In article <[EMAIL PROTECTED]>,
Mike Franke <[EMAIL PROTECTED]> wrote:
>I teach high school level AP Statistics, and I'm working a problem in
>AMSCO's "AP Statistics" study guide, by James Bohan, for my students.
>It's problem number 4 in chapter 6.4 (page 164), entitled "Sampling
>Distributions of a Difference of Two Proportions." It's multiple
>choice. The problem is stated as follows:
>
>"A school district had anticipated that the percentages of boys and
>girls who planned no further education would be the same,
>approximately 44% for all of the students. Two independent random
>samples of the seniors at a high school are taken; the first was a
>sample of 10 boys and the second ws a sample of 25 girls. The boys'
>sample indicated that 50% of them planned no further education after
>graduation, while the gurls' sample indicated that only 40% of them
>planned no further education after graduation. Which of the following
>is valid for this information?"
> A) The sampling distribution is approximately normal with mean 0
> and approximate standard deviation .1857.
> B) The sampling distribution is approximately normal with mean .1
> and approximate standard deviation .0345
> C) No conclusion can be drawn regarding the sampling distribution
> since the samples are taken from the same population.
> D) No conclusion can be drawn regarding the sampling distribution
> since the sample size of the boys' sample is too small
> E) None of these statements is valid.
This is a terrible question. First of all, what quantity are we
supposed to find "the sampling distribution" of? I think the only
clue is the title of the question, from which I conclude that what is
referred to is the sampling distribution of the difference in the
observed proportions of boys and girls planning further education.
Second, what data distribution is to be used in finding this sampling
distribution? The only thing that seems to make sense, given that the
answer is said to be (A) is that they are envisioning a test of the
null hypothesis that both proportions are 44%. This is a stupid null
hypothesis, unfortunately typical of examples in textbooks. Nobody
would really have any good reason to test such a null hypothesis.
However, under this null hypothesis, the sampling distribution of the
difference in proportions does indeed have mean zero and standard
deviation of
Sqrt[0.44(1-0.44)/10 + 0.44(1-0.44)/25] = 0.1857
Whether the distribution is "approximately normal" or not depends
entirely on your personal conception of "approximately".
Radford
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Radford M. Neal [EMAIL PROTECTED]
Dept. of Statistics and Dept. of Computer Science [EMAIL PROTECTED]
University of Toronto http://www.cs.utoronto.ca/~radford
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