2nd part of question: part of an analytical chemistry laboratory course, a stude
ID: 3353433 • Letter: 2
Question
2nd part of question:
part of an analytical chemistry laboratory course, a student measured the Ca 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 content in two water parts per million (ppm). Each sample was measured five times (n 5) by each method City-Supplied Drinking Water(x± s) 58.26 ± 0.71 ppm 59.01 ± 0.98 ppm Method FAAS EDTA Titration | Well-Supplied Drinking Water (x± s) 67.41 ± 0.73 ppm 68.26 ± 1.0 ppm A) Method Comparison: For each drinking water sample (city and well), compare the Ca2t content measured with FAAS and Number O Yes O No EDTA titration. Calculate the t value (tcalc) for each sample. Do the methods produce statistically different results at the 95% confidence level when measuring the City: teal,- O Yes O No Number Ca content of the two drinking water samples (Yes or No)? A list of Študent'st Well values at several confidence levels can be found in the Student's t table calcExplanation / Answer
Solution:-
1)
a)
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: 1 - 2 = 0
Alternative hypothesis: 1 - 2 0
Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.
Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a two-sample t-test of the null hypothesis.
Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).
SE = sqrt[(s12/n1) + (s22/n2)]
SE = 0.5412
DF = 8
t = [ (x1 - x2) - d ] / SE
t = - 1.39
where s1 is the standard deviation of sample 1, s2 is the standard deviation of sample 2, n1 is the size of sample 1, n2 is the size of sample 2, x1 is the mean of sample 1, x2 is the mean of sample 2, d is the hypothesized difference between the population means, and SE is the standard error.
Since we have a two-tailed test, the P-value is the probability that a t statistic having 8 degrees of freedom is more extreme than -1.39; that is, less than -1.39 or greater than 1.39.
Thus, the P-value = 0.202
Interpret results. Since the P-value (0.202) is more than the significance level (0.05), we have to accept the null hypothesis.
2)
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: 1 - 2 = 0
Alternative hypothesis: 1 - 2 0
Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.
Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a two-sample t-test of the null hypothesis.
Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).
SE = sqrt[(s12/n1) + (s22/n2)]
SE = 0.4523
DF = 8
t = [ (x1 - x2) - d ] / SE
t = - 20.23
where s1 is the standard deviation of sample 1, s2 is the standard deviation of sample 2, n1 is the size of sample 1, n2 is the size of sample 2, x1 is the mean of sample 1, x2 is the mean of sample 2, d is the hypothesized difference between the population means, and SE is the standard error.
Since we have a two-tailed test, the P-value is the probability that a t statistic having 8 degrees of freedom is more extreme than -20.2; that is, less than -20.2 or greater than 20.2
Thus, the P-value = 0.0001
Interpret results. Since the P-value (less than 0) is less than the significance level (0.05), we cannot accept the null hypothesis.
b)
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: 1 - 2 = 0
Alternative hypothesis: 1 - 2 0
Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.
Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a two-sample t-test of the null hypothesis.
Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).
SE = sqrt[(s12/n1) + (s22/n2)]
SE = 0.5412
DF = 8
t = [ (x1 - x2) - d ] / SE
t = - 1.39
where s1 is the standard deviation of sample 1, s2 is the standard deviation of sample 2, n1 is the size of sample 1, n2 is the size of sample 2, x1 is the mean of sample 1, x2 is the mean of sample 2, d is the hypothesized difference between the population means, and SE is the standard error.
Since we have a two-tailed test, the P-value is the probability that a t statistic having 8 degrees of freedom is more extreme than -1.39; that is, less than -1.39 or greater than 1.39.
Thus, the P-value = 0.202
Interpret results. Since the P-value (0.202) is more than the significance level (0.05), we have to accept the null hypothesis.
2)
State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.
Null hypothesis: 1 - 2 = 0
Alternative hypothesis: 1 - 2 0
Note that these hypotheses constitute a two-tailed test. The null hypothesis will be rejected if the difference between sample means is too big or if it is too small.
Formulate an analysis plan. For this analysis, the significance level is 0.05. Using sample data, we will conduct a two-sample t-test of the null hypothesis.
Analyze sample data. Using sample data, we compute the standard error (SE), degrees of freedom (DF), and the t statistic test statistic (t).
SE = sqrt[(s12/n1) + (s22/n2)]
SE = 0.5537
DF = 8
t = [ (x1 - x2) - d ] / SE
t = - 1.54
where s1 is the standard deviation of sample 1, s2 is the standard deviation of sample 2, n1 is the size of sample 1, n2 is the size of sample 2, x1 is the mean of sample 1, x2 is the mean of sample 2, d is the hypothesized difference between the population means, and SE is the standard error.
Since we have a two-tailed test, the P-value is the probability that a t statistic having 8 degrees of freedom is more extreme than -1.54; that is, less than -1.54 or greater than 1.54
Thus, the P-value = 0.162
Interpret results. Since the P-value (0.162) is greater than the significance level (0.05), we have to accept the null hypothesis.
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