Statistics Projects for High School Students

Find the equation of the line perpendicular to ${\displaystyle y=-\frac{7}{16}x}$ that passes through (5,4)

What are the advantages and disadvantages of standard deviation ?

You collect data from some population with mean $\mathrm{\mu <!--\; \mu \; -->}$ and standard deviation $\mathrm{\sigma <!--\; \sigma \; -->}$ to test the following hypotheses: $H_0:\mathrm{\mu <!--\; \mu \; -->}=14vs.H_A:\mathrm{\mu <!--\; \mu \; -->}\ne <!--\; \ne \; -->14$. You obtain a p-value of 0.053. Which of the following is true?
a. A 95% confidence interval for $\mathrm{\mu <!--\; \mu \; -->}$ will include the value 14.
b. A 95% confidence interval for $\mathrm{\mu <!--\; \mu \; -->}$ will include the value 1.
c. A 90% confidence interval for $\mathrm{\mu <!--\; \mu \; -->}$ will include the value 1.
d. A 90% confidence interval for $\mathrm{\mu <!--\; \mu \; -->}$ will include the value 14.

i'm seeking out thoughts for a 15-hour mathematical enrichment course in a chinese language high faculty. What (pretty) simple concern would you advocate as a subject for any such course?
historical past/issues:
My students are generally pretty good at math, but many of them have no longer been uncovered to rigorous or summary mathematical reasoning. an amazing topic would be one that could not be impossibly hard for students who have by no means written or study proofs in English.
i have taught this magnificence three times earlier than. (a part of the purpose that i'm posting that is that i have used up all my thoughts!) the primary semester I taught an introductory range theory elegance (which meandered its way toward a proof of quadratic reciprocity, though I think this become in the end too advanced/abstract for some of the students). the second one semester I taught fundamental graph idea and packages (with a focal point on planarity and coloring). The 1/3 semester I taught a class at the Rubik's dice.
the students' math backgrounds are pretty numerous: a number of them take part in contest math competitions, and so are familiar with IMO-fashion techniques, however many aren't. a number of them may additionally realize some calculus, however I cannot assume it. all of them are superb at what in the united states is on occasion termed "pre-calculus": trigonometry, conic sections, systems of linear equations (though, shockingly, no matrices), and the like. They realize what a binomial coefficient is.
So, any ideas? preferably, i'd like to find some thing a bit "sexy" (like the Rubik's cube) -- tries to encourage wide variety theory through cryptography seemed to fall on deaf ears, however being capable of "see" institution idea on the cube became pretty popular.
(Responses specifically welcome from folks who grew up in the percent -- any mathematical subjects you desire were protected within the excessive college curriculum?)

I'm a graduate student studying quantum mechanics/quantum information and would like to consolidate my understanding of linear algebra. What are some good math books for that purpose?

What's the difference between $(-<!--\; -\; -->x{)}^2,-<!--\; -\; -->x^2$ and $(-<!--\; -\; -->x{)}^2$ ?
$(-<!--\; -\; -->x{)}^2$ is definately equal to $(-<!--\; -\; -->1{)}^2(x{)}^2$, right?
But $-<!--\; -\; -->(x{)}^2$ and $-<!--\; -\; -->x^2$ are confusing me, do they mean $-<!--\; -\; -->(x^2)$ or do they mean $(-<!--\; -\; -->1{)}^2(x{)}^2$?

i'm going to start an test approximately drying leaves. I need to study how a few special kind of leaves are going to dry themselves in 2 distinctive scenario. a group of them will stay in a high density, and the some other institution i can spread out in a larger region (as an instance, 1 m2 for the primary, and 3 m2 for the second). Of route, i can repeat this experiment numerous times. in the end, i will get:
Individual weight of the leaves (selecting samples) in the beginning
Individual weight along 4 days
Individual weight in the end
Nutritional composition in the start and in the end
I am remembering my knowledge in stats and R, but I am still lost. My question is, which is the best comparison method to analyse this data? I want to know:
How the humidity content change along the days
To dry this kind of leaves, which drying area is better?
Is it statistical significance between both method about losing water and nutrient content?

How to deal with lack of peers?
I am pursuing a B.S in Mathematics(freshman) in a country where people rarely study math for the sake of it and in a new university with good professors. Yet, it is new and I suffer from the lack of peers and it is literally impossible to pick up mathematical conversations on ideas with students . As a result, I find it difficult to exchange ideas with others.
Is there something which can be done to make sure that lack of peers or the pressure to push oneself doesn't harm me much?

Understanding Empirical Data Distribution
"We can approximate $p(x,y)=p(x)p(y|x)$ using the empirical data distribution
$p(x,y)=\frac{1}{N}\underset{n=1}{\overset{N}{\sum <!--\; \sum \; -->}}{\mathrm{\delta <!--\; \delta \; -->}}_{x_n}(x){\mathrm{\delta <!--\; \delta \; -->}}_{y_n}(y)"$
In another part of the paper they say $p(y|y_n)={\mathrm{\delta <!--\; \delta \; -->}}_{y_n}(y)$.
I have some background in probability but none in statistics; I was able to figure out what an Empirical CDF is, but not a pdf like here, so I'm not sure exactly what the authors are doing. Does the $\mathrm{\delta <!--\; \delta \; -->}$ refer to the Dirac delta distribution?

