Statistics Projects for High School Students

What is $1\mathrm{\%<!--\; \%\; -->}$ of 1 billion?

Finding the function of a parabolic curve between two tangents
Alright folks, first question here so let me make the situation and background clear.
I'm attempting to start studying for aerospace engineering, so I'm working on improving on my math skills as we speak, but couldn't help "jumping in" to some of the design and starting to look at it. I'm not adverse to algebra, although my calculus legs haven't been walked on in awhile, which is why I haven't been able to get farther myself on my own question.
Another side reason is that I'm attempting to get the gist of equation-based curves in CAD, so rocket engine nozzle curves are perfect for learning that...if I can figure out the equations!
If the format or my thoughts seem a bit off, I have an idea of the concepts in play here, my math skills have just atrophied a little too much for my own comfort.
What I'm working with is the G.V.R. Rao approximation of a bell nozzle curve; essentially,
$$f_c=\{\begin{array}{ll}0.382R_t& \text{For divergent throat curve}\\ f_p& \text{Main body}\end{array}$$
Where $f_p$ starts from a point with a tangent of angle ${\mathrm{\theta <!--\; \theta \; -->}}_n$ and ends at a point with a tangent of angle ${\mathrm{\theta <!--\; \theta \; -->}}_e$. $f_p$ also has to fit in a region equal to $L_f-<!--\; -\; -->0.382R_t$, where $L_f$ is the complete distance between the throat and exit plane, so the displacement in the x- or y-axis, depending on how you view the rocket (orientation-wise).
I do know how to differentiate the curve $f_p$ to get $f_p^{\prime }$ and then find the angle of the slope at a point, but this is backsolving from two slopes to find the region in between.
If it's any help, ideally I'd be constructing the nozzle in CAD vertically, that is, $y_n>y_e$
What I'm looking for is help toward the derivation of a formula that allows me to construct a curve that is smooth between the two points. One of the reasons I've had a hard time figuring out the exact parameters is because it feels a lot like curve-fitting, which I haven't had much experience with.
If anyone can help break it down for me, it'd be much appreciated but if the question turns out to be too vague, references to places where I can get the requisite learning would be also appreciated.

What reason is there to conjecture that every finite string is really in the decimal expansion of $\mathrm{\pi <!--\; \pi \; -->}$?
One of my students asked me this, and it occurred to me that I had never really questioned it.
Apparently, it is only conjectured but widely believed that the decimal expansion in base 10 of π contains all finite strings of the numerals 0 through 9.
Am I even accurate that the conjecture is widely accepted? Regardless, what is the rationale for this belief? Do any good heuristics exist? It seems perfectly logical (dare I say likely) that, just maybe, the string 2347529384759748975847523462346435664900060906, for example, never occurs. The conjecture seems absurdly strong, to me.
And just because a separate question would be ridiculous: does this conjecture extend to other famous transcendental numbers? Is it indeed conjectured that this is a property of transcendental numbers in general?

What Are R-Modules Used For?
Kind of a simple question, but what exactly are R-modules used for? Do they have any engineering applications?
EDIT:
I am a graduate student researcher in computer architecture, a subfield of computer engineering. Specifically, I do research on the best way to build future general purpose processors
One thing I am looking into is if it is possible to apply mathematics to improve the design of CPUs. That is, can we use concepts from mathematics to improve the execution of general purpose programs on hardware. CPUs are a massive engineering design problem, and where exactly we could improve the design by applying math isn't entirely clear.
What I don't have is a very deep mathematical background. I have taken an introductory abstract algebra course and one in coding theory. I've also read a number of coding theory papers...
I know that other electrical engineering subfields like communications and compressed sensing have successfully applied elements of linear algebra and abstract algebra and have gotten very good results.
The fact that this particular question spans both engineering and mathematics makes it both hard to formulate and to discuss with people. I'd be happy to talk about it in more detail, but I'm not entirely sure what the best forum would be for that.
At least for now, I figured a good place to start would be to see if other people have successfully used some of the more abstract math concepts in engineering systems. One of the few I am aware of are R-modules, so I figured I'd ask if anyone knows of some engineering uses of them...

