Improve Your Understanding of College Level Statistics Problems

How do you find the exact value of $\sin <!--\; \; -->67.5$ degrees?

How are the measure of central tendency and measure of dispersion complementary to each other in describing data?

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.

How do you find the exact value of $\cos <!--\; \; -->({120}^{\circ <!--\; \circ \; -->})+\cos <!--\; \; -->({45}^{\circ <!--\; \circ \; -->})$?

How are the measures of central tendency and measures of dispersion complementary?

What sampling technique –random, stratified, cluster, systematic, or convenience –was used? Explain. a. the first 30 people we see are asked a question b. all patients from 4 randomly selected hospitals are selected

A large high school wants to select a group of 20 students in order to see how well the cafeteria is serving the student body. The school randomly selects 5 freshmen, 5 sophomores, 5 juniors and 5 seniors from a list of students.
- Indicate the type of sampling strategy involved.
- This sampling strategy is stratified sampling.
- This sampling strategy is systematic sampling.
- This sampling strategy is convenience sampling.
- This sampling strategy is clustered sampling.
- This sampling strategy is SRS.

Simplification step by step (exponential and logarithm) - regarding BLOOM FILTER - FALSE POSITIVE probability
$(1-<!--\; -\; -->\frac{1}{m}{)}^{kn}\approx <!--\; \approx \; -->e^{-<!--\; -\; -->kn/m}$
I don't get how to reach $e^{-<!--\; -\; -->kn/m}$ from $(1-<!--\; -\; -->\frac{1}{m}{)}^{kn}$
I have reviewed logarithm and exponent rules but I always get stuck, here is what I have tried; Starting from $(1-<!--\; -\; -->\frac{1}{m}{)}^{kn}$, I can write:
- $e^{(ln(1-<!--\; -\; -->\frac{1}{m}{)}^{kn})}$
- $e^{(kn\times <!--\; \times \; -->ln(1-<!--\; -\; -->\frac{1}{m}))}$
Or the other way around:
$ln(e^{(1-<!--\; -\; -->\frac{1}{m}{)}^{kn}})$
$ln(e^{(1-<!--\; -\; -->\frac{1}{m})\times <!--\; \times \; -->kn})$
$ln(e^{(1-<!--\; -\; -->\frac{1}{m})})+ln(kn)$
$(1-<!--\; -\; -->\frac{1}{m})+ln(kn)$
After few steps I fall in an unsolvable loop hole playing with $ln$ and $e$ trying to reach $e^{-<!--\; -\; -->kn/m}$.

What is the mean, median, and mode of 31, 28, 30, 31, 30?

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?

The proof to follow is taken from a research paper in risk analysis which is in reviewing. I have been tasked with repeating the results in the paper, under a different risk measure, but I am not sure if the main argument in the paper is correct as it is presented, although I believe the main result should hold.
The context is the following. We are optimizing a risk function over a set of parameters $(a_i{)}_{i=1}^m,(b_i{)}_{i=1}^m$, with particular choices of $(a_i{)}_{i=1}^m,(b_i{)}_{i=1}^m$ denoted as $a^{\ast <!--\; \ast \; -->},b^{\ast <!--\; \ast \; -->}$ respectively. The aim is to establish a criterion for optimality of the parameters.
The argument in the paper is as follows:
Begin by assuming that $a^{\ast <!--\; \ast \; -->}$ is chosen such that $(1)$ holds, where $(1)$ is a condition on $a^{\ast <!--\; \ast \; -->}$. It is then possible to show that there exists some $b^{\ast <!--\; \ast \; -->}$ such that a condition $(2)$ on $b^{\ast <!--\; \ast \; -->}$ holds. We then show that any $b^{\ast <!--\; \ast \; -->}$ satisfying $(2)$ is optimal given that $a^{\ast <!--\; \ast \; -->}$ is chosen according to $(1)$. Assume next that $b^{\ast <!--\; \ast \; -->}$ is chosen such that $(2)$ holds. Then, one shows that $a^{\ast <!--\; \ast \; -->}$ can only be optimal parameters if $(1)$ holds. The conclusion is that choosing $b^{\ast <!--\; \ast \; -->}$ such that $(2)$ holds is sufficient for obtaining optimal $(a^{\ast <!--\; \ast \; -->},b^{\ast <!--\; \ast \; -->})$.
There are two things that make me uneasy about this argument. Firstly, as we begin by assuming a condition on $a^{\ast <!--\; \ast \; -->}$, my instinct is to look next at what happens if $a^{\ast <!--\; \ast \; -->}$ does not satisfy $(1)$.
Secondly, it is easy to find $b^{\ast <!--\; \ast \; -->}$ such that $(2)$ does not hold. While existence of $b^{\ast <!--\; \ast \; -->}$ satisfying $(2)$ given $a^{\ast <!--\; \ast \; -->}$ such that $(1)$ holds, it seems to me like we assume existance in the second part of the proof. Then, we seemingly use the assumed existence of $b^{\ast <!--\; \ast \; -->}$ such that $(2)$ to show that we must choose $a^{\ast <!--\; \ast \; -->}$ such that $(1)$, and hence $b^{\ast <!--\; \ast \; -->}$ will exists such that $(2)$. Is this not a circular argument?

How do you find the median of 6 numbers?

Spread and direction of the cosmic background radiation
Something I can never understand is that where the cosmic background radiation spreads?
If I know well, the cosmic background radiation is actually the light of the Big Bang. If it happened exactly in the same time, it must have spread into the theoretic center of the universe. Which would mean that it already reached every parts of it, in case if it happened in the same time with the Big Bang.
It be possible that it spreads to this direction, but universe expands faster. In this case, the radiation approaches but also moves off - just like everything in the universe. But it's possible only if the universe expands faster than the speed of light. Is this possible?
If it would spread to the direction of the "edge" of the universe, we shouldn't be able to know about its existence, because it would never reach us.
Also, these theories are true with an important conclusion: universe has a beginning in time, which means that once upon a time, it started to expand - consequently it must have a size limit. The reason is that universe has 4 dimensions: length, width, height, and time. One of them (time) is not infinite, so none of the others can be infinite.

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

Let $X\sim <!--\; \sim \; -->Binomial(n,p)$
$H+0:p\ge <!--\; \ge \; -->.5$
$H_1:p<.5$
How do we evaluate the p-value?
ie. $P(X\ge <!--\; \ge \; -->x\mid <!--\; \mid \; -->p>=.5)$
I have seen settings where this is just interpreted as $P(X\ge <!--\; \ge \; -->x\mid <!--\; \mid \; -->p=.5)$, but I'm not sure how this is the same thing.

$\mathrm{\hslash <!--\; \hslash \; -->}\approx <!--\; \approx \; -->0$ and the spread of QM wave function
Is there a direct mathematical method to show that if a quantum wave funtion is initially sharply localized, then it will stay sharply localized if $\mathrm{\hslash <!--\; \hslash \; -->}\approx <!--\; \approx \; -->0$? In that case the Ehrenfest theorem implies the transition from quantum mechanics to classical mechanics.
Of course, we are dealing with the propagation of a wave function, but let's not mess with path integrals. Thus, the structure of the general solution of Schrödinger equation should imply the result - if possible.

How would actuarial science refer to the idea that the premium should increase as number of insured entities decreases.
In other words, what is the technical term for the intuition that I would charge a higher premium per car to insure just one car than to insure 1000 cars, if such intuition is even justified.
As a sketch of my thinking on this, I have defined risk-adjusted premium as the total premium to cover losses divided by the standard deviation of the total loss on the portfolio of policies. I believe both the numerator and the denominator can be deduced from the law of large numbers.