Statistics Projects for High School Students

Find k for the following probability distributions: P(x) = k(x + 2) for x = 1, 2, 3

I'm trying to learn linear algebra on my own. When I taught myself calculus I-III a year ago, I used Stewart's Concepts and Contexts 4th edition and I absolutely loved it. Theorems and definitions were displayed actually, the graphs were distinctive, and the problem selection was quite large.
Is there a linear algebra textbook just like how Stewart's textbook changed into designed?
moreover, i am sooner or later planning to research differential equations and i'm aware that pretty a few texts overlap when coaching those topics. i would favor to study them in my view, so i might want to keep away from texts with an amazing amount of differential equation, unless there's a text that covers each fields absolutely in the identical manner that Stewart's textual content included unmarried and multivariable calculus collectively.
P.S I don't imply to be praising Stewart's. it's the only arithmetic textbook i've used to self-have a look at so I do not have a lot else to relate to.

What are some applications of sets & relations in science/business/tech that a highschooler can understand? To kindle a young mind, what examples can be given?

If the median of a distribution is 1900, the mean is 2144.58, and the sample standard deviation is 884.07, what is the sample variance?

Statistics (Regression Analysis): Show that the residuals from a linear regression model can be expressed as $\mathbf{e}=(\mathbf{I}-<!--\; -\; -->\mathbf{H})\mathrm{ϵ<!--\; ϵ\; -->}$
The bold represents vectors or matrices.
I know that $\mathbf{e}=\mathbf{y}-<!--\; -\; -->\mathbf{H}\mathbf{y}$
So I tried expanding this to,
$\mathbf{e}=\mathbf{X}\mathrm{\beta <!--\; \beta \; -->}+\mathrm{ϵ<!--\; ϵ\; -->}-<!--\; -\; -->\mathbf{H}\mathbf{X}\mathrm{\beta <!--\; \beta \; -->}\mathbf{-<!--\; -\; -->}\mathbf{H}\mathrm{ϵ<!--\; ϵ\; -->}$
At this point I can see how to derive the more traditional,
$\mathbf{e}\mathbf{=}\mathbf{(}\mathbf{I}\mathbf{-<!--\; -\; -->}\mathbf{H}\mathbf{)}\mathbf{y}$
how to solve the original problem?

Beneficial to touch theoretically deeper texts earlier in core areas e.g. analysis, algebra?
Desired future direction: Dynamical System(Chaos), PDE
More beneficial to read theoretically deep, modern and masterpiece texts earlier, (e.g. levels like UTX/GTM/GSM/LNM/CSAM) ? Especially in core areas (e.g. Analysis, Algebra, Topology etc.)
i.e. Read briefly less for intro-level, then quickly drop to the deeper/modern texts, although basic skill's not trained enough, but to be trained again/better in 'deeper-stage'. (Sometimes it seems learning new material in deeper/complex 'environment/context' gives better/faster maturity.)
e.g: Calculus/Analysis: Apostol -> Hardy, Courant, Stein, Rudin, Amann
Linear Algebra: Axler -> Dym, Grueb, Lax, Halmos
Algebra: van der Waerden -> Lang, Hungerford, Jacobson, Bourbaki
Topology: Kelley -> Milnor

Determine whether the given statement is true or false. All normal distributions have a mean of 0.

i'm hoping to self-have a look at Geometry, Algebra, Calculus, Vector Calculus, Linear Algebra, probability and information, and different intermediate maths. i've located the pleasant way for me to analyze is to work on problems. regrettably, many math textbooks don't provide all of the answers so that i'm able to test my work.
Does absolutely everyone have any recommendations for materials that do provide all solutions to problems so that i will feel at ease that i'm solving the problems efficaciously?

I am a pure math major. I am taking a one-semester course covering the following topics:
Inner Product Spaces,
Direct Sums of Subspaces,
Primary Decomposition Theorem,
Reduction - (triangular, Jordan, rational, classical),
Dual Spaces,
Orthogonal Direct Sums,
Quadratic Forms
Our lecturer is various theorist so he'll probably make a few links with number theory. He does now not advise any textbooks for the course but I research excellent alone out of books designed for self-study. Lecture theaters full of whispering and distractions are hell to me! i love books which are written in a private, quirky fashion that set off deep reflection and inspire impartial questioning. i like colourful motivating cloth!! I dislike mention of any "real-international applications" (ie physics/engineering). I hate when writers pass over big chunks in proofs and say the lacking steps are "trivial" whilst they're now not. i love it while answers are available to the physical activities.

Determine the values of $a$ and $b$ for which the line $(a+2b-<!--\; -\; -->3)x+(2a-<!--\; -\; -->b+1)y+6a+9=0$ is parallel to $x$ axis and $y$-intercept is $-<!--\; -\; -->3$. Also write the equation of the line.
here is what i have tried.
$eq:(a+2b-<!--\; -\; -->3)x+(2a-<!--\; -\; -->b+1)y+6a+9=0$ Let the point be $(0,-<!--\; -\; -->3)$
Since the line is parallel to $x$ axis therefore
slope of line $=$ slope of $x$-axis
$-<!--\; -\; -->(a+2b-<!--\; -\; -->3)/(2a-<!--\; -\; -->b+1)$
$2a-<!--\; -\; -->b+1=0$
Now what?

Determine whether the following statement is true: When random samples of 50 men and 50 women are obtained and we want to test the claim that men and women have different mean annual incomes, there is no need to confirm that the samples are from populations with normal distributions.

