Master the Art of Research Methodology: Comprehensive Examples and Expert Advice

I am confused about creating a Venn diagram from the question below from one of the past papers as there is a negative number involved. Is this possible in Venn diagrams?
A survey of students in a lecture revealed that a total of 15 played rugby, 30 played football and 35 played cricket. 4 people played all three sports, while 2 played only football and rugby, 14 played only cricket and football, and 6 played only cricket and rugby.
If 4 people played all three sports and 2 played only football and rugby, how is it possible to create the Venn diagram? Are negative numbers allowed, as 2−4=−2?

Ice cream flavorsA survey was taken among 600 middle school students for their preferred ice cream flavours.- 250 like strawberry- 100 liked both strawberry and vanilla- 130 like strawberry but not chocolate- 120 like only vanilla- 300 like only one flavor- 270 students said they liked vanilla- 30 students like all three flavorsQuestions:- How many do not like any of the 3 flavors?- How many like at least one of the 3 flavors?- How many like at least 2 flavors?- How many like chocolate?

The average value of a diagnostic test from $10$ students is $55$. If we add $5$ more student so the total number of students is $15$, the average value changes to $53$
How much is the average value from the last $5$ students?

An automotive service center would like feedback on its customer service. Each customer receives a printed card after they pay for services with a short survey on which customer service is ranked on a scale from 1 (least satisfied) to 5 (most satisfied). Customers can then choose to submit the card in a box on their way out of the store. The store manager finds that 140 cards were filled out in the past week, with an average customer satisfaction score of 2.47. Which type of bias is most likely to be present in the survey results?
A. This is undercoverage bias because some types of customers may not have received cards.
B.This is nonresponse bias because many customers may choose to not fill out the cards.
D.This is question wording bias because the customer service score levels may not be clear to customers.
C. This is response bias because customers may not be truthful about their experience with customer service. hp

Let's say you wanted to calculate, for example, the average (i.e. mean) cost per day for a hotel to accommodate a tourist. If only aggregate cost data is available, is dividing the average cumulative cost for one person's stay by the average person's duration of stay an acceptable method? Would using the median cost and median duration of stay be better? Worse? The same?

In a study of the effectiveness of wearing seat belts, a group of people who had survived car accidents in which they had not worn seat belts reported that seat belts would not have helped them. specify the type of bias involved

What is the argument for not using the average of an average?

New kind of identities?
I found a new kind of identities which are half logic and half algebraic while working on a proof of NP-completeness. They are like this:
$\frac{a+mb}{n+m}<\frac{a}{n}⟺<!--\; ⟺\; -->b<\frac{a}{n}$
It's the first time, I encounter such an identity. It has the amusing and counter-intuitive property that the right member is not modified, but the two left members are not algebraically equivalents.
Can you tell me if you ever encountereed such identities before? These ones in particular? (There are 9 in total: equality, strict inequality or not, changing way if $n+m$ and m have same sign or not.) If not would you have suggestions for naming this kind of identities?
Moreover it would be interesting to know if there is a finite number of identities of this kind, an infinite number but recursively enumerable, etc.
I do not ask for a proof [!] but only whether somebody knows of a systematic treatment of equivalences of this kind in the literature.
Feel free to search one but let me a few months to search by myself. If I don't have any idea to prove it, I'll update my question to let you know anyone is welcome to publish on the subject.
Update 2013/07/26:
egreg definitely found the good idea behind these identities. Let me update my question as follows : Does there exist an identity of the type
$a?b⟺<!--\; ⟺\; -->a?f(a,b)$
where f(a,b) is not a weighted mean of a and b (? may be =, <, >, <=, >=)?
Update 2013/07/30:
Thanks to zyx we now have a very simple way to construct an infinity of such identities. There remains a "few" questions:
Since it doesn't appear anyone know an article or a book that explicitely remarked these kind of identities as particular, would you have suggestions for naming this kind of identities? (Question A)
Are all such identities as zyx described (in the meaning $(b-<!--\; -\; -->a{)}^n\times <!--\; \times \; -->\dots <!--\; \dots \; -->$)? (Question B) If not, are these identities recursively enumerable? (Question C)
What are the examples of such identities that are important, useful for demonstrating mathematical results? (Question D) We already know that these identities from weigthed means are useful. Do you have other examples from classical proofs? (Question D')
Note that the answer to question B is no if you change (or extend) the rules ;). For example on rings instead of fields and if the inequality may be strict or not in the same identity. As an example, if we take the ring 2Z of even integers, we have the identity
$b<a⟺<!--\; ⟺\; -->(b-<!--\; -\; -->a{)}^3+8+a\le <!--\; \le \; -->a.$
So far, I only had the following idea for naming this kind of identities: "semi-invariant identities". It shouldn't be hard to find a better name.
Update 2013/07/31 Clarification :
For now, I would like to answer the previous questions on fields only BUT the final word on these identities is out of reach. Why?
The broader framework I see for these identities is the framework of universal algebras. Given such an algebra $\mathcal{A}$ of domain A and two binary relations ${?}_1$ and ${?}_2$ on A such that ${?}_1$ and ${?}_2$ are orders or equivalence relations (yes, you can mix an equivalence relation and an order relation if you want but I have no idea if it could yield something interesting on some algebra), the idea is to study the identities that are as follows:
Let $n\in <!--\; \in \; -->\mathbb{N}$, an n-${?}_1$-${?}_2$-semi-invariant identity on $\mathcal{A}$ is defined by three terms $f_1$, $f_2$, $f_3$ on $\mathcal{A}$ such that:
$\mathrm{\forall <!--\; \forall \; -->}(a_1,\dots <!--\; \dots \; -->,a_n)\in <!--\; \in \; -->A^n,f_1(a_1,\dots <!--\; \dots \; -->,a_n){?}_1{f}_2(a_1,\dots <!--\; \dots \; -->,a_n)⟺<!--\; ⟺\; -->f_1(a_1,\dots <!--\; \dots \; -->,a_n){?}_2{f}_3(a_1,\dots <!--\; \dots \; -->,a_n)$
where $f_2$ and $f_3$ are not algebraically equivalents (as I noted in the original question).
Clearly, this is too broad to ask a question in this framework right now. These identities may be interesting on algebras used to construct graphs, etc. But not many people would see an interest in it or see what I'm talking about. I would like to see what can be useful with these identities on the most standard algebras (fields is a good starting point) and it could benefit to many people not only a few dozens of specialists of some particular research domain. Following part of zyx idea, one could go even further by considering arbitrary relations.

