Wikipedia:Reference desk/Archives/Mathematics/2012 August 25

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August 25

Reliability, validity, but what is more specific

I got interested in the impact that chance, contingent factors, etc. can have on student grades. I reduced my doubts to limit cases in which either an impossible question is added to an exam, or a giveaway question is added. I was surprised to discover that for good students a free question does not make up for an impossible question.

I then made a more thorough study to try to discover how many questions would needed to reduce the effect of the impossible or give-away question to having no important influence. Following that I devised a way of computing how "luck of the draw" would influence a student's grades if the sample size was very small. The method I developed is a simple application of probability theory, but as I have done it the process involves a great deal of computation.

Working the whole thing out would probably not be as much work as calculating a table of trig functions, but I am hopeful that somebody has already done the work or attacked the problem in some mathematically more efficient way. I have posted an explanation in more detail, and a spreadsheet based on a limited sample size used to evaluate spelling competence for a year's work, i.e., a vocabulary list of several hundred words.

http://www.china-learn.info/MathConcepts.xhtml

Can anybody point me at other work on this level? I am aware that work has been done on the subjects of "reliability" and "validity," but the math seems much more intense, and the preparatory work would be much more arduous. I am looking for materials that could easily be understood by secondary school teachers who may think that ten spelling words on a final exam will give all students fair grades.

Thanks.P0M (talk) 03:51, 25 August 2012 (UTC)

does requiring exactly one digit reduce or increase entropy?

Assuming an 8-character password that is either

1) all lower-case letters (e.g. fjwkdosj, or jefokds)

2) all lower-case letters, but one single digit in one place (e.g. fjds4sds)

which ruile has higher entropy? This is not homework. --80.99.254.208 (talk) 07:24, 25 August 2012 (UTC)

Looking at the number of possibilities, it's 26⁸ in the first case and 26⁷(10)(8) in the later case, assuming that the digit can be in any position, which means a tad more than 3 times as many possibilities, using a single digit. See if you can figure out entropy from there. StuRat (talk) 08:42, 25 August 2012 (UTC)

Dirichlet kernel

Show that ${\frac {1}{2}}\int _{0}^{x}\left({\frac {\sin((N+1/2)t)}{\sin(t/2)}}-1\right)dt$ is uniformly bounded in $N$ and $x\in [-\pi ,\pi ]$ using the fact that $\int _{0}^{\infty }{\frac {\sin t}{t}}dt<\infty$.

Obviously the strategy is to manipulate the first expression into $\int _{0}^{\infty }{\frac {\sin t}{t}}dt$ or something similar.

So, ${\frac {1}{2}}\int _{0}^{x}\left({\frac {\sin(Nt)\cos(t/2)+\cos(Nt)\sin(t/2)}{\sin(t/2)}}-1\right)dt$

$={\frac {1}{2}}\int _{0}^{x}\left({\frac {\sin(Nt)\cos(t/2)}{\sin(t/2)}}+\cos(Nt)-1\right)dt$

$={\frac {1}{2}}\int _{0}^{x}\left({\frac {\sin(Nt)\cos(t/2)}{\sin(t/2)}}+\cos(Nt)-1\right)dt$

Widener (talk) 13:47, 25 August 2012 (UTC)
$\int _{0}^{x}(\cos(Nt)-1)dt$ is uniformly bounded in N and $x\in [-\pi ,\pi ]$ at least. Widener (talk) 13:50, 25 August 2012 (UTC)

That isn't going to help. The first thing you need to observe is that $\int _{0}^{\infty }\sin t{\frac {dt}{t}}=\int _{0}^{\infty }\sin(N+1/2)t\,{\frac {dt}{t}}$. Sławomir Biały (talk) 14:54, 25 August 2012 (UTC)

How are they equal? Widener (talk) 15:39, 25 August 2012 (UTC)

dt/t is the multiplicative Haar measure for the group $\mathbb {R} _{+}$ (or just do a substitution). Sławomir Biały (talk) 15:55, 25 August 2012 (UTC)

Manifolds and implicit function theorem

Am I correct in saying that a surface is a manifold if, at every point on the surface, there exists a coordinate that can locally be written as a function of the others? Widener (talk) 14:03, 25 August 2012 (UTC)

I assume you mean a smooth manifold and smooth function. If so, the answer is "yes" (by the implicit function theorem). Sławomir Biały (talk) 14:55, 25 August 2012 (UTC)

I don't understand... the Euclidean plane is a smooth manifold, but its coordinates aren't locally degenerate. --Tango (talk) 15:01, 25 August 2012 (UTC)

Here I'm assuming that the question is specifically about surfaces in $\mathbb {R} ^{3}$, in which case the Euclidean plane is just $z=0$. (Of course, there are surfaces that cannot be realized as manifolds in $\mathbb {R} ^{3}$, but I don't think that's what the question was about.) Sławomir Biały (talk) 15:18, 25 August 2012 (UTC)

Ah, I didn't realise we were talking about an embedding of a surface. The question and answer make far more sense now! Thanks. --Tango (talk) 15:20, 25 August 2012 (UTC)