As part of an analytical chemistry laboratory course, a student measured the Ca2
ID: 3268577 • Letter: A
Question
As part of an analytical chemistry laboratory course, a student measured the Ca2 content in two water samples, city-supplied drinking water and well-supplied drinking water, using two different analytical methods, flame atomic absorption spectrometry (FAAS) and EDTA complexometric titration. The results of this experiment are shown below as the mean Ca2 concentration (x) ± standard deviation (s) in units of parts per million (ppm). Each sample was measured five times (n = 5) by each method.
City-Supplied Drinking Water (X± s) 58.06 ± 0.72 ppm 58.81 ± 0.95 ppm I | Well-Supplied Drinking Water (X± Method FAAS EDTA Titration 67.48 ± 0.73 ppm 68.33 ± 0.96 ppm A) Method Comparison: For each drinking water sample (city and well), compare the Ca content measured with FAAS and EDTA titration. Calculate the t value (tal City. eale for each sample. Do the methods produce statistically different results at the 95% confidence level when measuring the Ca2 content of the two drinking water samples (Yes or No)? A list of Student's ell Number O Yes Number Yes O No values at several confidence levels can be found in the Student's t table cale B) Sample Comparison: For each method (FAAS and EDTA titration), compare the Ca2 content measured in the city-supplied and well-supplied drinking water samples. Calculate the t value (tcalc) for each method. Do the drinking water samples contain statistically different amounts of Ca at the 95% confidence level when measured by each of the methods (Yes or No) Number Yes O No FAAS calc Number O Yes 0 EDTA t calcExplanation / Answer
A Part (1) For FAAS Method
Let X = Amount of Ca2 in City-supplied drinking water
Y = Amount of Ca2 in well-supplied drinking water
We assume X ~ N(µ1, 12) and Y ~ N(µ2, 22) where 12 = 22 = 2 ,say and it is unknown.
We have a sample of n (5) observations on each of X and Y
Claim:
Drinking water samples contain statistically different amounts of Ca2
Hypotheses:
H0: µ1 = µ2 Vs HA: µ1 µ2
Test Statistic:
t = (Xbar - Ybar)/{s(2/n)}
where Xbar and Ybar are sample means of X and Y respectively and
s2 = (s12 + s22}/2, s1 and s2 being the respective sample standard deviations.
Given Data and Computations
s2 = (0.722 + 0.732}/2 = 0.52565
t = (58.06 – 67.45)/{0.52565 x (2/5)} = - 9.39/0.21026 = - 20.478
Distribution and Critical Value:
Under H0, t ~ t2n - 2.
Critical value = upper (/2) percent point of t-Distribution with degrees of freedom = 2n – 2
= t8, 0.025 = 2.306 read off from the given table against 90% [90% confidence => = 5% or 0.05 ]
Decision:
Since| tcal |= |calculated value of test statistic| = 20.478l > t8, 0.025 = 2.306, H0 is rejected.
Conclusion
There is sufficient evidence to suggest that the claim ‘Drinking water samples contain statistically different amounts of Ca2’ is valid.
DONE
A Part (2) For EDTA Titration Method
Let X = Amount of Ca2 in City-supplied drinking water
Y = Amount of Ca2 in well-supplied drinking water
We assume X ~ N(µ1, 12) and Y ~ N(µ2, 22) where 12 = 22 = 2 ,say and it is unknown.
We have a sample of n (5) observations on each of X and Y
Claim:
Drinking water samples contain statistically different amounts of Ca2
Hypotheses:
H0: µ1 = µ2 Vs HA: µ1 µ2
Test Statistic:
t = (Xbar - Ybar)/{s(2/n)}
where Xbar and Ybar are sample means of X and Y respectively and
s2 = (s12 + s22}/2, s1 and s2 being the respective sample standard deviations.
Given Data and Computations
s2 = (0.952 + 0.962}/2 = 0.91205
t = (58.81 – 68.33)/{0.91205 x (2/5)} = - 9.52/0.36482 = - 15.761
Distribution and Critical Value:
Under H0, t ~ t2n - 2.
Critical value = upper (/2) percent point of t-Distribution with degrees of freedom = 2n – 2
= t8, 0.025 = 2.306 read off from the given table against 90% [90% confidence => = 5% or 0.05 ]
Decision:
Since| tcal |= |calculated value of test statistic| = 15.76l > t8, 0.025 = 2.306, H0 is rejected.
Conclusion
There is sufficient evidence to suggest that the claim ‘Drinking water samples contain statistically different amounts of Ca2’ is valid.
DONE
B part is identical to the above except that the tcal values will be different.
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