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

I have a few basic probability questions, concerning the following question:
An internet survey estimates that, when given the choice between David letterman and Jay Leno, 52% of the population prefers to watch Jay Leno. Three late night TV watchers are randomly selected and asked which of the two talk show hosts they prefer. Find the probability distribution of Y, the number of viewers in the sample who prefer Leno.
The answer is:
P(Y=0) = (0.48)${}^3$
P(Y=1) = 3 * (0.48)${}^2$ (0.52)
P(Y=2) = 3 * (0.48) (0.52)${}^2$
P(Y=3) = (0.52)${}^3$
I understand the solution mostly, except for where the 3 comes from in P(Y=1) and P(Y=2). Is it because there are 3 ways of getting 1 and 2 viewers who prefer Leno in the sample space? If so, is there a mathematical way of calculating this or is it simply writing out the entire sample size/combinations, and then counting?

Сalculate a windowed average where the newest value replaces the oldest value in a set of size $10$. In this case, I have a real-time number stream and have access to current value, current average and the previous value.Is there any way to calculate this value incrementally storing minimal amount of historical state?

Grassmannians are a pretty useful subject in numerous fields of mathematics (and physics). In fact, it was the first non-trivial higher-dimensional example that was given in an introductory projective geometry course during my education.
Later I learned you can use them to define universal bundles and that they are playing a role in higher-dimensional geometry and topology. Though I have never came across a book or a survey article on the geometry and topology of those beasts. The field is a little wide, so let me specify what I am interested in:
Topology and Geometry of Grassmannians $G_k({\mathbb{R}}^n)$ or $G_k({\mathbb{C}}^n)$
Connections with bundle and obstruction theory.
Differential Topology of $G_k({\mathbb{R}}^n)$ or $G_k({\mathbb{C}}^n)$ (for instance, are there exotic Grassmannians).
Homotopy Theory of $G_k({\mathbb{R}}^n)$ or $G_k({\mathbb{C}}^n)$.
Algebraic Geometry of $G_k(V)$, where V is a n-dimensional vectorspace over a (possible characteristic $\ne <!--\; \ne \; -->0$ field F)

Explain why taking an average of an average usually results in a wrong answer? Is there ever a case where the average of the average can be used correctly?

One of your employees has suggested that your company develop a new product. A survey is designed to study whether or not there is interest in the new product. The response is on a 1 to 5 scale with 1 indicating definitely would not purchase, · · ·, and 5 indicating definitely would purchase. For an initial analysis, you will record the responses 1, 2, and 3 as No, and 4 and 5 as Yes.
a. 5 people are surveyed. What is the probability that at least 3 of them answered Yes?
b. 100 people are surveyed. What is the approximate probability that between 45% to 52% of people answered Yes?
For part a) There are 5 choices that people can respond by 1, 2 ,3 ,4 and 5. Since 1, 2 and 3 are considered "No", the probability of someone answering "No" is 3/5. For choices 4 and 4, the probability of someone responding with that is 2/5.
This looks like it fallows a binomial distribution so I calculated the probability of P(3) + P(4) + P(5).
However for part b), I'm confused. I can calculate the probability of someone saying yes but I don't know how to calculate the probability that a percentage of people saying yes. Does anyone know how to approach this question? I though about using the Z table, but that already calculates area.

I'm writing a software algorithm at the moment which compares survey answers.
Questions have 5 possible answers, and a respondent could choose between 1 and 5 answers.
What I'd like to do, for each respondent, for each question, is calculate how strongly they feel about their answer.
I propose to do this by counting the number of answers they selected.
For example, suppose the question was:
Which of these colours do you like?
And the possible answers were:
a) Blue b) Red c) Green d) Orange e) Purple
Then someone who answers only b), feels more strongly about their answer than someone who selected all 5.
What I'm having trouble with, is how to account for the different numbers of possible answers. If a question has a yes or no answer, and the respondent only chooses one answer, we want to consider this as being less significant than if they only choose one answer on a question with more possible options.
So the more options to choose from, the higher the importance of each answer selected.
So far, I've come up with this:
$$i=\text{max}-<!--\; -\; -->(\frac{g}{n}\ast <!--\; \ast \; -->\text{max})$$
Where:
i (importance)
max (maximum importance %) =100
p (possible answers)
g (given answers)
Can anybody suggest how I could approach this problem differently or improve the formula?

What is the expected value of the numbers of calls for a survey?
I was asked the following question by a medical doctor:
He is working on a survey. Assume that he called 120 persons and 24 persons agreed to do the survey. The second time, he called 57 persons among those who refused the first time and 10 persons agreed to do the survey. What is the expected value of the number of phone calls for people to agree to do the survey?
I first thought it would be
$$\frac{24\times <!--\; \times \; -->1+10\times <!--\; \times \; -->2}{120}.$$
Then I thought it would be $$\frac{24}{120}\times <!--\; \times \; -->1+\frac{10}{57}\times <!--\; \times \; -->2.$$

A recent poll of 1,500 randomly selected eligible voters where asked the question "Voting is an important civil duty. Did you vote in the last election?" Only 525 (35 percent) said that they did not vote in the last election. However, a vote count showed that 80 percent of eligible voters actually did not vote in the last election. Which of the following types of bias is most likely to have occurred in the poll?
1)Nonresponse bias
2)Undercoverage bias
3)Voluntary Response bias
4)Response Bias

We have a current rating average: $3.3/5$
Remove a rating of $4$. How do I find the new average?