What statistical test to use to compare effectiveness of drugs used to fight a disease
So let's say that I want to collect some data on a sample of people who has the same type of disease.
One group will take a drug A
Another one will take drug B
Third type will take both of them.
I want to perform a statistical test to see whether taking both of the drugs simultaneously results in getting back to health quicker.
Can anyone tell me what test would I use and what hypothesis would I state? Assumptions that I would make?
I need to plan a research project (Without actually doing it)

Find the points at which the line $ax+by+c=0$ crosses the $x$ and $y$-axes. Assume that $a\ne <!--\; \ne \; -->0$ and $b\ne <!--\; \ne \; -->0$.
solve the equation $ax+by+c=0$ for $y$:
$ax+by+c=0$
$ax+by=-<!--\; -\; -->c$
$by=-<!--\; -\; -->ax-<!--\; -\; -->c$
$y=\frac{-<!--\; -\; -->ax-<!--\; -\; -->c}{b}$
$\because <!--\; \because \; -->x=0$ at $y$-intercept,
$\therefore <!--\; \therefore \; -->y=\frac{-<!--\; -\; -->a(0)}{b}-<!--\; -\; -->\frac{c}{b}$
$y=-<!--\; -\; -->\frac{c}{b}$
The point at which the line crosses the y-axis is $(0,-<!--\; -\; -->\frac{c}{b})$
Now we solve the equation $ax+by+c=0$ for $x$:
$ax+by+c=0$
$ax+by=-<!--\; -\; -->c$
$ax=-<!--\; -\; -->by-<!--\; -\; -->c$
$x=\frac{-<!--\; -\; -->by-<!--\; -\; -->c}{a}$
$\because <!--\; \because \; -->y=0$ at $x$-intercept
$\therefore <!--\; \therefore \; -->x=\frac{-<!--\; -\; -->b(0)}{a}-<!--\; -\; -->\frac{c}{a}$
$x=-<!--\; -\; -->\frac{c}{a}$
The point at which the line crosses the $x$-axis is $(-<!--\; -\; -->\frac{c}{a},0)$

How can to prove the variance of residuals in simple linear regression?
$\mathrm{var}<!--\; \; -->(r_i)={\mathrm{\sigma <!--\; \sigma \; -->}}^2[1-<!--\; -\; -->\frac{1}{n}-<!--\; -\; -->{\displaystyle \frac{(x_i-<!--\; -\; -->\stackrel{¯<!--\; ¯\; -->}{x}{)}^2}{\underset{l=1}{\overset{n}{\sum <!--\; \sum \; -->}}(x_l-<!--\; -\; -->\stackrel{¯<!--\; ¯\; -->}{x})}}]$
my try:
using
$r_i=y_i-<!--\; -\; -->\stackrel{^<!--\; ^\; -->}{y_i}$
$\mathrm{var}<!--\; \; -->(r_i)=\mathrm{var}<!--\; \; -->(y_i-<!--\; -\; -->\stackrel{^<!--\; ^\; -->}{y_i})=\mathrm{var}<!--\; \; -->(y_i-<!--\; -\; -->\stackrel{¯<!--\; ¯\; -->}{y})+\mathrm{var}<!--\; \; -->(\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_1}(x_i-<!--\; -\; -->\stackrel{¯<!--\; ¯\; -->}{x}))-<!--\; -\; -->2\mathrm{Cov}<!--\; \; -->((y_i-<!--\; -\; -->\stackrel{¯<!--\; ¯\; -->}{y}),\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_1}(x_i-<!--\; -\; -->\stackrel{¯<!--\; ¯\; -->}{x}))$
How can I go further?

