Learn Significance Test with Examples & Questions

Here's a problem I thought of that I don't know how to approach:
You have a fair coin that you keep on flipping. After every flip, you perform a hypothesis test based on all coin flips thus far, with significance level $\mathrm{\alpha <!--\; \alpha \; -->}$, where your null hypothesis is that the coin is fair and your alternative hypothesis is that the coin is not fair. In terms of $\mathrm{\alpha <!--\; \alpha \; -->}$, what is the expected number of flips before the first time that you reject the null hypothesis?
Edit based on comment below: For what values of α is the answer to the question above finite? For those values for which it is infinite, what is the probability that the null hypothesis will ever be rejected, in terms of $\mathrm{\alpha <!--\; \alpha \; -->}$?
Edit 2: My post was edited to say "You believe that you have a fair coin." The coin is in fact fair, and you know that. You do the hypothesis tests anyway. Otherwise the problem is unapproachable because you don't know the probability that any particular toss will come up a certain way.

Question:
A researcher believes that the stock market performance and the property market performance in Singapore are associated. Describe one (1) statistical approach the researcher would implement to determine the association between the performances of these two markets. Explain their association and the factors that affect their association.

A colleague of yours is faced with testing a relationship at the .05 level or significance or the .01 level of significance. What is your advice to your colleague? Why?

Learning Objective to be able to interpret the value of Pearson's correlation coefficient Match up the interpretations with the values of Pearson's correlation coefficient r:
r = -1
r = 0
r = 0.34
r = 8
r = 1
1. some degree of correlation; significance test recommended
2. impossible value for r
3. no evidence of linear relationship
4. perfect positive correlation
5. perfect inverse (negative)correlation

Life Expectancies Is there a relationship between the life expectancy for men and the life expectancy for women in a given country? A random sample of nonindustrialized countries was selected, and the life expectancy in years is listed for both men and women. Are the variables linearly related?
$\begin{array}{|ccccccc|}\hline \text{Men}& 56.8& 53.6& 70.6& 61.5& 44.6& 51.7\\ \text{Women}& 64.4& 46.2& 72.2& 20.2& 48.4& 47.1\\ \hline\end{array}$
(a) Compute the value of the correlation coefficient. Round your answer to at least three decimal places.
r=
(b) Stat the hypotheses.
$$\begin{gathered} H_0 \\ {H}_1 \end{gathered}$$
(c) Test the significance of the correlation coefficient at a=0.01 and 0.10 using The Critical Values for the PPMC Table.
Critical value(s): $\pm <!--\; \pm \; -->$
Reject/do not reject the null hypothesis
(d) Give a brief explanation of the type of relationship.
There is/is not (Choose one) a significant
(a) Compute the value of the correlation coefficient. Round your answer to at least three decimal places.
r=
(b) Stat the hypotheses.
$$\begin{gathered} H_0 \\ {H}_1 \end{gathered}$$
(c) Test the significance of the correlation coefficient at a=0.01 and 0.10 using The Critical Values for the PPMC Table.
Critical value(s): $\pm <!--\; \pm \; -->$
Reject/do not reject the null hypothesis
(d) Give a brief explanation of the type of relationship.
There is/is not (Choose one) a significant

According to a very large poll in 2015, about 90% of homes in California had access to the internet. Market researchers wonder if that proportion is now higher, so they take a random sample of 200 homes in California and find that 196 have access to the internet. They want to use this sample data to test H0:p = 0.9 versus Ha:p > 0.9, where p is the proportion of California homes with internet access. Which conditions for performing this type of significance test have been met?
Choose all answers that apply.
I. The data is a random sample from the population of interest.
II. np ≥ 10 and n (1 - p) ≥ 10
III. Individual observations can be considered independent.

Assume I'm thinking of investing in a small biotech company. In a phase 2 study with 120 patients split 80/40 between the new drug and old drug, the progression free survival (PFS) rate of the patients taking the new drug was greater than that of those taking the old drug at the 1% level (P = 0.01).
Now, the company is conducting a phase 3 trial with 300 patients split 200/100 between the new drug and the old drug. What is the probability that the PFS will be greater for the patients taking the new drug than for those taking the old drug at the 5% level (P = 0.05) or better?
In other words, what is the probability that the phase 3 trial will be successful so that the FDA will presumably approve the drug? I want to know if the drug company is a good investment, which of course also depends upon the market capitalization of the company and the size of the potential market, but the first question is what the probability is of the drug trial being successful? Is there any way to get a handle on that based only on the P2 results?
Edit: P.S. This is not a homework problem, though it (or something like it) would seem to be a standard and useful problem.