Find the probability of using debit card
Our professor gave this question as assignment, My question is does this question have all information needed to solve it? I asked her and she said it has all information but still I cannot figure out how I can solve it with just knowing the probability of using debit card in supermarket. Can someone please provide me a tip on how I can solve this question with this information or is it solvable?
According to a recent Interact survey, 28% of consumers use their debit cards in supermarkets. Find the probability that they use debit cards in only other stores or not at all.

Is it possible to calculate the regular average of a sequence of numbers when i dont know everything of the sequence, but just everytime i get a new number i know the total count of numbers and the average for the numbers $-<!--\; -\; -->1$.
for example: $2,3,10$ the average is of course: $5$
but in the last step to calculate i only have access to the previous average of $2$ and $3:2.5$ the next number: $10$ and the count of numbers: $3$
if this is possible, how?

The question goes ,"In a random survey of 1000 people, it is found that 7% have lung problem.Of those who have lung problems,40% are heavy smokers,50% are moderate smokers and 10% are non-smokers.For those who do not have any lung problems,10% are heavy smokers,70% are moderate smokers and 20% are non-smokers.If a person is found as a heavy smoker,what is the probability of that person having a lung problem?"
I don't know how exactly to tackle this problems,any help will be appreciated

Is there any precise rigorous definition of "Average of a statistical population" in statistics (regardless of our knowledge about mean, median or mode)?

There was a question in my exam discrete maths that I just couldn't figure out. I know it's supposed to be solved using the inclusion-exclusion principle. anyone able to help me solve and understand this question.
For a survey, 200 people are asked about which forms of transport the had used in the last month it was found that
150 used trains,
80 had cycled and used trains, and
180 had used one or the other of these two forms of transport.
Question: How many people had cycled in the last month?
Question: How many had not used either form of transport.

$2+3+\frac{4}{3}=3$(arithmetic mean)
$4miles-<!--\; -\; -->2miles/2hours-<!--\; -\; -->1hour=2milesperhour$(average rate of change)
What is the relationship between the average (arithmetic mean) and the average rate of change?

The average weight of a group of boys and girl is $38$ kg.
The average weight of boys is $42$ kg and that of girls is $33$ Kg.
If the number of boys is $25$, then the number of girls is?

In order to estimate the proportion of students at a college who spend more than 2 hours per day on Facebook, a random sample of students at the college is selected and each student is interviewed. The students conducting the survey are worried that people who spend a lot of time on Facebook may be embarrassed to admit it and their responses may not be honest.
What type of bias are the students conducting the survey worried about?
(a) Selection bias
(b) Nonresponse bias
(c) Measurement bias
(d) Bias due to confounding eonnsbrialduen a o Jososs ine
(e) There is no obvious source of bias

Confidence Interval question about house holds
I was wondering if my calculation is right for this question
A recent survey of 800 Irish households found that 74% of households have a fixed line telephone. Infer a 95% confidence interval for the percentage of the total population of Irish households that have a fixed line telephone.
N = 800
P = .74
Q = 1 - .74 = .26
$\sqrt{\frac{74\ast <!--\; \ast \; -->.26}{800}}=0.015$
95% Z_Score = 1.96
74+%/- 1.96 * 0.015 = 0.03
74%+/-3%

Problem where it gives us average value over $3$ sub intervals that are contained within an entire interval. To find the average value of the entire interval is it possible to just add the average values of the $3$ sub intervals or is that not right?

I have some practice homework questions. I did the first one I will go over the steps please tell me If I am doing it right.
a) As mentioned earlier, it is claimed that 70% of households in Ontario now own large-screen TVs. You would like to verify this statement for your class in mass communications. If you want your estimate to be within 5 percentage points, with 99 per cent level of confidence, how large of a sample must be acquired?
So for this i knew the targe parameter was p so the standard error is $\sqrt{\frac{pq}{n}}$ and that is equal to 0.05 and since we are finding the 99% C.I. our $Z_{\mathrm{\alpha <!--\; \alpha \; -->}}/2$ = 2.575
so my formula became
$0.05=2.575\cdot <!--\; \cdot \; -->\sqrt{\frac{(0.7)(0.3)}{n}}\Rightarrow <!--\; \Rightarrow \; -->n=556.07=557$
b) You are to conduct a sample survey to determine the mean annual family income in a rural area. The question is, “How many families should be sampled?” In a pilot study of just 10 families, the standard deviation of the sample was $500. The sponsor of the survey wants to use the 95 per cent confidence level. The estimate is to be within $100. How many families’ should be interviewed?
For this I kinda got confused because it said how many family should be interviewed but we are given 10 families? but I was going to follow my procedure as in a but I didnt know where the n would come from because we are give the S.D. ?
or would it be this
$100=1.96\cdot <!--\; \cdot \; -->\frac{500}{\sqrt{n}}$ and solve for n afterwards?
c) A student conducted a study and reported that an 80% confidence interval ranged from 48 to 52. He was sure the sample standard deviation was 16 and that the sample was at least 30, but could not remember the exact number (i.e., the size of the sample). Can you help him out?
for this since we are finding the 80% CI our area is 0.9 and our critical value now would be 1.285.
ans since we know it ranged from 48 to 52 our mean is $\frac{48+52}{2}=50$ and then that is equal to
$50=1.285\cdot <!--\; \cdot \; -->\frac{16}{\sqrt{n}}$ and then solve for n only confusion was is that number 50 right?