If C is the union of A and B, how do you calculate the standard deviation of population C if you know the standard deviations of A and B? What if A and B are of different sizes?

What is the lower bound of a random variable's variance?

While calculating percentile rank there are two formula
1st )
percentile ={ ( number of values below x + 0.5)/total number of observation } *100
2nd )
percentile ={ ( number of values below x )/total number of observation } *100
I have to know when to used 1st formula and when to used 2nd formula
propert explanation needed

Suppose that, for two populations, the distributions of the variable under consideration have the same shape. Further suppose that you want to perform a hypothesis test based on independent random samples to compare the two population means. In each case, decide whether you would use the pooled t-test or the Mann-Whitney test and give a reason for your answer. You know that the distributions of the variable are a. normal. b. not normal.

Question on designing a state observer for discrete time system
I came through this problem while studying for an exam in control systems:
Consider the following discrete time system
$$\stackrel{\to <!--\; \to \; -->}{x}(k+1)=A\stackrel{\to <!--\; \to \; -->}{x}(k)+b\stackrel{\to <!--\; \to \; -->}{u}(k),\stackrel{\to <!--\; \to \; -->}{y}(k)=c\stackrel{\to <!--\; \to \; -->}{x}(k)$$
where $b=(0,1{)}^T,c=(1,0),A=\left[\begin{array}{ccc}2& & 1\\ 0& & -<!--\; -\; -->g\end{array}\right]$ for some $g\in <!--\; \in \; -->\mathbb{R}$
Find a feedback regulation (if there is any) of the form $u(k)=-<!--\; -\; -->K\stackrel{^<!--\; ^\; -->}{x}(k)$ where $\stackrel{^<!--\; ^\; -->}{x}(k)$ is the country estimation vector that is produced via a linear complete-order state observer such that the nation of the system and the estimation blunders $e(k)=\stackrel{\to <!--\; \to \; -->}{x}(k)-<!--\; -\; -->\stackrel{^<!--\; ^\; -->}{x}(k)$ go to zero after a few finite time. layout the kingdom observer and the block diagram.
My method
it is clean that the eigenvalues of the machine are ${\mathrm{\lambda <!--\; \lambda \; -->}}_1=2,{\mathrm{\lambda <!--\; \lambda \; -->}}_2=-<!--\; -\; -->g$ (consequently it is not BIBO solid) and that the pair (A,b) is controllable for every fee of g, as nicely a the pair (A,c) is observable for all values of g. consequently we will shift the eigenvalues with the aid of deciding on a benefit matrix okay such that our device is strong, i.e. it has its eigenvalues inside the unit circle $|z|=1$.
The state observer equation is
$$[\stackrel{\to <!--\; \to \; -->}{x}(k+1)\stackrel{\to <!--\; \to \; -->}{e}(k+1){]}^T=\left[\begin{array}{ccc}A-<!--\; -\; -->bK& & Bk\\ O& & A-<!--\; -\; -->LC\end{array}\right][\stackrel{\to <!--\; \to \; -->}{x}(k)\stackrel{\to <!--\; \to \; -->}{e}(k){]}^T$$
With characteristic equation
$$\mathrm{\chi <!--\; \chi \; -->}(z)=|zI-<!--\; -\; -->A+bK||zI-<!--\; -\; -->A+LC|={\mathrm{\chi <!--\; \chi \; -->}}_K(z){\mathrm{\chi <!--\; \chi \; -->}}_L(z)$$
Also consider
$$K=\left[\begin{array}{ccc}{k}_1& & k_2\\ k_3& & k_4\end{array}\right]$$
and let $a=k_1+k_3,\mathrm{\beta <!--\; \beta \; -->}=k_2+k_4$
Then ${\mathrm{\chi <!--\; \chi \; -->}}_K(z)=(z-<!--\; -\; -->2)(z+g+\mathrm{\beta <!--\; \beta \; -->})+a$.
So we can select some eigenvalues inside the unit circle and determine $a,\mathrm{\beta <!--\; \beta \; -->}$ in terms of g. Choosing e.g. ${\mathrm{\lambda <!--\; \lambda \; -->}}_{1,2}=\pm <!--\; \pm \; -->1/2$ we get $a=3g+33/8,\mathrm{\beta <!--\; \beta \; -->}=9/4-<!--\; -\; -->g,g\in <!--\; \in \; -->\mathbb{R}$
Questions
I want to ask the following:
Is my approach correct? Should I select the eigenvalues myself since I am asked to design the observer or should I just solve the characteristic equation and impose $|{\mathrm{\lambda <!--\; \lambda \; -->}}_{1,2}|<1$?
Should I determine L matrix as well since the error must also vanish? (because it is not asked)

The normal distribution is a good estimate for which of the binomial distributions. n=30, p=0.4

NCAE shows that the average score of the students is 80 and sd is 15. What is the percentile rank of a score of 84?

This might be a really simple question, but I just didn't find an answer from anywhere.
I'm teaching linear algebra to myself and in my study material I came upon notation that I just don't understand. I can't find an explanation for it from my material and it is hard to find on the internet as well it seems.
Example:
$U=L((3,2,-<!--\; -\; -->6,4),(0,4,1,-<!--\; -\; -->5))$
What does the L() notation mean? U itself should be a subspace for $R^4$. I would assume that those are vectors within the L().