I'm having problems to understand the definition of the level of significance α. I thought I knew what $\mathrm{\alpha <!--\; \alpha \; -->}$ is but I realized I don't.When I stated to study statistics by myself I read this introductory book and everything was fine, the definition is very clear. He says on page 290:
You’re probably wondering, how small does a p-value have to be for us to achieve statistical significance? If we agree that a p-value of 0.0001 is clearly statistically significant and a p-value of 0.50 is not, there must be some point between 0.0001 and 0.50 where we cross the threshold between statistical significance and random chance. That point, measuring when something becomes rare enough to be called “unusual,” might vary a lot from person to person. We should agree in advance on a reasonable cutoff point.
Statisticians call this cutoff point the significance level of a test and usually denote it with the Greek letter $\mathrm{\alpha <!--\; \alpha \; -->}$ (alpha).
For example, if $\mathrm{\alpha <!--\; \alpha \; -->}$=0.05 we say we are doing a 5% test and will call the results statistically significant if the p-value for the sample is smaller than 0.05. Often, short hand notation such as P<0.05 is used to indicate that the p-value is less than 0.05, which means the results are significant at a 5% level.
Now, I'm studying about statistical inference, a more advanced subject, and I realized there are some concepts that don't exactly have the same definition as I studied before. The level of significance is an example.
I'm reading this book and on page 352 he introduces the Neyman-Pearson lemma as a method to find the UMP test.
Example:
On the basis of a random sample of size 1 from the p.d.f. $f(x;\mathrm{\theta <!--\; \theta \; -->})=\mathrm{\theta <!--\; \theta \; -->}{x}^{\mathrm{\theta <!--\; \theta \; -->}-<!--\; -\; -->1},0<x<1(\mathrm{\theta <!--\; \theta \; -->}>1)$
For ${\mathrm{\theta <!--\; \theta \; -->}}_1>{\mathrm{\theta <!--\; \theta \; -->}}_0$ , the cutoff point is calculated by:
... $C=(1-\mathrm{\alpha <!--\; \alpha \; -->}{)}^{\frac{1}{{\mathrm{\theta <!--\; \theta \; -->}}_0}}$
For ${\mathrm{\theta <!--\; \theta \; -->}}_1<{\mathrm{\theta <!--\; \theta \; -->}}_0$ , we have:
... $C={\mathrm{\alpha <!--\; \alpha \; -->}}^{\frac{1}{{\mathrm{\theta <!--\; \theta \; -->}}_0}}$
So in this second book, the cutoff point is not necessarily α, I'm confused.
MY ATTEMPT TO UNDERSTAND WITH THE HELP OF THE ANSWERS
The alpha is predetermined, but it doesn't mean I can't have a smaller rejection region. Then I end up having a smaller rejection region using NP lemma with the same level of significance alpha. Some introductory books let the cutoff point to be α by standard (why?), that's the reason of my confusion, I can shrink the rejection region keeping the value of α. Can someone say if I'm right?

Dantzig's unsolved homework problems
An event in George Dantzig's life became the origin of a famous story in 1939 while he was a graduate student at UC Berkeley. Near the beginning of a class for which Dantzig was late, professor Jerzy Neyman wrote two examples of famously unsolved statistics problems on the blackboard. When Dantzig arrived, he assumed that the two problems were a homework assignment and wrote them down. According to Dantzig, the problems "seemed to be a little harder than usual", but a few days later he handed in completed solutions for the two problems, still believing that they were an assignment that was overdue.
What were the two unsolved problems which Dantzig had solved?

Why do we still do symbolic math?
I just read that most practical problems (algebraic equations, differential equations) do not have a symbolic solution, but only a numerical one.
Numerical computations, to my understanding, never deal with irrational numbers, but only rational numbers.
Why do mathematicians construct the real numbers and go through the pain of using their complicated properties to develop real analysis, etc. to be able to solve just a few cases symbolically?
Why do non-mathematicians deal with real numbers and symbolic calculation if the majority of cases must use rationals and numerical computation only?

Explain Misconceptions about significance testing?

