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Sir Col
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impudens simia et macrologus profundus fabulae
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Exam Sham?
« on: Nov 12th, 2003, 11:55am » |
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A rather 'interesting' question featured recently on a GCSE Mathematics statistics module (14-16 year olds in England).
(only the essence of the question is presented to avoid publication copyrights)
A class consists of 10 boys and 18 girls. A teacher wishes to take a stratified sample of four pupils. Calculate the number of boys and the number of girls that should be chosen.
We proceed by finding 10/28 and 18/28 of 4:
10/28 x 4 ~= 1.43
18/28 x 4 ~= 2.57
So how many of each do we take?
It seems there are two possible answers:
(i) Round and take 1 boy and 3 girl. However, this is in the ratio 1:3 and the original distribution was not even in the ratio 1:2.
(ii) Take a closer approximation to the ratios and select 2 boys and 2 girls, but then we're ignoring the estimated proportions.
The (published) solution in the mark scheme gave the answer as 1:3.
I personally think that it is a very bad question, and I believe that the answer should be 2:2. My understanding of the method of taking statified samples is to first estimate by finding proportions, then modify groups to ensure that the total matches the desired sample size. This is done by increasing/decreasing groups that most/least 'deserve' it. Ultimately, the final values should reflect the original distribution as closely as possible.
I'd be pleased to hear any thoughts on this.
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As there is no "good" answer, I think the test answer was as close as possible. The boys comprise 35.7% of the total. You have a choice of increasing that percentage 14.3% to a 2:2 ratio or decreasing it 10.7% to a 1:3 ratio. The "correct" answer is the closer of the two.
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Eigenray
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Re: Exam Sham?
« Reply #2 on: Nov 12th, 2003, 7:27pm » |
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This is known as apportionment, the same problem that comes up when distributing, say, seats in the House of Representatives here in the States: each state receives an integral number of seats in direct proportion to its population.
Various methods and associated paradoxes are described here.
Unfortunately, the simplest method has several flaws: for example, increasing the total number of seats, or one state's population, can cause it to actually lose seats.
Hamilton's, Jefferson's, Webster's, and the Huntington-Hill method all give 1 boy and 3 girls, while Adams' gives 2 boys and 2 girls.
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Sir Col
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impudens simia et macrologus profundus fabulae
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Re: Exam Sham?
« Reply #3 on: Nov 13th, 2003, 12:37am » |
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Thank you both very much for your feedback. That link, Eigenray, was most interesting and very helpful. Cheers!
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