Statistics Projects for High School Students

What are the practical uses of e ?
How can e be used for practical mathematics? This is for a presentation on (among other numbers) e, aimed at people between the ages of 10 and 15.
To clarify what I want:
Not wanted: $e^{i\mathrm{\pi <!--\; \pi \; -->}}+1=0$ is cool, but (as far as I know) it can't be used for practical applications outside a classroom.
What I do want: e I think is used in calculations regarding compound interest. I'd like a simple explanation of how it is used (or links to simple explanations), and more examples like this.

Use the following values to answer the items below:
4, 4, 4, 5, 5, 6, 6, 6, 6, 7, 8, 8, 9, 9, 10, 11
What is the percentile rank of X = 6.5?

What parts of a pure mathematics undergraduate curriculum have been discovered since 1964?
What parts of an undergraduate curriculum in pure mathematics have been discovered since, say, 1964? (I'm choosing this because it's 50 years ago). Pure mathematics textbooks from before 1964 seem to contain everything in pure maths that is taught to undergraduates nowadays.
I would like to disallow applications, so I want to exclude new discoveries in theoretical physics or computer science. For example I would class cryptography as an application. I'm much more interested in finding out what (if any) fundamental shifts there have been in pure mathematics at the undergraduate level.
One reason I am asking is my suspicion is that there is very little or nothing which mathematics undergraduates learn which has been discovered since the 1960s, or even possibly earlier. Am I wrong?

Reminder : Given a set S of n elements (we will use [n] in the following for simplicity), a Latin square L is a function $L:[n]\times <!--\; \times \; -->[n]\to <!--\; \to \; -->S$, i.e., an $n\times <!--\; \times \; -->n$ array with elements in S, such that each element of S appears exactly once in each row and each column. For example,
Latin square
1, 2, 3
3, 1, 2
3, 1, 2
Let $L_1$ and $L_2$ be two Latin squares over the ground sets $S_2$ respectively. They are called orthogonal if for every $(x_1,x_2)\in <!--\; \in \; -->S_1\times <!--\; \times \; -->S_2$ there exists a unique $(i,j)\in <!--\; \in \; -->[n]\times <!--\; \times \; -->[n]$ such that $L_2(i,j)=x_2$. For example, the following are two orthogonal Latin squares of order 3.
1, 2, 3 2, 3, 1
3, 1, 2 3, 1, 2
1, 2, 1 1, 2, 3
It is known that there at most n−1 mutually orthogonal Latin squares of order n, and that the bound is achieved if and only there exist an affine plane of order n.
Graph : I'm building a graph $G_n$ with vertex set the latin squares of order n and two vertices are adjacent iff the Latin squares are orthogonal.
I want to understand some properties of this graph. For simplicity I consider the squares up to permutation of [n], hence w.l.o.g. all my squares have for first line $\{1,2,\dots <!--\; \dots \; -->,n\}$. Indeed if I call $H_n$ the graph not up to permutations, then $H_n$ is the n! graph blowup of $G_n$, or using the Tensor product
$$H_n=G_n\times <!--\; \times \; -->K_{n!}$$
As I'm mainly interested in the chromatic number of my graph, and we know that $\mathrm{\chi <!--\; \chi \; -->}(H_n)\le <!--\; \le \; -->min\{\mathrm{\chi <!--\; \chi \; -->}(G_n);n!\}$, I will study only $G_n$
For instance $G_3=K_2$
I know that :
It's trivial that $G_n$ is not complete.
If there exist an affine plane of order n then $G_n$ contains $K_{n-<!--\; -\; -->1}$ as a subgraph, and $\mathrm{\chi <!--\; \chi \; -->}(G_n)\ge <!--\; \ge \; -->n-<!--\; -\; -->1$
I wonder the following :
What is the maximum degree of $G_n$? We know that we have at most n−1 mutually orthogonal latin squares, but to how many squares can one square be orthogonal (still up to permutation)?
Do we have any other info on the chromatic number, not coming from the property $\mathrm{\chi <!--\; \chi \; -->}(G_n)\le <!--\; \le \; -->\mathrm{\Delta <!--\; \Delta \; -->}+1$
Can $G_n$ contains an induce k-cycle with k>3 (i.e. chordless cycle)?
Can it be conjectured that
Conjecture : for any n, $G_n$ is the disjoint union of complete subgraphs (of different sizes).
Edit After some simple Brute force and some additional reading, I can tell that
$G_4$ is made of 2 disjoint $K_3$ and 18 isolated vertices, for a total of 24 Latin squares up to permutations.
$G_5$ is made of 36 disjoint $K_4$ and 1200 isolated vertices, for a total of 1344 Latin squares up to permutation.
The case n=6 would be the first interesting case, as there are no affine plance of order 6, hence we will find no $K_5$ in $G_6$. It is known since 1901 (from Tarry hand checking all Latin squares of order 6) that no two were mutually orthogonal. So $G_6$ is made of only isolated vertices.
It is also know that the case n=2 and n=6 are the only one with only isolated vertices. (see design theory by Beth, Jingnickel and Lenz)
From the article "Monogamous Latin Square by Danziger, Wanless and Webb, available on Wanless website here. The authors show that for all n>6, if n is not of the form 2p for a prime $p\ge <!--\; \ge \; -->11$, then there exists a latin square of order n that possesses an orthogonal mate but is not in any triple of Mutually Orthogonal Latin Squares. Therefore our graph $G_n$ will have some isolated $K_2$

