Statistics Projects for High School Students

Should I go back and start with a more "proof" based approach?
I should go to a book like the one by Spivak which is entirely different from the book used for my course, and learn or in a way re-learn it the way it's presented in that book?
Would a more proof based approach help me in this understanding? Will I always lack some aspect of understanding if I don't know how to prove these problems?
To quote one of the comments on this question, the question can also be put
"will studying calculus in a proof based manner help in understanding the techniques I've already learned"

How to find the equation for a tangent line with a given y intercept.
An equation for a circle, $y^2+x^2={3959}^2$ and I have the y intercept for a tangent line $y=mx+3965$.

When is a definition via properties considered valid?
How one would define the validity of a definition of an object by its properties?
Little background: I'm trying to implement a kind of framework of mathematics in which the user is not restricted to any formal system, but can define anything at will just like in english but imposing logical constraints on the steps possible during a proof preventing mistakes. This obviously requires to prove that the definition is "valid".
Example: If we consider G to be a group and + its group operation, we can define $\mathrm{\forall <!--\; \forall \; -->}x\in <!--\; \in \; -->G:x+0=0+x=x$, we can prove that two sets with such property are equal $0_1=0_1+0_2=0_2$ so the definition is solid. Here equality is by definition $x=y:⟺<!--\; ⟺\; -->(\mathrm{\forall <!--\; \forall \; -->}z:(x\in <!--\; \in \; -->z⟺<!--\; ⟺\; -->y\in <!--\; \in \; -->z)\wedge <!--\; \wedge \; -->\mathrm{\forall <!--\; \forall \; -->}w:(w\in <!--\; \in \; -->x⟺<!--\; ⟺\; -->w\in <!--\; \in \; -->y)$, so basically in any formula of set theory we can freely replace x with y and vice versa. But what if we don't want to be restricted to set theory? Then what seems reasonable is to require for every definition a list of formulas(not necessarily of set theory) which constitutes the notion of "equality" and then proving that in these formulas they are indeed interchangeable. This issue does not arise when considering objects defined through expressions $3:=2+1$as they are just directly replaceable short-hand notations.

I am studying computer aided geometry and I have a background in mathematics. For me a (real) projective transformation is a map $f:{\mathbb{R}\mathbb{P}}^n\to <!--\; \to \; -->{\mathbb{R}\mathbb{P}}^n$ induced by a linear isomorphism $F:{\mathbb{R}}^{n+1}\to <!--\; \to \; -->{\mathbb{R}}^{n+1}$ since being injective it maps lines to lines (m-subspaces to m-subspaces). In the context of graphic design they usually never propey define projective maps, they usually project something onto a plane or use the following construction:
Given $(x,y,z)\in <!--\; \in \; -->{\mathbb{R}}^3$ we consider the point $(x,y,z,1)\in <!--\; \in \; -->{\mathbb{R}}^4$ (i.e. a copy of ${\mathbb{R}}^3$ on the affine hyperplane $\{w=1\}$, then we apply a bijective linear map on ${\mathbb{R}}^4$ and project the image back on the hyperplane (which causes some trouble if the image has forth coordinate equal to zero). This process on ${\mathbb{R}}^3$ is called a projective map. What is the link with my definition? I am sure it might involve considering homogeneous coordinates $(x,y,z)\to <!--\; \to \; -->[x:y:z:1]$, anyhow I don't get why the confused notation, they seem different things.

I am a student of Pure Mathematics and also interested in programming .I have learnt C++,SAGE .
Recently I have started learning "Cryptography" .But there are many definitions involved here like polynomial time algorithm,time complexity etc.
My question is it all right for a student in Pure Mathematics to study Cryptography or as time progresses I will eventually fall out of place and lose interest in this subject.
Is Cryptography more suitable for computer science graduates or it does not matter which background a student is from to study this?
Please share your thoughts here as i am still in my early days and may help to change the subject if necessary before it is too late

Checking answer Supposed that your Math test score, 60, has a percentile rank of 30 %; your English test score, 80, has a percentile rank of 70 %; your Sociology test score, 90, has a percentile rank of 50 %. For which test, the highest grade is expected if the grade is based on relative standing of your score compared to others’ scores? Math English Sociology Can not be determined.

When does the standard error of the mean decrease?

i'm completing my undergraduate diploma and one aspect i've noticed is how little weight has been placed upon the potential to examine proofs, in basically all of my math guides. In first year calculus you are shown the proofs for matters like the limit of sin(x)/x at 0, but in my experience there's no incentive to be able to apprehend them.
This pattern persisted even in more advanced undergraduate publications on foundations and real analysis. As one instance, the professor spent a whole lecture proving the schroeder-bernstein theorem, and only a few students made an effort to recognize it (they surely were not influenced to do so thru grades). typically speaking, my classes have followed a format in which the professor will show theorems for a giant portion of the lecture time but assessments are designed with packages and proof-writing in mind and truely maximum proofs carried out by using the professor are far too hard for a pupil to recreate independently, so there is no incentive to research the details of the greater complex proofs.
This seems unusual to me, thinking about the format of most guides requires you to apprehend the arguments backing up a specific proposition. is that this genuine of maximum university applications? need to a extra emphasis be positioned upon studying how to read complex proofs?