New largest prime number discovery - what's all the fuss [duplicate]
So I've read about the latest largest prime number discovery (M74207281), but I find it hand to understand what's the big deal because using Euclid's proof of the infinitude of primes we can generate primes as large as we want.
I'll be happy to know what I'm missing

New largest prime number discovery - what's all the fuss [duplicate]
So I've read about the latest largest prime number discovery (M74207281), but I find it hand to understand what's the big deal because using Euclid's proof of the infinitude of primes we can generate primes as large as we want.
I'll be happy to know what I'm missing

I have a question below revolving around two tailed tests, confidence intervals, etc, and I'm really struggling to figure out how to find the correct answers. It's 3am and this is the last question for an assignment due later in the day, and for some reason I can't seem to properly wrap my mind around t-tests and two tail tests and I'm slowly losing my mind. This is my last ditch-attempt and I would really, really appreciate some help.
Below you'll find the question and my attempt at some of the answers, which are probably way off.
Monthly profits (in million dollars) of two phone companies were collected from the past five years (60 months) and some statistics are given in the following table.
Company | Mean | SD
Rogers | 123.7|25.5
Bell | 242.7|15.4
(a) Find a point estimate for the difference in the average monthly profits of these two phone companies.
I subtracted the two means (242.7-123.7) and got 119.
(b) What is the margin of error for a 99% confidence interval for the difference in the average monthly profits of these two phone companies?
z = (1 - 0.99)/2 z = 0.01/2 z =0.005 1 - 0.005 = 0.995 p(z < 3.275) = 0.995
From looking online, I found this formula but I couldn't figure out how to fill in the values: ME = t(stdev/√n)
I don't know where to go from here. I know the formulas for the confidence intervals, but I don't understand what I'm supposed to do given that there are two standard devs and means? I don't know whether n is 60 or 2.
(c) Based on the constructed 99% confidence interval in part (b), can we conclude that there is a difference in the average monthly profits of these two phone companies?
(d) Given the level of significance α = 1%, test whether Bell's average monthly profit is more than Rogers'.

Determine the critical region for this test at significance level $\mathrm{\alpha <!--\; \alpha \; -->}=0.05$ .
Let the random variable X be the waiting time until the next student has to go to the toilet. Assume that X has an $Exp(\mathrm{\lambda <!--\; \lambda \; -->})$ distribution with unknown $\mathrm{\lambda <!--\; \lambda \; -->}$. We test $H_0:\mathrm{\lambda <!--\; \lambda \; -->}=0.2$ against $H_1:\mathrm{\lambda <!--\; \lambda \; -->}<0.2$, where we use X as test statistic. Determine the critical region for this test at significance level $\mathrm{\alpha <!--\; \alpha \; -->}=0.05$.
In my opinion I should calculate $P(X<C)|H_0)=1-<!--\; -\; -->e^{-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}x}=1-<!--\; -\; -->e^{-<!--\; -\; -->0.2x}$ So $1-<!--\; -\; -->e^{-<!--\; -\; -->0.2x}=0.05$ so x=0.25 and the critical region is $(-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->},0.25]$, is it the right method to solve this question?

How will you formulate the appropriate null and alternativehypotheses on a population mean.

If the results of the chi-squared test for independence turn out to be significant then we can guarantee the variables are dependent.
Group of answer choices:
-True
-False

I am struggling with the following question:
A test is constructed to see if a coin is biased. It is tossed 10 times and if there are 10 heads, 9 heads, 1 head or 0 heads, it is declared to be biased. Can 20 be the significance level of this test?
My thinking is as follows: $H_0:p=0.5$
$H_1:p\ne <!--\; \ne \; -->0.5$
Let X be the number of heads, under $H_0$, X$\sim <!--\; \sim \; -->$B(10,0.5).
If we look at a table of values, we get:
X=0,P=0.00098
X=1,P=0.0107
X=9,P=0.999
X=10,P=1
Since it is declared biased for these values, P<0.1 or P>0.9. Therefore, shouldn’t we be able to use 20% as the significance level, yet my book says we can’t. Any clarification?

What is the minimum number of tests to achieve statistical significance?
I'm dealing with a situation where in a large manufacturing facility we have approximately 2000 plumbing fittings of the same make and model. 3 of those fitting have failed within the last year. Each causing major property damage. We suspect that the plumbing components suffered degradation and we want to test a sample of the remaining plumbing components to see how wide spread the issue is (if at all).
What is the best way to decide how big of a sample we should choose so that we do not under test or over test the installed components.
Any idea where to begin?

If one of the categorical variables has 3 or more levels then we can consider the difference of proportions and use the normal distribution to determine the p-value.
Group of answer choices:
-True
-False