Question

In problem 23P in Chapter 4 of Quantitative Chemical Analysis the answer says that the standard deviations for the two measurement methods are different. But if I apply the the F test I find that the standard deviations for the two measurement methods are not statistically different at the 95 % confidence level for the cases of both rainwater and drinking water. For rainwater the value of Fcalc = 2.56 and the F table gives 4.53 so the standard deviations are not statistically different. Likewise, for drinking water the value of Fcalc = 1.31 and the F table gives 6.39 so these standard deviations are also not statistically different. Why does the answer say they are different and hence use equations 4-9b and 4-10 b to calculate the t values and dof's instead of 4-9a and 4-10a? It appears that the website answer just compared the values of the standard deviations and found them different instead of using the F-test to determine if they were statistically different. Why?

Answer 1

There are two standard deviations, one for rainwater and one for drinking water.

To use the t-test for two independent samples there are two different case of calculating degrees of freedom. One is when the standard deviation of the two groups differ significantly and in the other hand, when the standard deviation of the two groups are same/equal. To test if the standard deviation of the two groups are different we will use the F-test which will calculate the ratio of two standard deviaton and will calculate the F-calculated value. Hence, there will be only one F-calculated value which is estimated using the ratio of two standard deviations.

Whereas, in the question there are two different F-values calculated for two groups in which the testing was done to test the hypothesis that the the standard deviation of the groups are different from 0.

But for t-test we should test the hypothesis that the standard deviation of two groups are same, ie., Null hypothesis: H0: s1 = s2.

where s1 is the standard deviation of rainwater and s2 is the standard deviation of drinking water.

In the question, below two hypohtesis were considered which has different meaning for it and it is not of interest for using t-test.

Null hypothesis: H01: s1 = 0

Null hypothesis: H02: s2 = 0

So, if F-calculated value is estimated using the ratio of the standard deviations, you may find the standard deviations of the two groups are different from each other or the standard deviations of the two groups are not equal.