Statistics Projects for High School Students

What questions can you answer by using statistics and normal distributions?

Assume normal distributions in the following exercises. Find the value of z. An estimated 15%of the scores are to the right of z.

In what sense is Lebesgue integral the "most general"?
[ UPDATE: This question is apparently really easy to misinterpret. I already know about things like the Henstock-Kurzweil integral, etc. I'm asking if the Lebesgue integral (i.e. the general measure-theoretic integral) can be precisely characterized as "the most general integral that can be defined naturally on an arbitrary measurable space." ]
Apologies as I don't have that much of a background in real analysis. These questions may be stupid from the point of view of someone who knows this stuff; if so, just say so.
My layman's understanding of various integrals is that the Lebesgue integral is the "most general" integral you can define when your domain is an arbitrary measure space, but that if your domain has some extra structure, like for instance if it's Rn, then you can define integrals for a wider class of functions than the measurable ones, e.g. the Henstock-Kurzweil or Khintchine integrals.
[ EDIT: To clarify, by the "Lebesgue integral" I mean the general measure-theoretic integral defined for arbitrary Borel-measurable functions on arbitrary measure spaces, rather than just the special case defined when the domain is equipped with the Lebesgue measure. ]
My question: is there a theorem saying that no sufficiently natural integral defined on arbitrary measure spaces can (1) satisfy the usual conditions and integral should, (2) agree with the Lebesgue measure for measurable functions, and (3) integrate at least one non-measurable function? Or, conversely, is this false? Of course this hinges on the correct definition of "sufficiently natural," but I assume it's not too hard to render that statement into abstract nonsense.
Such an integral would, I guess, have to somehow "detect" sigma algebras looking like those of ${\mathbb{R}}^n$ and somehow act on this.
[ EDIT 2: To clarify, by an "integral" I mean a function that takes as input $((\mathrm{\Omega <!--\; \Omega \; -->},\mathrm{\Sigma <!--\; \Sigma \; -->},\mathrm{\mu <!--\; \mu \; -->}),f)$, where $(\mathrm{\Omega <!--\; \Omega \; -->},\mathrm{\Sigma <!--\; \Sigma \; -->},\mathrm{\mu <!--\; \mu \; -->})$ is any measure space and $f:(\mathrm{\Omega <!--\; \Omega \; -->},\mathrm{\Sigma <!--\; \Sigma \; -->})\to <!--\; \to \; -->(\mathbb{R},\mathcal{B})$ is a measurable function, and outputs a number, subject to the obvious conditions. ]
UPDATE: The following silly example would satisfy all of my criteria except naturality:
Let $(\mathrm{\Omega <!--\; \Omega \; -->},\mathrm{\Sigma <!--\; \Sigma \; -->},\mathrm{\mu <!--\; \mu \; -->})$ be a measure space, let $\mathcal{B}$ denote the Borel measure on R, and let $f:(\mathrm{\Omega <!--\; \Omega \; -->},\mathrm{\Sigma <!--\; \Sigma \; -->})\to <!--\; \to \; -->(\mathbb{R},\mathcal{B})$ be a Borel-measurable function. Define
$${\int <!--\; \int \; -->}_{\mathrm{\Omega <!--\; \Omega \; -->}}fd\mathrm{\mu <!--\; \mu \; -->}:=\{\begin{array}{ll}\text{the Khintchine integral}& \text{if }(\mathrm{\Omega <!--\; \Omega \; -->},\mathrm{\Sigma <!--\; \Sigma \; -->},\mathrm{\mu <!--\; \mu \; -->})=(\mathbb{R},\mathcal{L},{\mathrm{\mu <!--\; \mu \; -->}}_{\text{Lebesgue}});\\ \text{the Lebesgue integral}& \text{otherwise.}\end{array}$$

Let's suppose a regression between earnings and age (and suppose I do not know the distribution of earnings). Would it be possible for the residuals to be normally distributed?
I am thinking it would not be possible since earnings only takes on positive values and since the support of the normal is from $-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->}$ to $\mathrm{\infty <!--\; \infty \; -->}$, it would not be normal. However, since residuals are errors, they can be both positive and negative, so I am starting to question my hypothesis here.

In a normal linear model (with intercept), show that if the residuals satisfy $e_i=a+\mathrm{\beta <!--\; \beta \; -->}{x}_i$, for $i=1\dots <!--\; \dots \; -->n$, where $x$ is a predictor in the model, then each residual is equal to zero.
How to do this

The table shows Jenny's mark on the last math test. What percentile is her mark of 82? Her mark is included in the table. Round to a whole number.
$\begin{array}{|cccccc|}\hline 82& 77& 54& 39& 58& 65\\ 67& 81& 95& 92& 74& 63\\ 58& 75& 84& 88& 94& 90\\ 71& 76& 82& 91& 97& 53\\ \hline\end{array}$

How do confidence intervals change with standard deviations?

Having trouble finding the solution to $e^x-<!--\; -\; -->3e^{-<!--\; -\; -->x}-<!--\; -\; -->4x=0$. The answer is roughly $2.2$ but am not sure how to get there?

$f(x)=\frac{4x-<!--\; -\; -->1}{2x+1}$ How do u find the x and y intercept? for wich values of x is f(x) less than or equal to 0

Is second derivative of a function related to curve smoothness?
If there exist a first derivative of a function at any point then the funtion is continuous at that point.
What if the second derivative of that function is also exist at that point ? Does this mean that function is smooth at that point ? Means instead of taking sharp bend the function is having curved shape ?