Help me understand what is and isn't exponential growth.
If the price at which something grows is proportional to itself, then you definately could name it exponential increase. do not quote me on this, however I suppose it has something of the shape $y=e^x$
Now take for example this. if you want to calculate the extent of a sphere, the formula is $v=4/3\pi r^3$. The by-product with recognize to the radius is $dv/dr=4\pi r^2$ This suggests the rate of change of the volume with appreciate to the radius.
yet the extent of the field is proportional to the radius, which in flip defines the dimensions of the field. The rate at which the volume of the sector increases is proportional to itself in a manner. Intuitively, i might assume to discover someplace in there an exponential yet there isn't always. Why is that?

Finding a basis of an infinite-dimensional vector space?
the opposite day, my trainer become speakme limitless-dimensional vector spaces and headaches that arise when trying to find a foundation for the ones. He referred to that it is been demonstrated that a few (or all, do no longer pretty remember) infinite-dimensional vector areas have a foundation (the end result uses an Axiom of choice, if I recall efficiently), that is, an endless listing of linearly independent vectors, such that any detail within the area can be written as a finite linear aggregate of them. however, my instructor stated that honestly finding one is simply complicated, and i were given a experience that it changed into essentially not possible, which jogged my memory of Banach-Tarski paradox, where it is technically 'viable' to decompose the sector in a given paradoxical way, however this can not be truly exhibited. So my question is, is the basis state of affairs analogous to that, or is it surely viable to explicitly discover a basis for endless-dimensional vector areas?

Consider the two species competition model given by
$$\frac{da}{dt}=[{\lambda }_{1}a/(a+K1)]-<!--\; -\; -->r_{ab}\cdot <!--\; \cdot \; -->ab-<!--\; -\; -->da,(1)$$
$$\frac{db}{dt}=[{\lambda }_{2}b\ast <!--\; \ast \; -->(1-<!--\; -\; -->b/K2)]-<!--\; -\; -->r_{ba}\cdot <!--\; \cdot \; -->ab,t>0,(2)$$
for two interacting species denoted a=a(t) and b=b(t), with initial conditions a=a0 and b=b0 at t=0. Here $\mathrm{\lambda <!--\; \lambda \; -->},\mathrm{\lambda <!--\; \lambda \; -->}2,K_1,K_2,r_{ab},r_{ba}$ and d are all positive parameters. (a) Describe the biological meaning of each term in the two equations.
=> A series expansion of 1/(a+K1), gives
$\frac{1}{a+K1}\approx <!--\; \approx \; -->\frac{K1-<!--\; -\; -->a}{K{1}^2+O(a^2)}$
Now,
$\frac{da}{dt}=[\mathrm{\lambda <!--\; \lambda \; -->}1a\times <!--\; \times \; -->\frac{a+K1}{K{1}^2}]-<!--\; -\; -->r_{ab}ab-<!--\; -\; -->da,$
$\mathrm{\lambda <!--\; \lambda \; -->}$1 a represents the exponential growth of population
da represents the exponential decay of population
$\mathrm{\lambda <!--\; \lambda \; -->}$1 is the growth rate
d is the decay rate
what does r_(ab) ab represent?
The first term of RHS equation $1:[\mathrm{\lambda <!--\; \lambda \; -->}1a\times <!--\; \times \; -->\frac{a+K1}{K{1}^2}]$ represents logistic growth at a rate $\mathrm{\lambda <!--\; \lambda \; -->}$1 with carrying capacity K1.
$\frac{db}{dt}=[\mathrm{\lambda <!--\; \lambda \; -->}2b\times <!--\; \times \; -->\frac{1-<!--\; -\; -->b}{K2}]-<!--\; -\; -->r_{ba}ab,$
$\mathrm{\lambda <!--\; \lambda \; -->}$2 b represents the exponential growth of population
$\mathrm{\lambda <!--\; \lambda \; -->}$2 is the growth rate
what does r_(ba) ab represent?
The first term of RHS equation $2:[\mathrm{\lambda <!--\; \lambda \; -->}2b\times <!--\; \times \; -->1-<!--\; -\; -->bK2]$ represents logistic growth at a rate $\mathrm{\lambda <!--\; \lambda \; -->}$2 with carrying capacity K2.

In a linear model, we defined residuals as:
$e=y-<!--\; -\; -->\stackrel{^<!--\; ^\; -->}{y}=(I-<!--\; -\; -->H)y$ where $H$ is the hat matrix $X(X^{T}X{)}^{-<!--\; -\; -->1}{X}^T$
and we defined standardized residuals as:
$r_i=\frac{e_i}{s\sqrt{1-<!--\; -\; -->h_{ii}}}$, $i=1,...,n$
where $s^2$ is the usual estimate of ${\mathrm{\sigma <!--\; \sigma \; -->}}^2$, $var(e_i)={\mathrm{\sigma <!--\; \sigma \; -->}}^2{h}_{ii}$, and $h_{ii}$ is the diagonal entry of $H$ at the $i^{th}$ row and $i^{th}$ column
Why $r_i$ and $e_i$ are functions of $h_{ii}$ rather than the whole row $h_i$?