Show explicitly that the following identity holds under a Simple Linear Regression:
$\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{r}_i\stackrel{^<!--\; ^\; -->}{{\mathrm{\mu <!--\; \mu \; -->}}_i}=0$
with residuals $r_i=y_i-\stackrel{^<!--\; ^\; -->}{{\mathrm{\mu <!--\; \mu \; -->}}_i}$ and $\stackrel{^<!--\; ^\; -->}{{\mathrm{\mu <!--\; \mu \; -->}}_i}=\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_0}+\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_1}{x}_i$.
my steps:
$\begin{array}{rl}& \underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}{r}_i\stackrel{^<!--\; ^\; -->}{{\mathrm{\mu <!--\; \mu \; -->}}_i}\\ =& \underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}(y_i-\stackrel{^<!--\; ^\; -->}{{\mathrm{\mu <!--\; \mu \; -->}}_i})(\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_0}+\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_1}{x}_i)\\ =& \underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}(y_i-\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_0}-<!--\; -\; -->\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_1}{x}_i)(\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_0}+\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_1}{x}_i)\\ =& \underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}(\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_0}{y}_i+\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_1}{x}_i{y}_i-<!--\; -\; -->{\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_0}}^2-<!--\; -\; -->\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_0}\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_1}{x}_i-<!--\; -\; -->\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_0}\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_1}{x}_i-<!--\; -\; -->{\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_1}}^2{x_i}^2)\\ =& \underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}(\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_0}{y}_i+\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_1}{x}_i{y}_i-<!--\; -\; -->{(\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_0}+\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_1}{x}_i)}^2)\\ =& \underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}(y_i(\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_0}+\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_1}{x}_i)-<!--\; -\; -->{(\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_0}+\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_1}{x}_i)}^2)\\ =& \underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}(\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_0}+\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_1}{x}_i)(y_i-<!--\; -\; -->\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_0}+\stackrel{^<!--\; ^\; -->}{{\mathrm{\beta <!--\; \beta \; -->}}_1}{x}_i)\\ =& \underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}\stackrel{^<!--\; ^\; -->}{{\mathrm{\mu <!--\; \mu \; -->}}_i}{r}_i\\ & \end{array}$
how to proceed?

What is the percentile rank x=15 and what is the score for 75th percentile?

Show that $\stackrel{^<!--\; ^\; -->}{e}$ and $\stackrel{^<!--\; ^\; -->}{\mathrm{\beta <!--\; \beta \; -->}}$, the residuals and coefficient vector for the OLS problem $y=X\mathrm{\beta <!--\; \beta \; -->}+\mathrm{ϵ<!--\; ϵ\; -->}$, are uncorrelated.

An experiment is designed to test the potency of a drug on 40 rats. Previous animal studies have shown that a 10-mg dose is lethal 10% of the times within the first 4 hours.
What is the probability that between 2 and 8 rats die during the experiment due to the drug?
My attempt: Let X be the number of rats that die in the first 4 hours.
$P(2\le <!--\; \le \; -->x\le <!--\; \le \; -->8)=\underset{x=2}{\overset{8}{\sum <!--\; \sum \; -->}}(\genfrac{}{}{0ex}{}{40}{x})(0.1{)}^x(0.9{)}^{40-<!--\; -\; -->x}$
To make the drug more potent, the company came with a new formula. This reduced the chances of a new drug being lethal to 1%. The new drug is administered to 10000 rats. Approximate the exact probability that 5 rats die.
My attempt:
$P(x=5)=(\genfrac{}{}{0ex}{}{10000}{5})(0.01{)}^5(0.99{)}^{10000-<!--\; -\; -->5}$
Am I on the right path?

In general, when investigating a question of interest, you are not aware of the actual population statistic. However, by taking a simple random sample of an appropriate size, you can make inferences about the entire population. Also, the normal distributions are completely described by two statistics: the mean and the standard deviation. The standard deviation for a sampling distribution is given by
$\sqrt{\frac{p(1-<!--\; -\; -->p)}{n}}$
You used p = 0.20 and n = 100. To be more accurate, would you prefer to use n = 1000? Use the formula to evaluate standard deviations to support your answer.

The function $f$ is one-to-one. Prove that the sum of all the $x$- and $y$-intercepts of the graph of $f(x)$ is equal to the sum of all the $x$- and $y$-intercepts of the graph of $f^{-<!--\; -\; -->1}(x)$.

How do I know whether to use the Z-table or T-table?

Is there a "canonical" representation of integers using numbers other than primes?
Consider the (cumbersome) statement: "Every integer greater than 1 can be written as a unique product of integers belonging to a certain subset, S of integers.
When S is the set of primes, this is the Fundamental Theorem of Arithmetic. My question is this: Are there any other types of numbers, for which this is true.
EDIT: As the answers show, this obviously cannot be done. What if we relax the integer condition, i.e. can there be any other canonical representation of positive integers using complex numbers?

In a class of 10 students, James got 3rd position from the lowest. What is his percentile rank?