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

So in my stats book I am told the following: A type I error is a false positive meaning you reject the null hypothesis when it's true. A type II error is a false negative meaning you reject the alternative hypothesis when it's true. My question is what causes these errors to occur? Is it simply bad sampling methods resulting in your getting biased data that causes you to get a skewed test statistic value and make the wrong conclusions? What causes these types of errors in hypothesis testing?

How Can I calculate the standard deviation of a yes/no question?
Is it possible to calculate the standard deviation of a single yes/no question? For example, 100 people took the survey and 60 of them chose "yes", 40 of them chose "no". I'm thinking using binomial distribution sd formula: $$\sqrt{np(1-<!--\; -\; -->p)}$$ so it would be $\sqrt{100\ast <!--\; \ast \; -->(60/100)\ast <!--\; \ast \; -->(1-<!--\; -\; -->60/100)}$is this correct?

We want to calculate the average lifetime of our customers. For each customer we know when they joined, whether they are still with us and if not, what date they left us. How can this be calculated?

How do I 'reverse engineer' the standard deviation?
My problem is fairly concrete and direct.
My company loves to do major business decisions based on many reports available on the media. These reports relates how our products are fairing in comparison to the competitor's offerings.
The latest report had these scores (as percentages):
Input values (%) 73.5, 16.34, 1.2, 1.15, 0.97, 0.94, 0.9, 0.89, 0.81, 0.31
Our product in the 'long' tail of this list. I argued with them that besides spots #2 and #1 all the other following the tail are on a 'stand', since, probably, the standard deviation will be much bigger that the points that separates everyone in the tail.
So the question is: How may I calculate the standard deviation having only these percentual values available?

As the randomly selected telephone numbers are called repeatedly till answer is received, the chance of selection bias is reduced.
However, due to telephone survey, the section of the population without telephones cannot be reached, introducing selection bias from a different angle.
Moreover, if the survey is carried out within a short time of the year and not over the whole year, it might over-represent the opinions of individuals born at a certain time of the year, while under-representing those born at other times. This again leads to selection bias.
It is also known that people may not always respond truthfully, thus leading to nonresponse bias

Problem with complex eigenvalue in these Researches?
It's about a vibration model call Hoffmann with the equations of motion:
$\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]\left(\frac{\stackrel{¨<!--\; ¨\; -->}{x}}{\stackrel{¨<!--\; ¨\; -->}{y}}\right)+\left[\begin{array}{cc}2& 1-<!--\; -\; -->\mathrm{\Delta <!--\; \Delta \; -->}\\ 1& 2\end{array}\right]\left(\frac{x}{y}\right)=0$
And then they calculated the complex eigenvalue:
$s_{1,2}={\pm <!--\; \pm \; -->[2\pm <!--\; \pm \; -->\sqrt{1-<!--\; -\; -->\mathrm{\Delta <!--\; \Delta \; -->}}]}^{\frac{1}{2}}$
I really don't undersatand how did they calculate this formula. Because i think the complex eigenvalue in this situation must be looked like this:
$s_{1,2}=2\pm <!--\; \pm \; -->\sqrt{1-<!--\; -\; -->\mathrm{\Delta <!--\; \Delta \; -->}}$

A recent survey of post-secondary education students in Canada revealed that 73% know what type of job they want, when they graduate. You are to randomly pick 32 post-secondary education students across the country, and ask each the following question: Have you selected a particular career path?
You have defined the random variable X to represent the number, out of 32 post-secondary students chosen, who responded YES
If you continued to select additional students, what is the probability that the 50th student selected will be the 39th student to respond YES?