Studying math in the morning or in the evening has no effect on a student's performance.
I know this claim might be a little unconventional. Most examples I have read in books and online are that something either increases or decreases. However, for the research I am currently doing, what's relevant it's to see if there's no change.
For what I have read, the null hypotheses should always have an equality and the alternative hypotheses an inequality. However, in this case, I want precisely the opposite.
So, let's consider M as the test scores of people studying in the morning, and E the tests scores of people studying in the evening.
H0:M¯¯¯¯¯≠E¯¯¯¯
H1:M¯¯¯¯¯=E¯¯¯¯This formalization is what I though should be correct, since H1 is my claim (what I am trying to demonstrate) and H0 is the complement of that. However, it goes against the hypothesis formulation definition.
I think, in the end, what I am asking is how can one formalize hypotheses for an experiment that is designed to show no correlation between two levels of a factor?

What I have found is that I love theoretical discussions of mathematics and proving theorems, though the education system focuses on doing exercises and applications, and that is what grades are determined on. I have a hard time doing exercises mainly because I don't find them motivating, although, I admit, doing the exercises does cultivate one's intuition to a certain degree.
Up until now my main technique of learning was to take meticulous notes from textbooks(not in class as all they do is mundane exercises), think about those concepts and then practice exercises before exams, and it worked fine for discrete math, precalculus, and even the first course of calculus, which was mainly focused on derivatives. Further. I as able to maintain A's in all my classes with enough energy left over to even do very well in other classes.
However, this hasn't gone too well in the second course of calculus, which is mostly about integrals of various functions and their applications and techniques. The instructor usually does problems on the board step by step without any discussion for what we are doing. For example, both the fundamental theorems were introduced and given about 15 minutes of time to discuss. But, of course, this could be bad teaching style.
My question is this, should I learn math theoretically and understand the material well even though on tests this would mean getting about B's that I have gotten or should I only care about the grades and just do what I am told, like a human calculator, practice exercises and just focus on getting A's? What is the better for me in the long run if I want to get into a decent grad school? And have any of you ever had to decide between getting an A or spending more energy and time actually learning the theory, which is more satisfying? I could be looking at this completely wrong and perhaps there is a way to do both? Thank you and your input will be appreciated.

i'm essentially questioning in which there are a few sturdy applications to get a masters in arithmetic. whilst locating a list of these isn't so difficult, i am conscious that a number of them serve in particular to help the ones whose undergraduate math training is a bit susceptible to be a PhD candidate. i'm going to count on (perhaps wrongfully) that i am now not in that category. So my query is this: Assuming i am qualified to pursue a PhD in mathematics, but am barely uneasy about a 5 year commitment, which abbreviated graduate school alternatives are to be had to me?

Question: What is known about the role that binary representation of data plays in algorithmic complexity theory?
I define "analog representation of data" loosely as any method of representing data which is not a sequence of 1's and 0. A more precise definition would be nice to have obviously. The basic idea is that an analog representation of data contains additional information beyond what is contained in a binary representation.
A very trivial example of an "analog" representation of the natural numbers is to label all the primes $p_1,p_2,p_3\cdots <!--\; \cdots \; -->$ and then represent numbers by their prime factorization, instead of in binary form. It is obvious that with this representation of numbers, the problem of factoring large numbers is of polynomial complexity.
Less trivially, according the Shor's Alogirithm, if we represent numbers using an analog physical system (in this case, using quibits instead of bits), then it's possible to factor large numbers in polynomial time as well.
The natural conclusion to draw is that the choice to use binary or not for representing data likely plays an important role in determining if a given problem is in P or NP, and one would think that in order to prove P is not equal to NP it would be required to use the assumption that data is represented in a binary way at some point.

Getting ready for Calculus?
So I wanted to start a Masters software however they require that i've Calculus III. I need to take that path at the college, however I need to be prepared for it. As I take a look at Khan Academy and do some of the sports for pre-calc, I realise i am thus far out of faculty that I need to look at regions of math even previous to pre-calc. Is there an evaluation of some type on-line that could assist guide me to what I want to awareness on or have to I just work through Khan Academy material? Any thoughts?

