^22 ,n− 1 ) =α/ 2

andpr ob(χ^2 n− 1 ≤χ^21 −α/ 2 ,n− 1 )= 1 −α/ 2.

13.1.1.1 Hypothesis Test for a Mean or One-Sample T Test

To test the null hypothesisH 0 ≡μ=μ 0 against thetwo-sidedalternative hypoth-

esisH 1 ≡μ=μ 0 (H 0 can be rejected equally byμ<μ 0 or byμ>μ 0 ), we

can use the statistic:tcal =

̄x−μ 0

s/

√

n

that has a t-Student distribution withn− 1

df, ifH 0 is true. For a fixed value ofα,if|tcal|>t 1 −α/ 2 ,n− 1 ,H 0 ≡μ=μ 0 is

rejected andH 1 ≡μ=μ 0 is accepted (the test isstatistically significant at levelα);

otherwise (|tcal|≤t 1 −α/ 2 ,n− 1 ), there is no reason to rejectH 0 (the test isstatistically

nonsignificant). With the associated probability (P= 2 pr ob(tn− 1 >|tcal|) ), facili-

tated by the statistical programs, if thePvalue is less thanαthen the null hypothesis

(H 0 ≡μ=μ 0 ) is rejected, otherwise (P>α)μ=μ 0 is not rejected. In the case

of theone-sidedalternative hypothesis (e.g.H 1 ≡μ>μ 0 orH 1 ≡μ<μ 0 ), if

|tcal|>t 1 −α,n− 1 ,orifP=pr ob(tn− 1 >|tcal|)<αthenH 0 should be rejected.

13.1.1.2 Example of Application

The one-sample t test can be used to test for systematic errors in the analyti-

cal method for a standard material with a knownconcentrationμ 0 (H 0 ≡μ=

μ 0 vsH 1 ≡μ=μ 0 ), or to verify if the changes in the elaboration process

of a certain product affect the previous concentrationμ 0 of a compound (H 0 ≡

μ =μ 0 vsH 1 ≡μ>μ 0 ) (Massart et al. 1990; Miller and Miller 2000;

Mart ́ın-Alvarez 2000, 2006). Table 13.1 shows the results of the one-sample t test ́