What kind of mathematical "discoveries" have enabled mankind to build modern computers?
After studying the very thrilling Examples of mathematical discoveries which had been saved as a mystery I got here to think of something: maximum math discoveries appear to had been made centuries ago, and with Pascal and Leibniz we already had machines that would add and multiply (a few don't forget them as the first computer systems). Of direction modern computer systems needed energy and electric signals that can be transformed to at least one's and zero's, but those we've since the 19th century. First computer systems were also built with transistors, however we've the ones since the 1940's.
current computers are, at their center, not so extraordinary from the primary computer systems that followed the Von Neumann structure (with its arithmetic-logic unit). this is: they understand how move information from one place to any other, the way to upload and a way to multiply, in addition to perform logical operations (and, or, xor, now not), and from that they get all of the other operations (some can substract, divide and do some different stuff, however it is all very primitive in that sense). Of path, current computers paintings at a miles better level than their predecessors: while inside the beginning programming needed to be accomplished the use of commands composed of zero's and 1's or, in the event that they were lucky, the usage of meeting language, now we have high stage programming languages that (to say it kind of) enclose a group of these single commands into one high stage command.
And when I see pretty graphics in video-games, programs like photoshop, 3D rendering, CAD programs and such, I always think of all the calculus stuff I learned and how it must be applied to achieve those wonderful results.
And this is where my question arises: all of this mathematical knowledge has been available to us for way longer than computers existed. So, are there any modern mathematical discoveries that enabled the giant leap we took from the first computers we had in the 40's to what we have now? Or maybe old math started being applied in a different way at some point in time?

How are z-scores found for normal distributions where $$\mathrm{\mu <!--\; \mu \; -->}\ne <!--\; \ne \; -->0$$ or $$\mathrm{\sigma <!--\; \sigma \; -->}\ne <!--\; \ne \; -->1$$?

What is the variance of {-7, 42, -5, 1, 6, -3, 4, -2, 17, 53, -2, 4, 7, 88}?

I have heteroskedastic data of unequal sample sizes and would like to run a two way welch ANOVA.
1.) Is this appropriate? Why or why not?
2.) How do you do this in r?
3.) What are other ways of dealing with this situation?

Having run a regression I check the estimated kernel density of the residuals. They appear nearly normal.
What can I conclude based on this? Can I say that my regression is 'good' in some sense as a result?

Initial-value problem for non-linear partial differential equation $y_x^2=k/y_t^2-<!--\; -\; -->1$
For this problem, y is a function of two variables: one space variable x and one time variable t.
k>0 is some constant.
And x takes is value in the interval [0,1] and $t\ge <!--\; \ge \; -->0$
At the initial time, y follows a parabolic profile, like $y(x,0)=1-<!--\; -\; -->(x-<!--\; -\; -->\frac{1}{2}{)}^2$
Finally, y satisfies this PDE:
$${\left(\frac{\mathrm{\partial <!--\; \partial \; -->}y}{\mathrm{\partial <!--\; \partial \; -->}x}\right)}^2=\frac{k}{{\left(\frac{\mathrm{\partial <!--\; \partial \; -->}y}{\mathrm{\partial <!--\; \partial \; -->}t}\right)}^2}-<!--\; -\; -->1.$$
Does anyone have an idea how to solve this problem (and find the expression of y(x,t)) ?
About: The problem arise in physics, when studying the temporal shift of a front of iron particles in a magnetic field.

Why the residuals do not sum to 0 when we don't have the intercept in simple linear regression?

The Mean and Standard Deviation of Mathematics Achievement Test are 500 and 100, respectively. Find the 25th percentile rank (the score that total of 25% falls at or below)

You take a sample of 144 observations and have a value of x bar equal to 59. The population mean is 68 and the population standard deviation is 36. What is the z score of your sample mean?

Probability question. Find the probability that the data set falls within 45% to 52% of the data set
one of your employees has cautioned that your enterprise expand a brand new product. A survey is designed to have a look at whether or not or not there may be hobby inside the new product. The reaction is on a 1 to five scale with 1 indicating without a doubt could no longer buy, · · ·, and 5 indicating absolutely could purchase. For an preliminary analysis, you will document the responses 1, 2, and 3 as No, and 4 and 5 as yes.
a. five people are surveyed. what is the opportunity that as a minimum three of them replied sure?
b. 100 people are surveyed. what's the approximate possibility that between 45% to fifty two% of people answered sure?
For component a) There are 5 choices that human beings can respond by way of 1, 2 ,three ,4 and five. when you consider that 1, 2 and three are considered "No", the possibility of a person answering "No" is three/five. For choices four and 4, the probability of a person responding with this is 2/5.
This seems like it fallows a binomial distribution so I calculated the chance of P(three) + P(four) + P(five).
but for component b), i'm stressed. i will calculate the chance of a person pronouncing sure but I do not know how to calculate the chance that a percentage of people announcing sure. Does absolutely everyone recognize how to technique this question? I though approximately the use of the Z table, however that already calculates region.

How do I go about self-studying Maths? Do I create a routine? Answer a bunch of Q's?
Im 19F & I'm starting my Computer Science course this year. Before starting it, I wanted to make sure I had good GCSE & A level Maths knowledge.
The problem is I don't know how to go about self-studying Maths. Do I answer textbook questions section per section? Do I watch videos? How do I know when I'm comfortable enough with a topic to move on to the next? Do I do 1 section a day? How do I organise it all?
To add: I have plenty of time in my day to dedicate to self-studying. I have no commitments.