Whats the difference between a one way anova and two way anova?
I know that in one way anova you compare the difference between two or more means and the same in two way, but I'm unclear as to how the use of categorical variables differs between them.
Any help would be appreciated

Is there any abstract theory of electrical networks?
Designing electrical networks is among the highly mathematical engineering disciplines, which uses a vast scope of techniques from Fourier analysis and complex function theory, to logic, combinatorics and topology. But, at least to me, with my minor knowledge of electrical engineering, it seems that at the end of the day, what is physically built out of these theories--I mean in a manufacturer laboratory-- is always a finite graphs with nodes labeled with "simple functions", in a way that the whole thing is again a function, with desired characteristics. But, from a mathematical point of view, it is customary to investigate such structured functions, in a categorical context and exploit the language and power of category theory, much like what programmers do.
Here, I do not dare to further these vague ideas and pose my question:
What is the right and fruitful mathematical foundation for the theory of electrical networks? Is there any purely axiomatic approach to the subject, accessible to a mathematics enthusiast with minor background in electrical engineering.

I am writing to have some advice about what Algebra I should study to start tackling Algebraic Geometry and (advanced) Algebraic Topology.
My background consists in Linear Algebra and a 'fundational' course covering the basics of Groups, Rings and Fields.So I think I can learn about this topics, having all the necessary prerequisites, but do not know which are the most relevant and fundamental topics to the stated area. Any advice and reference to some compact text apt to self study would be great!

I'm taking real analysis I, and my professor gives us a previous exam with solution each time before the midterm/exam. I would do the previous exam and check if I get it right. But it's obvious that my professor are not going to give us the same questions.
So, I also read and tried to apprehend the proofs which are given at the ebook. once in a while, I proved the theorems and corollaries by myself. besides, i might memorize the theorems and corollaries.
For guides like calculus, we have a few preferred steps to resolve a kind of question. So it is simpler to prepare for the exam. however for proof-based guides like real evaluation, algebra and topology, and many others, how can we recognise whether we're nicely-organized for the examination? How did you observe the materials and prepare for the examination whilst you have been an undergraduate pupil? checks are designed to test our know-how of the substances, but how can we take a look at ourselves first?

Find the sample variance and standard deviation
23,11,5,9,12

Compute using residuals the integral of the following function over the positively oriented circle $|z|=3$
$f(z)={\displaystyle \frac{e^{-<!--\; -\; -->z}}{z^2}}$
My solution: The only singular point of $f$ in $\left|z\right|\le <!--\; \le \; -->3$ is $z=0$ (double pole) and its remainder is therefore
${\mathrm{Res}}_{z=0}<!--\; \; -->f(z)={\displaystyle \underset{z\to <!--\; \to \; -->0}{lim}{\displaystyle \frac{1}{(2-<!--\; -\; -->1)!}{\left({\displaystyle \frac{e^{-<!--\; -\; -->z}{z}^2}{z^2}}\right)}^{\mathrm{\prime <!--\; \prime \; -->}}={\displaystyle \underset{z\to <!--\; \to \; -->0}{lim}-<!--\; -\; -->e^{-<!--\; -\; -->z}=-<!--\; -\; -->1}}}$
Consequently, ${\displaystyle {\int <!--\; \int \; -->}_{|z=3|}f(z)=2\mathrm{\pi <!--\; \pi \; -->}i{\mathrm{Res}}_{z=0}<!--\; \; -->f(z)=-<!--\; -\; -->2\mathrm{\pi <!--\; \pi \; -->}i.}$
this right?

Why is important to perform a hypothesis test about a standard deviation?

Mrs. Campbell said there would be 100 people at the concert. Only 27 showed up. What was the percent error?