Statistics Projects for High School Students

The original function is given,
$y=2x^2+2x+7$
at the point $(4,47)$.
I found the tangent line from the original function which is:
$f^{\prime }(x)=4x+2.$
The question states that the tangent line can be written as $y=mx+b$ and asks me to find the slope ($m$) as well as the $y$-intercept ($b$).
I have found the slope which is $18$, by plugging in the $x$-value from the point $(4,47)$ into the tangent line: so I have $f(x)=4x+2$ which equals $4(4)+2$ which is $18$.
The way I'm trying to find the $y$-intercept is by doing: $y-<!--\; -\; -->47=18(x-<!--\; -\; -->3)$ which is $-<!--\; -\; -->7$, but it's wrong the answer is $-<!--\; -\; -->25$.
What mistake am I making that's leading me to get $-<!--\; -\; -->7$?

How to find the y of the vertex. How can I find the y? The x intercepts are (-1,0) (5,0) and the y intercept is (0,5) and the x of the vertex is 2.

Identifying the k points in 2D geographic space which are 'most distant' from each other
I have a set of DNA samples from Y plants in a given geographic area. I'm going to be doing DNA sequencing on individuals in this population (and a number of other, separate populations), however due to financial constraints I'm unable to perform sequencing on all Y individuals. I've decided that I can afford to sequence k (out of Y) in each separate area.
I'd like to select the k samples which are farthest apart/most geographically distributed within the Y samples I collected. Samples that are taken from plants in close proximity to each other are more likely to be closely related, and I'd like to sequence what are ultimately the most genetically diverse samples for my later analysis.
So, the question is: given a set of Y points/samples in 2d geographic space (lat/long coordinates), how do I select the k points that are most geographically distributed/distant from each other?
As I've explored this a bit, I think the problem I'm really having is how to define 'distance' or 'most distributed'. Some of the metrics I've though of (e.g. maximum average distance between the k points) result in really unintuitive point selection in certain cases (for example, if two points are right next to each other but far away from another cluster of all the other points, the two points will be included even if they're almost on top of each other).
I have a feeling that this is not an uncommon problem, and there must be good answers out there. I'll add that I have access to a large cluster and I can brute force the problem to some extent.
I'd appreciate any and all advice or discussion; this is an interesting theoretical topic to me.

Ambiguity between methods of calculating quartiles. Which method is more correct?
I am having confusion between two methods of calculating the first quartiles.
Let me explain by an example:
$A=[1,2,3,4,5]$
The method that I know:
To find the first quartile in the list above I first find the median = $3$ that is the 3rd element.
Now I split the list in the following fashion:
$(1,2)$
$3$
$(4,5)$
Now we take the list $(1,2)$ and we find the media between them that is $(1+2)/2=1.5$
So according to my calculation the first quartile $Q_1=1.5$
The nearest rank method
$n=\lceil <!--\; \lceil \; -->\frac{P}{100}\times <!--\; \times \; -->N\rceil <!--\; \rceil \; -->$
So for the above list the $0.25$ percentile or the first quartile $Q_1$ will be
$\lceil <!--\; \lceil \; -->\frac{25}{100}\times <!--\; \times \; -->5\rceil <!--\; \rceil \; -->=2$
which is 2nd position.
So which is correct $1.5$ or $2$ ? Or does chosing any of them is fine ?
If I am correct in my original calculation why am I getting a difference between the 2 methods.

Hypergeometric Distribution
I hate to be that guy who comes here and asks one question, however I've been stuck on this problem for quite some time, and I'm going to assume that I shouldn't even be using the binomial theorem at this point
I was given this question:
"A teacher was asked by her principal to select 7 students at random from her class to take a standardized math test, which will be used to determine how well students at that school are doing with respect to math (and correspondingly, how well the teacher is doing at teaching math). The teacher previously had rank ordered her students on the basis of their performance in her class on math tests, and divided the class into quartiles, such that there were 5 students in the upper quartile and 15 students in the lower three quartiles. When the teacher handed the principal the names of the students she had randomly selected from the class, the number of students from the upper quartile was 5, and the number of students from the lower three quartiles was 2. Is there any statistical evidence that the teacher is biased toward selecting students from the upper quartile?"
The first question given was:
"Compute the probability that, of the students selected, 0 are upper quartile students. Round off to 4 decimal places."
I was able to answer that question with the answer 0.0830, this i was able to answer by doing (15/20)(14/19)(13/18)(12/17)(11/16)(10/15)(9/14), however when it asks a question such as computing the probability that of the students selected, 2 are upper quartile students, I get stuck, I tried using the binomial theorem, but to no avail, perhaps I'm doing something wrong, I can't find a fast enough way to determine all of the sequences that will produce two upper quartile students of the 7 draws.