Which summary statistic and hypothesis test should be used?
A survey is conducted on 398 people that are either of type A (198) or type B (200) and answer 1 question on a 5 point scale 1=strongly agree, 2= agree,...5=strongly disagree.
What is the correct summary statistic and what hypothesis test should be used in answering: if the 2 types of people have differing opinions regarding the question?
attempt
Summarize using a 2 way table of type vs response (1,2,...,5), with a vector $\stackrel{^<!--\; ^\; -->}{p}=\left(\begin{array}{c}{p}_1\\ p_2\\ p_3\\ p_4\\ p_5\end{array}\right)$ as the summary statistic
This seems like a test of homogeneity is appropriate to answer the question but these people are drawn from the same population and cross-classified (right?) so then a test of independence (however a test of indep is unrelated to the question?)

Australia is having a national "postal survey" about same-sex marriage.
The options on the ballot paper are "Yes" and "No", but voting is not compulsory.
The survey is being conducted over 8 weeks during which eligible voters can return their response.
At the end of the third week, 60% of voters have returned their response.
Exit-polling indicates of those who have already voted, 60% voted "Yes" and 40% voted "No".
Question: All other things being equal (e.g. early/late responders no indication of which way they vote) what % of the remaining 40% who haven't returned their response would need to vote "No" in order to achieve parity with the "Yes" vote (here's the kicker...) taking into account that not all eligible voters will ultimately return their ballots?
Is this work-outable?

I recently have this question:
I have a bag of toys. 10% of the toys are balls. 10% of the toys are blue.
If I draw one toy at random, what're the odds I'll draw a blue ball?
One person provided an answer immediately and others suggested that more details were required before an answer could even be considered. But, there was a reason I asked this question the way that I did.
I was thinking about probabilities and I was coming up with a way to ask a more complicated question on math.stackexchange.com. I needed a basic example so I came up with the toys problem I posted here.
I wanted to run it by a friend of mine and I started by asking the above question the same way. When I thought of the problem, it seemed very clear to me that the question was "what is $\mathbb{P}(blue\cap <!--\; \cap \; -->ball)$." I thought the calculation was generally accepted to be
$$\mathbb{P}(blue\cap <!--\; \cap \; -->ball)=\mathbb{P}(blue)\cdot <!--\; \cdot \; -->\mathbb{P}(ball)$$
When I asked my friend, he said, "it's impossible to know without more information." I was baffled because I thought this is what one would call "a priori probability."
I remember taking statistics tests in high school with questions like "if you roll two dice, what're the odds of rolling a 7," "what is the probability of flipping a coin 3 times and getting three heads," or "if you discard one card from the top of the deck, what is the probability that the next card is an ace?"
Then, I met math.stackexchange.com and found that people tend to talk about "fair dice," "fair coins," and "standard decks." I always thought that was pedantic so I tested my theory with the question above and it appears you really need to specify that "the toys are randomly painted blue."
It's clear now that I don't know how to ask a question about probability.
Why do you need to specify that a coin is fair?
Why would a problem like this be "unsolvable?"
If this isn't an example of a priori probability, can you give one or explain why?
Why doesn't the Principle of Indifference allow you to assume that the toys were randomly painted blue?
Why is it that on math tests, you don't have to specify that the coin is fair or ideal but in real life you do?
Why doesn't anybody at the craps table ask, "are these dice fair?"
If this were a casino game that paid out 100 to 1, would you play?
This comment has continued being relevant so I'll put it in the post:
Here's a probability question I found online on a math education site: "A city survey found that 47% of teenagers have a part time job. The same survey found that 78% plan to attend college. If a teenager is chosen at random, what is the probability that the teenager has a part time job and plans to attend college?" If that was on your test, would you answer "none of the above" because you know the coincident rate between part time job holders and kids with college aspirations is probably not negligible or would you answer, "about 37%?"

A survey was conducted of graduates from 30 MBA programs. On the basis of the survey, assume that the mean annual salary for male and female graduates 10 years after graduation is $168,000 and $117,000, respectively. Assume the standard deviation for the male graduates is $40,000, and for the female graduates it is $25,000.
Question 1: What is the probability that a simple random sample of 40 male graduates will provide a sample mean within $10,000 of the population mean, $168,000?
Question 2: What is the probability that a simple random sample of 40 female graduates will provide a sample mean within $10,000 of the population mean of $117,000?

