Enthalpy is a measure of total energy in a thermodynamic sys

Enthalpy is a measure of total energy in a thermodynamic system. The following table presents enthalpy measurements on 16 water based systems. Four measurements are made on each of four systems which differ by concentration of NaCl (table salt)

Concentration (%)

Enthalpy

5

1.62

1.60

1.62

1.66

10

2.69

2.66

2.72

2.73

20

3.56

3.45

3.65

3.52

30

3.35

3.18

3.40

3.06

c) What pairs of concentrations, if any, can you conclude to have differing enthalpies? Use alpha = 0.05.

Solution

In the previous question related to same I have performed wrong calculations for ANOVA:

To test the hypothesis:

H0: Enthalpy is same for all concentration.

against

H1: Enthalpy differes for atleast one concentration.

ANOVA result is

One-way ANOVA: A1, B2, C1, D1

Source DF     SS     MS      F      P
Factor   3 7.451 2.484 15.66 0.000
Error   12 1.903 0.159
Total   15 9.354

S = 0.3982   R-Sq = 79.65%   R-Sq(adj) = 74.57%

                          Individual 95% CIs For Mean Based on
                          Pooled StDev
Level N    Mean   StDev ---+---------+---------+---------+------
A1     4 1.6250 0.0252 (-----*-----)
B2     4 2.7000 0.0316                 (------*-----)
C1     4 3.5450 0.0835                             (------*-----)
D1     4 2.7425 0.7910                  (-----*-----)
                          ---+---------+---------+---------+------
                           1.40      2.10      2.80      3.50

Pooled StDev = 0.3982

This is MINITAb solution for ANOVA. AS p-value = 0< 0.05 we will reject the null hypothesis and conclude that enthalpy differes for atleast one concentration.

Now to test for for which pair of concentration have different enthalpy
All Pairwise Comparisons
A1 subtracted from:

     Lower Center   Upper -----+---------+---------+---------+----
B2 0.4615 1.0750 1.6885                      (-----*-----)
C1 1.3065 1.9200 2.5335                              (-----*-----)
D1 0.5040 1.1175 1.7310                      (-----*-----)
                            -----+---------+---------+---------+----
                              -1.0       0.0       1.0       2.0

In this case 5% (A1) is compared with 10%(B2), 20%(c1) and 30% (D1) as comparison conclusion are as follows:

As CI of difference between 5% and10 % concentration does not include 0 this implies that there is significant difference between enthalphy level for 5% and 10%.

CI of difference between 5% and 20 % concentration does not include 0 this implies that there is significant difference between enthalphy level for 5% and 20%.

CI of difference between 5% and 30% concentration does not include 0 this implies that there is significant difference between enthalphy level for 5% and 30%.

B2 subtracted from:

      Lower Center   Upper -----+---------+---------+---------+----
C1   0.2315 0.8450 1.4585                   (-----*------)
D1 -0.5710 0.0425 0.6560           (-----*------)
                             -----+---------+---------+---------+----
                               -1.0       0.0       1.0       2.0

As CI of difference between 10% and 20 % concentration does not include 0 this implies that there is significant difference between enthalphy level for 10% and 20%.

CI of difference between 10% and 30% concentration does not include 0 this implies that there is significant difference between enthalphy level for 10% and 30%.

C1 subtracted from:

      Lower   Center    Upper -----+---------+---------+---------+----
D1 -1.4160 -0.8025 -0.1890   (-----*-----)
                               -----+---------+---------+---------+----
                                 -1.0       0.0       1.0       2.0

CI of difference between 20% and 30% concentration include 0 this implies that there is no significant difference between enthalphy level for 20% and 30%.