Daphne mails a survey to 50,000 recent college graduates across America. The responses indicate overwhelmingly that graduates felt their education helped secure them a better job, but only 6% of the surveys were returned. What type of bias is most strongly present in Daphne’s study?
Self-Interest Study
Voluntary Response Bias
Loaded questions
Perceived lack of anonymity
Response Bias
Nonresponse Bias

Linear multivariate recurrences with constant coefficients
In the theory of univariate linear recurrences with constant coefficients, there is a general method of solving initial value problems based on characteristic polynomials. I would like to ask, if any similar method is known for multivariate linear recurrences with constant coefficients. E.g., if there is a general method for solving recurrences like this:
$f(n+1,m+1)=2f(n+1,m)+3f(n,m)-<!--\; -\; -->f(n-<!--\; -\; -->1,m),f(n,0)=1,f(0,m)=m+2.$
Moreover, is their any method for solving recurrences in several variables, when the recurrence goes only by one of the variables? E.g., recurrences like this:
$f(n+1,m)=f(n,2m)+f(n-<!--\; -\; -->1,0),f(0,m)=m.$
This second question is equivalent to the question, if there is a method of solving infinite systems of linear univariate recurrences with constant coefficients. That is, using these optics, the second recurrence becomes $f_m(n+1)=f_{2m}(n)+f_0(n-<!--\; -\; -->1),f_m(0)=m,m=0,1,\dots <!--\; \dots \; -->.$
I am not interested in a solution of any specific recurrence, but in solving such recurrences in general, or at least in finding out some of the properties of possible solutions. For instance, for univariate linear recurrences, each solution has a form $c_1{p}_1(n)z_1^n+\dots <!--\; \dots \; -->+c_k{p}_k(n)z_k^n,$, where $c_i$'s are constants, $p_i$'s are polynomials and $z_i$'s are complex numbers. Does any similar property hold for some class of recurrences similar to what I have written?
I have been googling a lot, but have found only methods for some very special cases (in monographs on partial difference equations, etc.), but nothing general enough. I am not asking for a detailed explanation of any method, but references to the literature would be helpful. I don't know much about transforms (like discrete Fourier transform or z-transform), but I found certain hints that there could be a method based on these techniques. Is it possible to develop something general enough using transform, i.e., is the study of transforms worth an effort (in the context of solving these types of recurrences)? However, it seems to me that the generalization of the characteristic polynomial method (perhaps, some operator-theoretic approach) could lead to more general results. Has there been any research on this topic?

In an Math Exam there are $80$ more men than women. The result showed that the women's average is $20\mathrm{\%<!--\; \%\; -->}$ higher than men's, and that the total average is $75\mathrm{\%<!--\; \%\; -->}$. what is the women's average?

A 2015-2016 report stated that hiring over the past year of people with a bachelor's degree increased 8 %, for people with a PhD increased 28 %, and for people with an MBA decreased 25 %. This was based on a voluntary poll of all recent graduates who interacted with at least one of 350 career service centers on college campuses. The survey was answered by 7,500 graduates. What are potential sources of bias with this survey?
Select all that apply.
Nonresponse bias
Response bias
Sampling bias

a) A school wants to gather parent opinions on the current cell phone policy at the school. They have sent letters with a survey for the parent to return to the school board office.
b) A survey about teaching strategies used in the "Secondary Classes" (7-12th grades) for a school district was handed out to a random sample of teachers from the three high schools in the district. Less than 20% of the surveys were returned.
c) A survey with the question "Do you agree that a national system of health insurance should be favored because it would provide health insurance for everyone and reduce administrative costs?"