Statistical analysis of study with categorical and numerical variables
I am researching the effect of a certain innovation type on firm performance. The innovation type is measured through a 6-item survey with nominal answers (yes/no; 1/0) and is retrospective (e.g. Did you introduce XY in the last 5 years). For firm performance I have financial data for the 5 year period I'm interested in. Now, there are two possible approaches I could take:
1) I compute an "innovation" variable from the survey answers, to distinguish between adopters and non-adopters and I examine whether the adopter-group shows to have better firm performance than the non-adopter group. Which type of analysis would this be? And how would I control for firm size and time effects?
2) I investigate whether firms that answered more questions with "yes" perform better than firms that answered with fewer "yes". Which type of analysis would this then be? Regression?

Just want to check if my answer and reasoning is correct for the following problem (Not a homework problem - it is a sample question for a test I'm preparing for)
In a survey, viewers were given a list of 20 TV Shows and are asked to label 3 favourites not in any order. Then they must tick the ones that they have heard of before, if any. How many ways can the form be filled, assuming everyone has 3 favourites?
My reasoning:
1) Choose 3 shows out of 20: c(20,3)
2) Choosing 0-17 shows from 17 choices: $c(17,0)+c(17,1)+c(17,2)+...+c(17,16)+c(17,17)$
and add 1) and 2) together for the final answer.
Would this be correct? Is there a better way of doing the second part that doesn't involve so many calculations?

I currently have this question..
A survey was conducted that found 72% of respondents liked the new motorway. Of all respondents, 65% intend to drive more. Suppose that 81% of those who like the new motorway intend to drive more.
I get rather confused with how the 65% and 81% intertwine. I assume I'm working backwards to find out the percentage of those who don't like the new motorway but intend to drive more.
Let l = like, d = drive..
pr(l) = 0.72, pr(l') = 0.28
Would I be right in claiming that pr(d) = 0.65 therefore pr(d/l) = 0.65/0.72 ?

I have a bag of toys. 10% of the toys are balls. 10% of the toys are blue.
If I draw one toy at random, what're the odds I'll draw a blue ball?

I have a result that uses ordered fields. However, I am ignorant of the literature surrounding ordered fields. I have read the basic facts about them, such as those contained in the wikipedia page. However, in the introduction, I would like to provide some motivation.
My result has the property that if it holds for some ordered field k, then it holds for any ordered subfield of k. Does that mean it is enough to prove my result for real-closed ordered fields?
I also read the statement that to prove a 1st order logic statement for a real-closed ordered field, then it is enough to prove it for one real-closed ordered field, such as R for instance. Can someone please provide a reference for that?
A mathematician mentioned to me a classical link between iterated quadratic extensions and ordered fields. What is the precise statement please?
I would also like a number of interesting examples of ordered fields. I know for example of an interesting non-archimedean example using rational functions (that I have learned about from the wikipedia page on ordered fields).
Does anyone know of a survey on ordered fields, or a reference containing answers to my questions above?

