What is the standard error of the mean in analytical chemistry?

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The discussion centers on understanding the standard error of the mean in analytical chemistry, particularly regarding probability calculations related to standard deviations. The first part of the problem, concerning a single determination being below the mean by more than two standard deviations, yields a probability of 2.28%. The more complex second part involves calculating the probability for the average of four determinations being further below the mean than two standard deviations, which is stated to be 0.0032%. The user expresses confusion about the derivation of this latter value despite exploring mathematical resources and textbooks. Clarification on these statistical concepts is sought to resolve the confusion.
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in my analytical chemistry review, this question cropped up. the first part was simple, but the second part has me confused to no end. I've tried playing around with the math as well as scouring my textbook for answers. help!

what is the probability that a single determination of a value is further below the mean than 2 standard deviations?

this part is the simpler part, given the table at the bottom of http://64.233.167.104/custom?q=cache:bdW_K0aRhzEJ:www.palgrave.com/business/taylor/taylor1/lecturers/lectures/handouts/hChap5.doc+chart+area+beneath+normal+curve+standard+deviations&hl=en&ct=clnk&cd=7&client=pub-8993703457585266" page.
the answer turns out to be 2.28%

the second part of the problem is: what is the probability that the average of 4 determinations is further below the mean than 2 standard deviations?

the answer was given to be 0.0032% but i cannot figure out where this value comes from.



ps. my apologies for not reading the sticky, I'll post this in the proper forums now.
 
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