13 Statistical Techniques for the Interpretation of Analytical Data 681
Table 13.2Mean and the standard deviation (SD) values for octanoic acid in white and ros ́ewines
and the results of the two-sample t test
White wines Ros ́e wines Assuming equal No assuming equal
(n=8) (n=8) variances variances Test of variancesMean SD Mean SD t-value df P t-value df P F-value P
2.54 0.92 8.26 3.01 –5.14 14 .0002 –5.14 8.31. 0008 ∗∗ 10.61. 0059 ∗
t-value=value of the statistic tcal,df=degrees of freedom, P=associated probability
∗the two variances are different (P<0.05).
∗∗the two means are different (P<0.05).
13.1.2.3 Example of Application
The two-sample t test can be used to compare the results obtained by two labora-
tories for the same sample of reference, tocompare the concentrations of a certain
compound in wines elaborated with grapes of two varieties, etc., and in a general
way, to compare the mean values of two groups of independent observations (Miller
and Miller 2000; Massart et al. 1990; Mart ́ın-Alvarez 2000, 2006). Table 13.2 shows ́
the mean and the standard deviation values for octanoic acid in white and ros ́ewines
and the results of the two-sample t test, obtained with the STATISTICA program
(procedureT-Test for Independent samples,intheBasic Statistics and Tablesmod-
ule). The results of the test of variances are also included in the table, and since the
P-value is less than 0.05, the variancesare significantly different. The ros ́ewines
have a higher octanoic acid content than the white wines (P<0.05).
13.1.3 Statistical Treatment to Compare Two Related Samples
Let
{
(x 1 , 1 ,x 1 , 2 ),(x 2 , 1 ,x 2 , 2 ),...,(xn, 1 ,xn, 2 )}
be a random sample ofnpaired obser-vations of two continuous random variablesX 1 andX 2 , from a populationWwhere
the variables have mean valuesμ 1 ,μ 2 .Fromthesenpaired observations,ndif-
ferences may be calculated
{
di=xi, 1 −xi, 2}
, with descriptive valuesd ̄ andsd.Accepting normality of the differences, to test the null hypothesisH 0 ≡μd = 0
(orH 0 ≡μ 1 =μ 2 ), againstH 1 ≡μd=0(orH 1 ≡μ 1 =μ 2 ), the test statistic
istcal=
d ̄
sd/√
n, which has a t-distribution withn−1df.Forafixedvalueofα,if
|tcal|>t 1 −α/ 2 ,n− 1 ,orifP<α,H 0 ≡μd=0 should be rejected andH 1 ≡μd= 0
accepted, otherwise,|tcal|<t 1 −α/ 2 ,n− 1 orP>α,H 0 ≡μd=0 is not rejected.
If a normal distribution is not assumed,the Wilcoxon matched-pairs signed-ranks
test can be used to testH 0 ≡μd=0.
13.1.3.1 Applications
This t test for related samples can be used to compare a new analytical method
and the reference method (Massart et al. 1990), to compare the amino acid con-
tent between different stages of the elaboration processes for wines from the same
batches (Marcobal et al. 2006b), and more generally, to compare paired samples.