Statistics Projects for High School Students

Do you prove all theorems whilst studying?
When you come across a new theorem, do you always try to prove it first before reading the proof within the text? I'm a CS undergrad with a bit of an interest in maths. I've not gone very far in my studies -- sequence two of Calculus -- but what I'm trying to understand right now, though, is how one actually goes about studying so that when finished with a good text, there's more of an intuitive understanding than superficial.
After reading "The Art Of Problem Solving" from the Final Perspectives section of part eight in 'The Princeton Companion to Mathematics', it seems to hint at approaching studying in that very way. A quote in particular, from Eisenstein, that caught my attention was the following -- I'm not going to paraphrase much:
The approach utilized by the director turned into as follows: each pupil needed to show the theorems consecutively. No lecture took place at all. nobody became allowed to inform his answers to every person else and every scholar acquired the following theorem to show, impartial of the opposite college students, as soon as he had proved the previous one correctly, and as long as he had understood the reasoning. This changed into a completely new interest for me, and one that I grasped with outstanding enthusiasm and a zeal for knowledge. Already, with the first theorem, i was a long way ahead of the others, and whilst my friends had been still struggling with the eleventh or 12th, I had already proved the hundredth. there has been simplest one younger fellow, now a medicine pupil, who could come close to me. even as this technique is excellent, strengthening, as it does, the powers of deduction and inspiring self sustaining wondering and opposition among college students, generally speakme, it can likely no longer be adapted. For as lots as i can see its advantages, one ought to admit that it isolates a positive power, and one does not obtain an overview of the complete concern, that may simplest be done with the aid of an amazing lecture. as soon as one has acquired a amazing variety of material thru [...] for college kids, this technique is manageable only if it deals with small fields of effortlessly, comprehensible information, in particular geometric theorems, which do now not require new insights and thoughts.
I feel that this type of environment is something you don't often see, especially in the US -- perhaps that's why so many of our greats are foreign born. As I understand it, he does go on to say that he wouldn't particularly recommend that method of study for higher mathematics, though.
A similar question was posed to mathoverflow where Tim Gowers (Fields Medal) went on to say that he recommended similar methods to study: link
I'm not quite certain that I understood the context of it all, though. Upon asking a few people whose opinion mattered to me, I was told that it if time were precious to me, it would be a waste going about studying mathematics in that way, so I'd like to get some perspective from you math.stackexchange. How do you go about studying your texts?

Let's imagine I have a function $f$(x) whose evaluation cost (monetary or in time) is not constant: for instance, evaluating f(x) costs $c(x)\in <!--\; \in \; -->\mathbb{R}$ for a known function c. I am not interested in minimizing it in few iterations, but rather in minimizing it using "cheap" evaluations. (E.g. maybe a few cheap evaluations are more "informative" to locate the minima, rather than using closer-to-the-minima and costly evaluations.)
My questions are: Has this been studied or formalized? If so, can you point me to some references or the name under which this kind of optimization is known?
Just to clarify, I am not interested in either of these:
Standard optimization, since its "convergence rate" is linked to the total number of function evaluations.
Constrained optimization, since my constraint on minimizing $f$(x) is not defined in terms of the subspace allowed to x, but in terms of the added costs of all the evaluations during the minimization of $f$.
Goal optimization, since using few resources to minimize $f$ can not be expressed (I think) as a goal.
Edit
Surrogate-based optimization has been proposed as a solution. While it faces the fact that the function is expensive to compute, it does not face (straightforwardly) the fact that different inputs imply different evaluations costs. I absolutely think that surrogate-based optimization could be used to solve the problem. In this line, is there a study of how to select points xi to estimate the "optimization gain"? In terms of surrogate-based optimization: what surrrogate model and what kind of Design of Experiment (DoE) could let me estimate the expected improvement?
Also, some remarks to contextualize. Originally, I was thinking on tuning the hyperparameters of machine learning algorithms. E.g., training with lots of data and lots of epochs is costly, but different architectures may be ranked based on cheaper trainings and then fine-tuned. Therefore:
Let us assume that c(x) can be estimated at zero cost.
There is some structure in c(x): large regions have higher cost, large regions have cheaper cost, the cost is smooth, the minima may be in either region.
Let us assume that $f$(x) is smooth. Also, if possible, let us assume it has some stochastic error that we can model (such as Gaussian error with zero mean and known variance).