Though there are several posts discussing the reference books for topology, for example best book for topology. But as far as I looked up to, all of them are for the purpose of learning topology or rather on introductory level.
I am wondering if there is a book or a set of books on topology like Rudin's analysis books, S.Lang's algebra, Halmos' measure (or to be more updated, Bogachev's measure theory), etc, serving as standard references.
Probably, such a book should hold characteristics such as being self-contained, covering the most of classical results, and other good properties you can name.
In the end, I only have the interest on general-topology (topological space, metrization, compactification...), and optionally differential topology (manifolds). So please don't divert into algebraic context.

After studying general a linear algebra course, how would an advanced linear algebra course differ from the general course?
And would an advanced linear algebra course be taught in graduate schools?

Algorithm design: $\mathrm{\theta <!--\; \theta \; -->}$(f(n)) - help explaining my answer
I am studying algorithm design and need some help explaining my (correct) answer to the following question:
Assume that $T(n)=\mathrm{\Theta <!--\; \Theta \; -->}(n^2)$. Can we say that for every input size n, our algorithm will perform a maximum of $10n^3$ operations? Why?
My answer, is as follows:
No, because by definition: T(n) is $\mathrm{\Theta <!--\; \Theta \; -->}(f(n))$ only if T(n) is both O(n) and $\mathrm{\Omega <!--\; \Omega \; -->}(n)$
$c_1\cdot <!--\; \cdot \; -->f(n)\le <!--\; \le \; -->T(n)\le <!--\; \le \; -->c_2\cdot <!--\; \cdot \; -->f(n)$
I was told that whilst the answer is correct, the explanation is not. Can somebody explain to my why this is the case?

In its first 10 years the Janus Flexible income Fund produced an average annual return of 9.58%. Assume that money invested in this fund continues to earn 9.58% compounded annually. How long will it take money invested in this fund to double?

Is there any academic fields where pure mathematics (Abstract Algebra or Number theory) are applied to medical/biological studies?
I am currently looking for applications of pure mathematics ( Especially abstract algebra or number theory) to medical studies, currently I can only think of the explore of deep learning (used in medical image analysis), design of Graph Neural Networks, and a little bit use of integer programming (Not sure if it is pure math)... Thanks so much in advance!

A metal alloy weighing 4 ounces and containing 40% nickel is melted and added with 1 ounce of pure nickel. What percent of the resulting metal is nickel?

I am looking for a book designed for self teaching applied/engineering mathematics. Can someone help me find a few?
I have used Shaum's differential equations. Are there others that I could use that are designed for self-study?

Do a negative Gaussian curvature help with the stability of a surface?
I found the statement "[...] ruled surfaces are statically efficient, especially in the case of skewed ruled surfaces, which are very stable due to a generally negative Gaussian curvature".
Why would a negative Gaussian curvature imply (or help with) the stability of a surface? I did not find a mathematical argument for the claim in the magazine, nor any mention of the same fact elsewhere. Is it really true?
I'm interested in this question because I'm studying ruled surfaces (which have non-positive Gaussian curvature) and their applications in architecture and product design in general, and if such surfaces have strength or stability advantages, I'd be very interested to know.

Find out the angular speed in terms of time.
Here is the equation that describes the motion of a planet under the gravitational field generated by a fixed star: $u=\frac{e}{l}\cos <!--\; \; -->\mathrm{\theta <!--\; \theta \; -->}+\frac{1}{l}$, where u is the reciprocal of the radial distance between the planet and the star, e is the eccentricity of the orbit, l is the semi latus rectum, and let h denote the angular momentum per unit mass, $\mathrm{\theta <!--\; \theta \; -->}$ is the angular coordinate. e,h,l turn out to be independent from one another, and they are independent from t and $\mathrm{\theta <!--\; \theta \; -->}$. At time t=0, we let the radial speed vanish, and we also let the angular coordinate vanish. To find the relationship between time and angular speed $\mathrm{\omega <!--\; \omega \; -->}$, we assume that u is a smooth fuction of t, and differentiate u w.r.t. t, and use $\mathrm{\omega <!--\; \omega \; -->}=h{u}^2$ to find out an expression for $\mathrm{\omega <!--\; \omega \; -->}$. To do this we can differentiate u w.r.t. $\mathrm{\theta <!--\; \theta \; -->}$ first then multiply it by ω, which equals to $h{u}^2$.
Then differentiate the first derivate of u w.r.t $\mathrm{\theta <!--\; \theta \; -->}$ first, then multiply the result by hu2 and so on. Since the whole process involves the differentiation w.r.t. θ only, we can assume that e=0.5,l=h=1. We set e=0.5 only because we wish to study bounded orbits so that we can apply Kepler's law and verify our result. However, the whole process is time consuming since the formula for the derivatives of u becomes complicated very quickly, even if we assume explicit values for e,l,h. The only effective way, therefore, is to design an algorithm for this process. But I do not have any knowlege about computer science, if anyone knows how to design algorithms or know about some other ways to find out ω in terms of t, please share.

An error 2% in excess is made while measuring the side of a square. The percentage of error in the calculated area of the square is