What is the difference between the normal average and weighted average?

In survey question, what is Nonresponse bias?

We talk about Average speed and Average time, why not Average Distance?

Is anybody researching "ternary" groups?
As someone who has an undergraduate education in mathematics, but didn't take it any further, I have often wondered something.
Of course mathematicians like to generalize ideas. i.e. it is often better to define and write proofs for a wider scope of objects than for a specific type of object. A kind of "paradox" if you will - the more general your ideas, often the deeper the proofs (quoting a professor).
Anyway I used to often wonder about group theory especially the idea of it being a set with a list of axioms and a binary function $a\cdot <!--\; \cdot \; -->b=c$. But has anybody done research on tertiary (or is that trinary/ternary) groups? As in, the same definition of a group, but with a $\cdot <!--\; \cdot \; -->(a,b,c)=d$ function.
Is there such a discipline? Perhaps it reduces to standard group theory or triviality and is provably of no interest. But since many results in Finite Groups are very difficult, notably the classification of simple groups, has anybody studied a way to generalize a group in such a way that a classification theorem becomes simpler? As a trite example: Algebra was pretty tricky before the study of imaginary numbers. Or to be even more trite: the Riemann $\mathrm{\zeta <!--\; \zeta \; -->}$ function wasn't doing much before it was extended to the whole complex plane.
EDIT: Just to expand what I mean. In standard groups there is an operation $\cdot <!--\; \cdot \; -->:G\times <!--\; \times \; -->G\to <!--\; \to \; -->G$ I am asking about the case with an operation $\cdot <!--\; \cdot \; -->:G\times <!--\; \times \; -->G\times <!--\; \times \; -->G\to <!--\; \to \; -->G$

How do I determine sample size for a test?
Say you have a die with n number of sides. Assume the die is weighted properly and each side has an equal chance of coming up. How do I determine the minimum number of rolls needed so that results show an equal distribution, within an expected margin of error?
I assume there is a formula for this, but I am not a math person, so I don't know what to look for. I have been searching online, but haven't found the right thing.

Let $A$ and $B$ be some continuous random variables. We proceed as follows: we take a coin with bias $b$ and flip it. If heads, we inspect $A$, if tails we inspect $B$. Call this resulting random variable $C$.
Now say I can observe $C$ and want to figure out if the coin was heads or tails, i.e. I want to compute $Pr[$
One thought I had was to approximate the coin toss with a continuous random variable $K$ with pdf:
$k(x)=b,\text{ for }x\in <!--\; \in \; -->[-<!--\; -\; -->1,0]$
$k(x)=1-<!--\; -\; -->b,\text{ for }x\in <!--\; \in \; -->[0,1]$
Then one could compute the joint density function for $K$ and $C$ and compute $Pr[K<0|C=c]$ from there. But this looks clunky and ugly. Is there a better way?

Study of an infinite product
During some research, I obtained the following convergent product $P_a(x):=\underset{j=1}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\prod <!--\; \prod \; -->}}\cos <!--\; \; -->\left(\frac{x}{j^a}\right)(x\in <!--\; \in \; -->\mathbb{R},a>1).$.
Considering how I got it, I know it's convergent and continuous at 0 (for any fixed a), but if I look at $P_a$ now, it doesn't seem so obvious for me.
Try: I showed that $P_a\in <!--\; \in \; -->L^1(\mathbb{R})$, i.e. it is absolutely integrable on $\mathbb{R}$. Indeed, by using the linearization of the cosine function and the inequalities $\ln <!--\; \; -->(1-<!--\; -\; -->y)⩽<!--\; ⩽\; -->-<!--\; -\; -->y$ (for any $y<1$) and $1-<!--\; -\; -->\cos z⩾<!--\; ⩾\; -->z^2/2$ (for any real z), we obtain
$\begin{array}{rcl}{P}_a(x{)}^2& =& \underset{j=1}{\overset{\mathrm{\infty <!--\; \infty \; -->}}{\prod <!--\; \prod \; -->}}(1-<!--\; -\; -->\frac{1-<!--\; -\; -->\cos <!--\; \; -->(x{j}^{-<!--\; -\; -->a})}{2})⩽<!--\; ⩽\; -->\underset{j>|x{|}^{1/a}}{\prod <!--\; \prod \; -->}(1-<!--\; -\; -->\frac{1-<!--\; -\; -->\cos <!--\; \; -->(x{j}^{-<!--\; -\; -->a})}{2})⩽<!--\; ⩽\; -->\exp <!--\; \; -->(-<!--\; -\; -->C|x{|}^{1/a}),\end{array}$ for some absolute constant $C>0$. However, I have not been able to use this upper bound to prove continuity at 0 (via uniform convergence for example, if we can).
Question : I wanted to know how to study $P_a$ (e.g. its convergence and continuity at 0), if you think it has an other form "without product" (or other nice properties) and finally if anyone has already seen this type of product (in some references/articles), please.