Enthalpy is a measure of total energy in a thermodynamic system. The following t
ID: 3223011 • Letter: E
Question
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
a) Construct an ANOVA table for this experiment
b) Is it plausible that the enthalpy is the same at all concentrations? Explain.
c) What pairs of concentrations, if any, can you conclude to have differing enthalpies? Use alpha = 0.05.
d) Compute the quantity s = (MSE)0.5, the estimate of the standard deviation sigma
e) Assuming s to be the error standard deviation, find the sample size necessary in each treatment to provide a power of 0.80 to detect a maximum difference of 0.1 in the treatment means at the 5% level.
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
Explanation / Answer
a) Null Hypothesis H0:mew1=mew2=mew3=mew4 i.e there is no significance difference between the mean plausible that the enthalpy is the same at all concentrations
Alternative Hypothesis H1: atleast two are not equal (two-tailed test)
Descriptives
Treatment
95% Confidence Interval for Mean
N Mean Std. Deviation Std. Error
Concentration (%) 4 1.6250 .02517 .01258
Concentration (%) 4 2.7000 .03162 .01581
Concentration (%) 4 3.5450 .08347 .04173
Concentration (%) 4 3.2475 .15650 .07825
Total 16 2.7794 .76053 .19013
ANOVA One-way classification table
Treatment
Sum of Squares df Mean Square F Sig.
Between Groups 8.577 3 2.859 345.578 .000
Within Groups .099 12 .008
Total 8.676 15
from F tables the critical value of F at 5% level with (3,12) df is 3.885
therefore calculated value > table value(345.578>3.885)
b) therefore H0 is rejected i.e we may conclude that there is significance difference between the mean plausible that the enthalpy is the not same at all concentrations.
c) Multiple Comparisons
Treatment
Tukey HSD
(I) Factors (J) Factors Mean Difference (I-J) Std. Error Sig.
Concentration (%) Concentration (%) -1.07500* .06432 .000
Concentration (%) -1.92000* .06432 .000
Concentration (%) -1.62250* .06432 .000
Concentration (%) Concentration (%) 1.07500* .06432 .000
Concentration (%) -.84500* .06432 .000
Concentration (%) -.54750* .06432 .000
Concentration (%) Concentration (%) 1.92000* .06432 .000
Concentration (%) .84500* .06432 .000
Concentration (%) .29750* .06432 .003
Concentration (%) Concentration (%) 1.62250* .06432 .000
Concentration (%) .54750* .06432 .000
Concentration (%) -.29750* .06432 .003
*. The mean difference is significant at the 0.05 level.
from the above table all pairs of concentrations are significant at 0.05 level
d) the mean square of standard error is 0.19013 from the descrptive table.
e) here s= 0.19013, maximum error = 0.1= E, critical value of Z at 5% level = 1.96
the sample size (n) = (Zalfa*sigma/E)2 = (1.96*0.19013/0.1)2 = 13.88~14
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