How one can calculate the sample size in any random sampling? Is it varies with sampling method or it is fixed for all methods? Explain it.

Method for finding efficient algorithms?
TL;DR
What can you recommend to get better at finding efficient solutions to math problems?
Background
The first challenge on Project Euler says:
Find the sum of all the multiples of 3 or 5 below 1000.
The first, and only solution that I could think of was the brute force way:
target = 999
sum = 0
for i = 1 to target do
if (i mod 3 = 0) or (i mod 5 = 0) then sum := sum + i
output sum
This does give me the correct result, but it becomes exponentially slower the bigger the target is. So then I saw this solution:
target=999
Function SumDivisibleBy(n)
p = target div n
return n * (p * (p + 1)) div 2
EndFunction
Output SumDivisibleBy(3) + SumDivisibleBy(5) - SumDivisibleBy(15)
I don't have trouble understanding how this math works, and upon seeing it I feel as though I could have realised that myself. The problem is just that I never do. I always end up with some exponential, brute force like solution.
Obviously there is a huge difference between understanding a presented solution, and actually realising that solution yourself. And I'm not asking how to be Euler himself.
What I do ask tho is, are there methods and or steps, you can apply to solve math problems to find the best (or at least a good) solution?
If yes, can you guys recommend any books/videos/lectures that teach these methods? And what do you do yourself when attempting to find such solutions?

What happens to the sample standard deviation when the sample size is increased?

I am carrying out a category on discrete arithmetic and i'm inquisitive about skipping my faculties transition courses so that you can take a rigorous theory path next semester (topology, analysis, abstract algebra). What are a few properly transition books for me to examine that provide troubles and a few solutions so i will monitor my development, as well as being very , almost laboriously, particular in every step of evidence such as theorem packages. as an instance, i have observed abbots expertise analysis to be pretty cogent but Laczkovich Conjecture and evidence to be lacking some data important for me to understand some proofs as much as I would love. thank you for any assist

Given below are the annual number of divorce certificates registered in Oman by year between 2010 and 2019 as published by the NCSI. Find the percentile rank of 3663 divorces.
$\begin{array}{|cc|}\hline \text{Year}& \text{Divorces}\\ 2010& 2736\\ 2011& 3805\\ 2012& 3570\\ 2013& 3550\\ 2014& 3622\\ 2015& 3619\\ 2016& 3736\\ 2017& 3867\\ 2018& 3663\\ 2019& 3728\\ \hline\end{array}$

Explain how to find the equation of a line given the point (5,3) and it has a y-intercept of 13.

what is the best book for Pre-Calculus?
i've overlooked pre-calculus know-how in my school however i used to be excellent at maths, and now i am a pc science scholar, i am feeling terrible being bad in maths, so i'm seeking out the quality Pre-Calculus e-book, i love maths, i need the right nicely of precalculus books.

Given a set of data with $11$ observations of two variables (response and predictor), I've been asked to "calculate the fitted values ${\stackrel{^<!--\; ^\; -->}{y}}_i=\stackrel{^<!--\; ^\; -->}{\alpha }+\stackrel{^<!--\; ^\; -->}{\beta }{x}_i^{\prime }$ and residuals $e_i=y_i-{\stackrel{^<!--\; ^\; -->}{y}}_i$ by hand".
What is the question asking me to do here? I have thus far estimated the regression line for the data in the form ${\stackrel{^<!--\; ^\; -->}{y}}_i=\stackrel{^<!--\; ^\; -->}{\alpha }+\stackrel{^<!--\; ^\; -->}{\beta }{x}_i^{\prime }$ by calculating the coefficients $\alpha \mathrm{\&<!--\; \&\; -->}\beta ,,\; but\; this\; doesn\text{'}t\; answer\; the\; original\; question\; alone.\; Where\; do\; I\; go\; from\; here?$

Smoothest function which passes through given points?
I am trying to interpolate/extrapolate on the basis of a known collection of (finitely many) points. I'm wondering if there is a way to formalize this intuitive notion: find a 'smoothest' function which passes through each of the points. Of course I general the function would not have a nice form.
The idea is similar to that of Bézier curves (ad would be closer still, if I used a parametric curve rather than a function) and essentially opposite to the Lagrange interpolating polynomial, where fitting more than a few points usually produces wild oscillations.
Any idea how to make an idea like this work?

What is the variance of {-4, 5, 8 ,-1, 0 ,4 ,-12, 4}?

On a placement examination, Fatimah scored lower than 420 of the 980 students who took the exam. Find the percentile rounded to the nearest percent for Fatimah’s score. Show your solution.

What is the formal or general term for the y-intercept? Is there a general term for the value at which a function intercepts the vertical axis, in the Cartesian plane?