Statistics Projects for High School Students

The normal distribution is a good estimate for which of the binomial distributions. n=40, p=0.9

Suppose there is a Quadratic relationship between a predictor, which exhibits a trend over time, and the response but we included only a linear term for that predictor in the linear regression model. Which of the following will happen if we use linear model for this regression model?
Will the diagnostics show autocorrelation in the residuals?
Do you think the residuals will add up to 0?

The given are
Two $x$-intercepts
$y$-intercept math $(0,-<!--\; -\; -->4)$
Maximum at $(2,4)$
How to create quadratic equation given $y$ intercept, and maximum and $B=8$?

where do combinatorics play a role in computer science?
I am graduated from computer science. I need some advice in Combinatorics material related to master fields of computer science. I already know that discrete mathematics have a key role in some concepts which I need a little more elaboration. In which fields could I trace combinatorics concepts and theorems and techniques?

Explain why t distributions tend to be flatter and more spread out than the normal distribution.

The average weekly earnings in dollars for various industries are listed below. Find the percentile rank of each value. Round to the nearest whole percentile.
548, 450, 475, 460, 687, 514 641
The percentile rank of the value 548 is ?

Finding singularities and residuals
Let $f(z)=\frac{1}{(z-<!--\; -\; -->1)(2z-<!--\; -\; -->1)}$. Then
$f(z)=\frac{1}{z-<!--\; -\; -->1}-<!--\; -\; -->\frac{2}{2z-<!--\; -\; -->1}=-<!--\; -\; -->\underset{k=0}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\sum <!--\; \sum \; -->}}{z}^k-<!--\; -\; -->\frac{1}{z(1-<!--\; -\; -->\frac{1}{2z})}$
$=-<!--\; -\; -->(1+z+z^2+z^3+\cdots <!--\; \cdots \; -->+z^k+\dots <!--\; \dots \; -->)-<!--\; -\; -->\frac{1}{z}\ast <!--\; \ast \; -->(1+\frac{1}{2z}+\frac{1}{(2z{)}^2}+\cdots <!--\; \cdots \; -->+\frac{1}{(2z{)}^k}+\dots <!--\; \dots \; -->)$
$=-<!--\; -\; -->(1+z+z^2+z^3+\cdots <!--\; \cdots \; -->+z^k+\dots <!--\; \dots \; -->)-<!--\; -\; -->\frac{1}{z}\ast <!--\; \ast \; -->(1+\frac{1}{2z}+\frac{1}{(2z{)}^2}+\cdots <!--\; \cdots \; -->+\frac{1}{(2z{)}^k}+\dots <!--\; \dots \; -->)$
Therefore, the origin is an essential singularity of $f(z)$ with residue $-<!--\; -\; -->1$.

Suppose we have model $Y=X\mathrm{\beta <!--\; \beta \; -->}$ and first collumn of $X$ consists of 1$1$. Why the sum of residuals in the this regression model equals $0$? And why in generall it doesn't, when there is no constant term?

Assume normal distributions in the following exercises. Find the value of z. The probability that a score is between z and -z is 0.82.

Let $L$ be the tangent line to $y=\tan <!--\; \; -->(2x)$ at $(\frac{\mathrm{\pi <!--\; \pi \; -->}}{2},0)$. What is the $y$-intercept of $L$?
(a) $0$
(b) $\frac{\mathrm{\pi <!--\; \pi \; -->}}{2}$
(c) $-<!--\; -\; -->\mathrm{\pi <!--\; \pi \; -->}$
(d) $1$
(e) $2$

When you take the Scholastic Assessment Test (SAT), your score is recorded as a percentile score. If you scored in the 92nd percentile, it means that you scored better than approximately 92% of those who took the test.
(a) If Lisa's score was 83 and that score was the 22nd score from the top in a class of 260 scores, what is Lisa's percentile rank? (Round your answer to the nearest whole number.)
(b) Lee has received a percentile rank of 83% in a class of 40 students. What is Lee's rank in the class? (Round your answer to the nearest whole number.)

3500 people competed in an ironman race, John finished 805th What was John's percentile rank?

Assume normal distributions in the following exercises. Find the value of z. An estimated 86% of the scores are to the left of z.

How can I deepen my knowledge in Mathematics without getting a degree in Mathematics?
I'm an undergraduate student, I'm pursuing a Bachelor Degree in Computer Engineering. I had some exams about Mathematics (calculus 1/2/3, probability and statistics, algorithms and data structures, complex functional analysis).
While I don't want to become a researcher/professor (I would like to be a Software Engineer) I also love Mathematics, and I feel like I always want to know more.
Question: Which is the best way of studying on my own? Should I get some textbooks? Aren't textbooks suitable for those who are going to have an exam?
Actually I don't know which part I would like to deepen the most, but I'm really amazed by the Riemann hypothesis. What would it take to understand a wrong proof?

A quartic polynomial has $2$ distinct real roots at $x=1$ and $x=-<!--\; -\; -->3/5$. If the function has a $y$-intercept at $-<!--\; -\; -->1$ and has $f(2)=2$ and $f(3)=3$. How to determine the remaining roots when two distinct real roots and y-intercept are given.

Euclid's view and Klein's view of Geometry and Associativity in Group
One common item in the have a look at of Euclidean geometry (Euclid's view) is "congruence" relation- specifically ""congruence of triangles"". We recognize that this congruence relation is an equivalence relation
Every triangle is congruent to itself
If triangle $T_1$is congruent to triangle $T_2$then $T_2$is congruent to $T_1$.
If $T_1$is congruent to $T_2$and $T_2$is congruent to $T_3$, then $T_1$is congruent to $T_3$.
This congruence relation (from Euclid's view) can be translated right into a relation coming from "organizations". allow $Iso({\mathbb{R}}^2)$ denote the set of all isometries of Euclidean plan (=distance maintaining maps from plane to itself). Then the above family members may be understood from Klein's view as:
∃ an identity element in $Iso({\mathbb{R}}^2)$ which takes every triangle to itself.
If $g\in <!--\; \in \; -->Iso({\mathbb{R}}^2)$ is an element taking triangle $T_1$to $T_2$, then $g^{-<!--\; -\; -->1}\in <!--\; \in \; -->Iso({\mathbb{R}}^2)$ which takes $T_2$to $T_1$.
If $g\in <!--\; \in \; -->Iso({\mathbb{R}}^2)$ takes $T_1$to $T_2$and $g\in <!--\; \in \; -->Iso({\mathbb{R}}^2)$ takes $T_2$ to $T_3$ then $hg\in <!--\; \in \; -->Iso({\mathbb{R}}^2)$ which takes $T_1$ to $T_3$.
One can see that in Klein's view, three axioms in the definition of group appear. But in the definition of "Group" there is "associativity", which is not needed in above formulation of Euclids view to Kleins view of grometry.
Question: What is the reason of introducing associativity in the definition of group? If we look geometry from Klein's view, does "associativity" of group puts restriction on geometry?

What is the mean and standard deviation of 5, 10, 12, 15, 20?

Assume normal distributions in the following exercises. Find the value of z. The probability that a score is to the left of z is 0.96.

In each of the following situations, which of the two distributions would be more spread out? Standard normal distribution or t-distribution with df = 10. t-distribution with df = 5 or t-distribution with df = 25. t-distribution with df = 100 or normal distribution with standard deviation = 100.