Dive Deep into Survey Questions with Comprehensive Examples and Expert Assistance

In a random survey 250 people participated. Out of 250 people who took part in the survey, 40 people listen to Pink Floyd. 30 people listen to Metallica and 20 people listen to John Denver. If 10 people listen to all three then find the no. of people who listen only Pink Floyd.

What is key in user research?

Is the invariant subspace problem open for invertible maps?
Let $T:H\to <!--\; \to \; -->H$ be a bounded linear operator with bounded inverse on the separable complex Hilbert space. Does T preserve a closed proper non-trival invariant subspace?
I'm aware the question is (famously) open for bounded linear maps, and of partial results, but no survey (or Tao's blog, etc) seem to address the invertible case.
If it is open, does a positive or negative answer imply the answer in the non-invertible case?

Finite Math (Probability/Venn Diagram)
13) A survey revealed that 25% of people are entertained by reading books, 39% are entertained by watching TV, and 36% are entertained by both books and TV. What is the probability that a person will be entertained by either books or TV? Express the answer as a percentage.
Is this problem stated correctly? How can 36% of the people be entertained by both books and TV, when only 25% of the people are entertained by reading books?
EDIT
Here are two other questions from the exam, that the instructor said followed the same logic as the question above.
14) Of the coffee makers sold in an appliance store, 5.0% have either a faulty switch or a defective cord, 1.6% have a faulty switch, and 0.2% have both defects. What is the probability that a coffee maker will have a defective cord? Express the answer as a percentage.
15) A survey of senior citizens at a doctor's office shows that 42% take blood pressure-lowering medication, 45% take cholesterol-lowering medication, and 13% take both medications. What is the probability that senior citizen takes either blood pressure-lowering or cholesterol-lowering medication? Express the answer as a percentage.

Homotopy equivalence between two mapping tori of compositions
For any maps $s:X\to <!--\; \to \; -->K$ there is defined a homotopy equivalence
$T(d\circ <!--\; \circ \; -->s:X\to <!--\; \to \; -->X)\to <!--\; \to \; -->T(s\circ <!--\; \circ \; -->d:K\to <!--\; \to \; -->K);(x,t)\mapsto <!--\; \mapsto \; -->(s(x),t).$
Here, T(f) denotes the mapping torus of a self-map $f:Z\to <!--\; \to \; -->Z$ (not necessarily a homeomorphism). It is very surprising to me that this holds with no extra conditions on d and s. I'm guessing that the homotopy inverse is the map:
$T(s\circ <!--\; \circ \; -->d)\to <!--\; \to \; -->T(d\circ <!--\; \circ \; -->s),(k,t)\mapsto <!--\; \mapsto \; -->(d(k),t).$
If the above is a genuine homotopy inverse, then the map:
$(x,t)\mapsto <!--\; \mapsto \; -->(d(s(x)),t)$
would have to be homotopic to the identity somehow. However, after banging my head against the wall on this for a while I can't come up with a valid homotopy. So my questions are:
Is the map $T(s\circ <!--\; \circ \; -->d)\to <!--\; \to \; -->T(d\circ <!--\; \circ \; -->s)$ I've defined above actually a homotopy inverse? If so, what is the homotopy from the composition I wrote down above to the identity map?
Is there a better one that makes the homotopy obvious?

How to identify and distinguish a sample and population mean?
In a village the mean rent was 1830; a rental company comes and surveys with a sample size of 25 for a hypothetical testing to test if the μ is equals 1830 or not.
This new survey gets a mean value of 1700 with a standard deviation of 200.
For the company to deduce t-value(for hypothesis testing) which is sample mean and which will be population mean? I assumed 1830 to be the population mean but it turned out to be the sample mean and 1700 to be the population mean.

I am self studying, and have the answer to this question at the back of the book. The question is as follows (paraphrase):
A survey of chemical research workers has shown on average that each man requires no cupboard 60% of the time, one cupboard 30% of the time and two cupboards 10% of the time; three or more were never needed. If four chemists work independently, how many cupboards should be provided so that there are adequate facilities at least 95% of the time?
The answer is that 4 cupboards are enough for 95.85% of the time. I just do not know how to get the answer.

A survey is conducted and 3 people are to be chosen from a group of 20.
a) In how many different ways could the 3 be chosen? (1140)
b) If a group contains 8 men and 12 women, how many groups containing exactly 1 man are possible? (528)
A solution for question b) is wanted. How can I solve this problem?

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?

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)

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 $$\frac{24}{120}\times <!--\; \times \; -->1+\frac{10}{57}\times <!--\; \times \; -->2.$$

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.

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

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.

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%

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?

I am studying for the p exam and realized I really need to brush up on my basic set theory, and am having trouble with this question.
In a survey on Popsicle flavor preferences of kids aged 3-5, it was found that:
- 22 like strawberry.
- 25 like blueberry.
- 39 like grape.
- 9 like blueberry and strawberry.
- 17 like strawberry and grape.
- 20 like blueberry and grape.
- 6 like all flavors.
- 4 like none.
Apparently the answer is 50, but I can't seem to figure out how to arrive at this

A space X is unicoherent if whenever A,B are closed connected subsets of X such that $A\cup <!--\; \cup \; -->B=X$, their intersection $A\cap <!--\; \cap \; -->B$is connected. The survey "A Survey on Unicoherence and Related Properties" by Garcia-Maynez and Illanes says that intuitively, a unicoherent space is a space with no "holes". For example, the closed disk in the plane is unicoherent, but the circle $S^1$ is not. This sounds like simple connectedness. For simple connectedness, by van Kampen's theorem, we have the following: if a space X has simply connected open subsets U,V such that $U\cup <!--\; \cup \; -->V=X$ and $U\cap <!--\; \cap \; -->V$ is nonempty and path-connected, then X is simply connected. It is then natural to ask a similar question for unicoherence. So, my question is: if a space X has closed unicoherent subsets A,B such that $A\cup <!--\; \cup \; -->B=X$ and $𝐴\cap 𝐵$ is connected, is X necessarily unicoherent? If necessary, you can assume that A,B are open rather than closed, or replace unicoherence conditions with open unicoherence conditions. Also, you can assume that $𝐴\cap 𝐵$ is nonempty if needed. Thank you in advance!