Find the $y$-intercept of a given parametric curve.
$x=t(t^2-<!--\; -\; -->3)$
$y=3(t^2-<!--\; -\; -->3)$

Suppose we have data set: $19,21,22,22,28,31,33,44,50$. Find the interquartile range of this set.
First solution: Firstly, we should find the $75$th percentile of this set. $0,75\cdot <!--\; \cdot \; -->9=6,75$ and rounding up this number to nearest whole we get $7$. So the $75$th percentile of this set is $33$. Secondly, should find the $25$th percentile of this set. $0,25\cdot <!--\; \cdot \; -->9=2,25$ and rounding up this number to nearest whole we get $3$. So the $25$th percentile of this set is $22$. Hence,
$\text{interquartile range}=Q_3-<!--\; -\; -->Q_1=33-<!--\; -\; -->22=11.$
Second solution: The median of this set is $28$. First quartile is the median of lower set. Hence $Q_1={\displaystyle \frac{21+22}{2}}=21.5$, third quartile is the median of upper set. Hence $Q_3={\displaystyle \frac{33+44}{2}}=38.5$
$\text{interquartile range}=Q_3-<!--\; -\; -->Q_1=38.5-<!--\; -\; -->21.5=17.$
Which one is correct? Please explain why one of the solutions is false.

Consider the surface given by the function $f(x,y)=\frac{4}{\sqrt{x^2+y^2-<!--\; -\; -->9}}$. What is the range of $f$? Determine any $y$-intercept(s) of $f$.
For range, I realize that $\sqrt{x^2+y^2-<!--\; -\; -->9}$ can be any number, so long as $x^2+y^2-<!--\; -\; -->9$ is positive or equal to zero. However, when looking at $f$, we know that the denominator cannot equal zero, which means that $\sqrt{x^2+y^2-<!--\; -\; -->9}$ can be any number greater than $0$. Would this mean that the range of $f$ is $(0,\mathrm{\infty <!--\; \infty \; -->})$?
For the $y$-intercept, I know that both $x$ and $z$ need to equal zero. Would this mean that there are no $y$-intercepts because $0$ is not in the range of $f$?

The graph of $r=\frac{-<!--\; -\; -->5}{2\cos <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->})+\sin <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->})}$ is a line. Find the y-intercept of this line.

What is the equation of the line with 5 and -3 as x- and y-intercept, respectively?

Standard Deviation after removing outlier.
I am totally new to statistics. I'm learning the basics.
I came upon this question while solving Erwin Kreyszig's exercise on statistics. The problem is simple. It asks to calculate standard deviation after removing outliers from the dataset.
The dataset is as follows: 1, 2, 3, 4, 10. What I did is, I found out qm = 3. Then $q$l $=\frac{1+2}{2}=1.5$ and $q$m $=\frac{4+10}{2}=7$
Now, $IQR=7-<!--\; -\; -->1.5=5.5$ and $1.5\ast <!--\; \ast \; -->IQR=8.25$
So, we can say numbers beyond $1.5-<!--\; -\; -->5.5=-<!--\; -\; -->4$ and $7+5.5=12.5$ will be an outlier.
Since there is no outlier, I found out the Standard Deviation of the set which is 3.53.
But, the answer provided is 1.29 which is different from the standard deviation of the set.
Can anyone help me what I missed?
Also, I have another question - we can see with plain eyes 10 is an outlier. But it is not detected here - why?

Quartiles and appending elements to an end of a stack
"The company at which Mark is employed has 80 employees, each of whom has a different salary. Mark’s salary of $43,700 is the second-highest salary in the first quartile of the 80 salaries. If the company were to hire 8 new employees at salaries that are less than the lowest of the 80 salaries, what would Mark’s salary be with respect to the quartiles of the 88 salaries at the company, assuming no other changes in the salaries?"
The answer is
"The fifth-lowest salary in the second quartile."
This doesn't make sense.
Imagine the employees ranked in order from highest salary to lowest salary. If we add 8 new employees who have salaries lower than the pre-existing 80, then we append 8 salaries to the bottom of the list. How does Mark's position in the queue change? Shouldn't his position stay the same?

What are the slopes and $y$-coordinates of the $y$-intercepts of the following lines.
$a)y=-<!--\; -\; -->3x+2$
$b)2y=-<!--\; -\; -->3x+2$
$c)5y=2-<!--\; -\; -->3x$
$d)y+3x-<!--\; -\; -->2=0$
And as the slope intercept form is $y=mx+b$.