Bayes' Rule with a binomial distribution
The question is as follows:
Each morning, an intern is supposed to call 10 people and ask them to take part in a survey. Every person called has a 1/4 probability of agreeing to take the survey, independent of any other people’s decisions. Eighty percent of the days, the intern does what they are supposed to; however, the other twenty percent of the time, the intern is lazy and only calls 6 people. One day, exactly 1 person agree to take the survey. What is the probability the intern was lazy that day?
My solution is partially shown using Bayes' Rule with the binomial distribution. Let L be the event he is lazy and let O be the event of one person agreeing to the survey. Then,
$$\mathbb{P}(L\mid <!--\; \mid \; -->O)=\frac{\mathbb{P}(O\mid <!--\; \mid \; -->L)\mathbb{P}(L)}{\mathbb{P}(O\mid <!--\; \mid \; -->L)\mathbb{P}(L)+\mathbb{P}(O\mid <!--\; \mid \; -->L^C)\mathbb{P}(L^C)}.$$
We know
$$\mathbb{P}(O\mid <!--\; \mid \; -->L)=\frac{\mathbb{P}(O\cap <!--\; \cap \; -->L)}{\mathbb{P}(L)}=\frac{(\genfrac{}{}{0ex}{}{6}{1})\left(\frac{1}{4}\right){\left(\frac{3}{4}\right)}^5}{\frac{1}{5}}\approx <!--\; \approx \; -->1.7798$$
and
$$\mathbb{P}(O\mid <!--\; \mid \; -->L^C)=\frac{\mathbb{P}(O\cap <!--\; \cap \; -->L^C)}{\mathbb{P}(L^C)}=\frac{(\genfrac{}{}{0ex}{}{10}{1})\left(\frac{1}{4}\right){\left(\frac{3}{4}\right)}^9}{\frac{4}{5}}\approx <!--\; \approx \; -->.2346.$$
I would plug this into the Bayes' Rule equation, but I got a probability above 1 in one of the cases. Why?

In a survey searching at reading conduct at a college, it become observed that 48% study mag A, 46% read mag B, 55% examine magazine C, 18% examine magazines A and B, 20% examine A and C, 23% examine B and C, ad 8% examine all three. what number read atleast one of the three magazines.
Then what I did was state:
$$\begin{gathered} n(U)=100 \\ n(A)=48 \\ n(B)=46 \\ n(C)=55 \\ n(AB)=18 \\ n(AC)=20 \\ n(BC)=23 \\ n(ABC)=8. \end{gathered}$$
Noting that $AB=A\cap <!--\; \cap \; -->B$
Then what I did was:
$$n(A\cup <!--\; \cup \; -->B\cup <!--\; \cup \; -->C)=n(A)+n(B)+n(C)-<!--\; -\; -->n(AC)-<!--\; -\; -->n(AB)-<!--\; -\; -->n(BC)+n(ABC)$$
when I did this I got that 96% read either A, B, or C and I was just wondering if I had made an error because it seemed a bit high when I first did it.

I'm reading Carlsson's "A survey of equivariant homotopy theory" and I have a question.
Let G be a topological group and X be a G-space (for a nice notion of "space"). He defines
$$X_{hG}=EG{\times <!--\; \times \; -->}_{G}X$$
the homotopy orbit space, and
$$X^{hG}=F(EG,X{)}^G$$
the homotopy fixed point space.
He claims there are spectral sequences
$$E_{p,q}^2=H_p(G;H_q(X))\Rightarrow <!--\; \Rightarrow \; -->H_{p+q}(X_{hG})$$
and
$$E_2^{p,q}=H^{-<!--\; -\; -->p}(G;{\mathrm{\pi <!--\; \pi \; -->}}_q(X))\Rightarrow <!--\; \Rightarrow \; -->{\mathrm{\pi <!--\; \pi \; -->}}_{p+q}(X^{hG}).$$
Now, if I'm not mistaken the first one follows from the Serre spectral sequence applied to the Borel fibration $X\to <!--\; \to \; -->X_{hG}\to <!--\; \to \; -->BG$ (take the fiber bundle with fiber X associated to the action of G on X and to the G-principal bundle $EG\to <!--\; \to \; -->BG$).
But where does the second one come from?

So the question looks like this A survey finds that 30% of all Canadians enjoy jogging. Of those Canadians who enjoy jogging, it is known that 40% of them enjoy swimming. For the 70% of Canadians who do not enjoy jogging, it is known that 60% of them enjoy swimming.
Suppose a Canadian is randomly selected. What is the probability that they enjoy exactly one of the two activities? Select the answer closest to yours.
I solved this using $$P(JoggingorSwimming)=P(Jogging)+P(Swimming)-P(Joggingandswimming)P(Jogging)=0.3P(Swimming)=0.54P(Joggingandswimming)=0.12$$
my answer was 0.72 but it was wrong. Please, what am I missing?