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

Multiple Choice Question
Identify the bias that is most present in the following scenario:
Students are asked by their teacher whether they had ever cheated on a test and are required to respond.
A. Response
B. Nonresponse
C. Selection

The average number of students per class at school $X$ is $25$ and the average number of students per class at school $Y$ is $33$. Given that there are more classes in school $X$ than in school $Y$, is the average number of students per class for both schools combined less than $29$?

In Traditional Survey Reseach, elaborate what are;
1. Nonresponse Bias
2. Refusal rate
3. Response Bias

A random sample of 5,000 people is selected from local telephone book to participate in financial planning survey to help infer about typical saving and spending habits. three thousand return the questionnaire. Describe the types of bias that may result in at least 4-5 complete sentences.

From $10$ numbers $a,b,c,...j$ all sets of $4$ numbers are chosen and their averages computed. Will the average of these averages be equal to the average of the $10$ numbers?

Is a weighted average, assuming the weights sum up to one, always smaller than the unweighted arithmetic average? As if the weights are smaller than one the weighted mean is a convex combination of the individual points. Is there any condition on the individual observation for the statement to hold true?

If I found that a series converges, how can I know to what number it's converging to?
I started learning series in calculus and I have trouble catching a basic concept. When I try to find if a series converges or diverges I have many ways to go about it. If I see that the series diverges than I stop there. If I see that the series converges than there is a number it's converging to right?
For example: $\sum <!--\; \sum \; -->\frac{2}{n^3+4}$. I do the limit comparison test with the series $\sum <!--\; \sum \; -->\frac{1}{n^3}$ and get a finite number 2. I know that $\sum <!--\; \sum \; -->\frac{1}{n^3}$ converges, so now I know that $\sum <!--\; \sum \; -->\frac{2}{n^3+4}$ converges also. How do I know to what number it converges to?

There is a box with $12$ dice which all look the same. However there are actually three types of dice:
$6$ normal dice. The probability to get a $6$ is $1/6$ for each dice.
$3$ biased dice. The probability to get a $6$ is $0.85$.$3$ biased dice. The probability to get a $6$ is $0.05$.You take a die from the box at random and roll it.What is the conditional probability that it is of type $b$, given that it gives a $6$?

Which statement is true with regard to non-response bias?
1.Respondents have strong opinions about the subject matter.
2.Respondents change their answers to influence the results.
3.Respondents give the answer they think the questioner wants.
4.Respondents do not give their honest opinion.

Find the find the quadratic average and the geometric average. To do this I have these informations :
The standart deviation, the arithmetic average and the number of values.

Probability of 1 billion monkeys typing a sentence if they type for 10 billion years
Suppose a billion monkeys type on word processors at a rate of 10 symbols per second. Assume that the word processors produce 27 symbols, namely, 26 letters of the English alphabet and a space. These monkeys type for 10 billion years. What is the probability that they can type the first sentence of Lincoln’s “Gettysburg Address”?
Four score and seven years ago our fathers brought forth on this continent a new nation conceived in liberty and dedicated to the proposition that all men are created equal.
Hint: Look up Boole’s inequality to provide an upper bound for the probability!
This is a homework question. I just want some pointers how to move forward from what I have done so far. Below I will explain my research so far.
First I calculated the probability of the monkey 1 typing the sentence (this question helped me do that); let's say that probability is p:
$P(\text{Monkey 1 types our sentence})=P(M_1)=p$
Now let's say that the monkeys are labeled $M_1$ to $M_{{10}^9}$, so given the hint in the question I calculated the upper bound for the probabilities of union of all $P(M_i)$ (the probability that i-th monkey types the sentence) using Boole's inequality.
Since $P(M_i)=P(M_1)=p$,
$P\left(\underset{i}{\bigcup <!--\; \bigcup \; -->}{M}_i\right)\le <!--\; \le \; -->\underset{i=1}{\overset{{10}^9}{\sum <!--\; \sum \; -->}}P(M_i)=\stackrel{{10}^9}{\sum <!--\; \sum \; -->}p={10}^{9}p$
Am I correct till this point? If yes, what can I do more in this question? I tried to study Bonferroni inequality for lower bounds but was unsuccessful to obtain a logical step. If not, how to approach the problem?

Calculating conditional probability given Poisson variable
I encountered a set of problems while studying statistics for research which I have combined to get a broader question. I want to know if this is a solvable problem with enough information specifically under what assumptions or approach.
Given that a minesweeper has encountered exactly 5 landmines in a particular 10 mile stretch, what is the probability that he will encounter exactly 6 landmines on the next 10 mile stretch. (Average number of landmines is 0.6 per mile in the 50 mile stretch)
I have figured that the approach involves finding out the Poisson probabilities of the discrete random variable with the combination of Bayes Conditional probability. But am stuck with proceeding on applying the Bayes rule. i.e $Pr(X=6\mid <!--\; \mid \; -->X=5)$.
I know that $Pr(X=5)=e^{-<!--\; -\; -->6}{5}^6/5!.$ Here $\mathrm{\lambda <!--\; \lambda \; -->}=0.6\cdot <!--\; \cdot \; -->10$ and $X=5$) Similarly for $Pr(X=6)$. Is Bayes rule useful here: $P(Y\mid <!--\; \mid \; -->A)=Pr(A\mid <!--\; \mid \; -->Y)Pr(Y)/(Pr(A\mid <!--\; \mid \; -->Y)Pr(Y)+Pr(A\mid <!--\; \mid \; -->N)Pr(N))$?
Would appreciate any hints on proceeding with these types of formulations for broadening my understanding.