The heights of people of the same gender and similar ages follow Normal distributions reasonably closely. How about body weights? The weights of women aged 20 to 29 have mean 141.7 pounds and median 133.2 pounds. The first and third quartiles are 118.3 pounds and 157.3 pounds. Is it reasonable to believe that the distribution of body weights for women aged 20 to 29 is approximately Normal? Explain your answer.

I am studying control systems, and I want to solve following problem.
Given full rank state matrix A (with all unstable eigenvalues), design input matrix B, such that cost function J=trace(B′XB) is minimized, where X is the solution to discrete-time Ricatti equation (DARE). I have contraint that (A,B) is stabilizable, i.e.
For a given full rank $A\in <!--\; \in \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->n}$, with ${\mathrm{\lambda <!--\; \lambda \; -->}}_i(A)>1$, solve the following
$$\begin{array}{ll}\underset{X\in <!--\; \in \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->n},B\in <!--\; \in \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->m}}{\text{minimize}}& \mathrm{t}\mathrm{r}\left(B^{\prime }XB\right)\\ \text{subject to}& X=A^{\prime }X(I+B{B}^{\prime }X{)}^{-<!--\; -\; -->1}A\\ & (A,B)\text{ is stabilizable}\end{array}$$
From my understanding, since all eigenvalues of A are outside of unit circle (discrete-time system), we can change condition (A,B) is stabilizable with (A,B) is controllable, which is equivalent to rank([$\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}([BAB{A}^{2}B\dots <!--\; \dots \; -->A^{n-<!--\; -\; -->1}B])=n$
The problem is for sure feasible, since for any full rank A, there is B such that rank condition is satisfied and we can solve DARE.

What can you say about distributions A and B if they have the same mean but the standard deviation of A is greater than that of B?

Consider two binomial distributions for 1,000 repeated Bernoulli trials-the first for trials with p=.15, and the second for trials with p=.85. How are the histograms for the two distributions related? Explain.

Let A and B be two positive definite matrices of orders n and d, respectively. Let X be a matrix of type (n,d) (n rows and d columns).
Is it true that A−1XB−1X⊺ have eigenvalues between zero and one?
I am unsure if the result holds with this generality, so I am giving more context below.
This problem arises in the study of "contingency tables" (i.e. statistics) where X is a design matrix of zeros and ones, where each row represents an observation and each column represents the category or cell where this observation falls. Thus, xij=1 means that the ith object appeared in the jth cell and if xij=0 it means taht the object did not appear. It is assumed that at least one entry of every row and every column will be one (the possibility of different objects being in different categories is OK). The matrices A and B are obtained by normalising the rows and columns of X. That is, definexi⋅=∑j=1dxij,x⋅j=∑i=1nxij
and set A=diag(xi⋅), B=diag(x⋅j).
I have made more than a few attemps without success but neither of these have really used the structure of the problem. For example, if λ is an eigenvalue of the matrix A−1XB−1X⊺ then any eigenvector of λ will satisfy XB−1X⊺=λAv. It is tempting to multiply on the left by v⊺ and us the fact that boths sides are now weighted norms of v. This already shows that λ≥0. But the algebra becomes messy and I could not see a path forward to show λ≤1. On top of this, I am not really using the structure of X,A and B as I already mentioned.
A perhaps useful fact (that I have not been able to exploit) is that the vector 1 (full of ones) has eigenvalue 1 and any other eigenvector is orthogonal to this one!

When comparing two data sets, which is better: frequency distributions or relative frequency distributions? Why?

i'm presently reading Multivariate Calculus (Larson and Edwards book). I want to do a project in pc technology to look some first-class programs of factors i am mastering.
Any specific source of papers/journals/books? thank you

The role of dual space of a normed space in functional analysis
We have known that dual space of a normed space is very important in functional analysis.
I would like to ask two questions related dual space of a normed space:
What is the motivation of constructing dual space of a normed space?
What is the main role of dual space of a normed space in functional analysis?

What are the allowed axioms to solve the Riemann Hypothesis?
First, is it even possible to state a well known problem and know what axioms are allowed for it? I'd guess that one must look at the branches of math involved, but I'm no mathematician.
Therefore, I would like to ask the people working on this problem if this there is a fixed list of allowed axioms for this problem.

Assume normal distributions in the following exercises. Find the value of z. 8% of the scores are to the right of z.

What does the sampling distribution of the sample variance look like if we sample from a population with an approximately normal distribution? Find out using the applet Sampling Distribution of the Variance (Mound Shaped Population) (at academic.cengage.com/statistics/wackerly) to complete the following. In the previous exercises in this section, you obtained simulated sampling distributions for the sample mean. All these sampling distributions were well approximated (for large sample sizes) by a normal distribution. Although the distribution that you obtained is mound-shaped, does the sampling distribution of the sample variance seem to be symmetric (like the normal distribution)?

Explain why it is likely that the distributions of the following variables will be normal: the diameter of bolts immediately after manufacture.

Fourier and Laplace transforms together, is this possible?
Answering on some posts on MSE about Laplace transform and Fourier transform I stumbled upon a question to which I cannot answer myself (not having a good ground in pure mathematics).
The question is the following:
Is there some mathematical constraint that doesn't let us use both Fourier and Laplace transform on the same equation?
I'm not saying that it would be useful in any case, I was just wondering if it's feasible! Just as an example I could use both transforms to solve the one dimensional (or three, doesn't change much) wave equation with some external force
$$\{\begin{array}{l}{\mathrm{\partial <!--\; \partial \; -->}}_t^{2}u(x,t)-<!--\; -\; -->c^2{\mathrm{\partial <!--\; \partial \; -->}}_x^{2}u(x,t)=f(x,t)\\ u(x,0)={\mathrm{\partial <!--\; \partial \; -->}}_{t}u(x,t){|}_{t=0}=0\\ -<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->}<<!--\; <\; -->x<<!--\; <\; -->\mathrm{\infty <!--\; \infty \; -->}t><!--\; >\; -->0\end{array}$$

Two populations have normal distributions. The first has population standard deviation 2 and the second has population standard deviation 3. A random sample of 16 measurements from the first population had a sample mean of 20. An independent random sample of 9 measurements from the second population had a sample mean of 19. Test the claim that the population mean of the first population exceeds that of the second. Use a 5% level of significance. State the hypotheses.

The normal distribution is a good estimate for which of the binomial distributions. n=40, p=0.1

I have recently started studying graphs and their different traversal algorithms, and can't seem to be able to come up with an answer to this question. I really need your help, I don't even know where to start. P.S. It was my birthday yesterday and I don't want to cry because of this problem.
A company in New York manufactures blue halogen bulbs for cars. Unfortunately, it is very difficult to color bulbs consistently. Naturally, it is also very important to package bulbs that look alike in pairs. To package bulbs in pairs, bulbs coming out of the assembly line are first partitioned into two sets of bulbs sharing similar colors (e.g., one set of darker bulbs, another set of lighter bulbs), and then pairs are formed within each set.
Because of increasing demand, the company wishes to hire more workers to partition bulbs into two sets. To determine whether applicants have appropriate skills to perform this rather delicate task, they are asked to take this simple test: Given a set of n bulbs, compare each pair of bulbs and determine whether two bulbs have “similar” or “different” colors. An applicant also has an option to say “indeterminate” for each pair. The company wishes to hire applicants who are consistent in their judgment. We say m judgments (resulted in either “similar” or “different”) are consistent if it is possible to partition n bulbs into two sets such that (i) for each pair {a,b} determined to be “similar”, a and b indeed belong to the same set, and (ii) for each pair {a,b} determined to be “different”, a and b indeed belong to different sets.
For a given test result for n bulbs with m judgments, design an O(n+m)-time algorithm to determine whether the judgments are consistent. In practice, there should be a minimum number of judgments applicants are required to make, but your algorithm should work for any integer $m\ge <!--\; \ge \; -->0$

i've recently started studying up on design concept, with the ultimate reason of doing some authentic studies in that region. I understand the arithmetic pretty properly, but am now not understanding the application. Can a person here give an explanation for that? instead are there a few articles, web-websites, papers explaining the software of layout concept in agriculture, biological experiments and so forth.
for example after establishing the lifestyles of/constructing a certain sort of design mathematically, how can we use that expertise in exercise?

Consider two normal distributions, one with mean -4 and standard deviation 3, and the other with mean 6 and standard deviation 3. Answer true or false to each statement and explain your answers. a. The two normal distributions have the same shape. b. The two normal distributions are centered at the same place.