Improve Your Understanding of College Level Statistics Problems

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.

How do you find the range and interquartile range for 3, 7, 4, 6, and 5?

Under what conditions might the median be a preferable measure of center to use, rather than the mean?

Hence, $v(6,000)<v(4,000)+v(2,000)$ and $v(-<!--\; -\; -->6,000)>v(-<!--\; -\; -->4,000)+v(-<!--\; -\; -->2,000)$. These preferences are in accord with the hypothesis that the value function is concave for gains and convex for losses.
What this means and by how we can know if these align with convex and concave functions?

Can quantum particles spread out over large distances?
While trying to understand quantum mechanics I was wondering about this: since free quantum particles naturally spread out until the wave function collapses (if I understand correctly); does there exist an abundance of extremely spread out particles in outer space where interaction with other particles is rare or do the particles collapse before this happens?
To be more specific:
1.Does it occur often that particles in outer space reach macroscopic spreads of let's say multiple kilometers? Or does quantum decoherence occur before this happens? With spreading I mean ${\mathrm{\sigma <!--\; \sigma \; -->}}_x$ or the uncertainty in position.
2.If a particle reaches such a big spread, does this accelerate or inhibit wave function collapse?
A spread out particle covers more area making it interact with more matter but at the same the probability amplitude per area decreases making the chance of interaction smaller.

How do I find the average of $36,995, $32,507, and $37,286?

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.

What is the mean, median, and mode of 72, 75, 78, 72, 73?

In the hypotehses testing, I would like to see an intuition how the operator $min$ and the constant $2$ in the computation of $p$-value of a two side alternative $p=2min(F(T_0),1-<!--\; -\; -->F(T_0))$ can be derived and understood for a statistics $T_0$. Here $F$ is the c.d.f.

The data generating process is
$f(x\mid <!--\; \mid \; -->\mathrm{\theta <!--\; \theta \; -->})=\frac{1}{\mathrm{\theta <!--\; \theta \; -->}},0\le <!--\; \le \; -->x\le <!--\; \le \; -->\mathrm{\theta <!--\; \theta \; -->}$
Let $H_1:\mathrm{\theta <!--\; \theta \; -->}=5$ and $H_2:\mathrm{\theta <!--\; \theta \; -->}\ne <!--\; \ne \; -->5$. Prior probabilities of $\frac{1}{2}$ are assigned to the two hypotheses, and a sample of size $5$ yields a maximum value of $4$. Find the $p$-value here.
According to the definition of $p$-value,
$p=\mathbb{P}(LR\le <!--\; \le \; -->L{R}^{\ast <!--\; \ast \; -->}\mid <!--\; \mid \; -->H_1),$
and $L{R}^{\ast <!--\; \ast \; -->}={\left(\frac{4}{5}\right)}^5$. I am not sure what to do next. Furthermore, intuitively, if the maxima is equal to $5$ exactly, shall we get an extremely big $p$ or a small one?

What is the average of the 3 temperatures -10°C, -30°C, and -55°C?

I have the following data on duration that different aged adults can remain standing.
$\begin{array}{|cccc|}\hline \text{ Age }& \text{ Sample Size }& \text{ Sample Mean }& \text{ Sample Std Dev }\\ \text{ Old }& 28& 801& 117\\ \text{ Young }& 16& 780& 72\\ \hline\end{array}$
The data I'm using has a normal distribution with the same variances. I want to do a test of hypotheses however at a significance level of 5% (a=0.05) to be able to confirm whether or not the average duration that older adults can remain standing is larger than among younger adults.

I'm not sure which of the different formulas I should use however to determine this due to my sample size being relatively small. Should I be using the following test.

to compute pooled standard deviation: $s^2=\frac{(1-<!--\; -\; -->1)s{1}^2+(n2-<!--\; -\; -->1)s{2}^2}{n1+n2-<!--\; -\; -->2}$

compute test statistics: $t=\frac{y1-<!--\; -\; -->y2-<!--\; -\; -->0}{s\sqrt{\frac{1}{n1}+\frac{1}{n2}}}$

I am trying to create an algorithm that could calculate the p-value given the chi-square statistic and the degrees of freedom. Can anyone please point me in the right direction on how I could go about to evaluate the formula and what prerequisites I need to learn before I could do it.

Do cracks in solids spread at a characteristic speed?
I have a feeling this is actually a pretty complicated problem in detail as I know a tremendous amount of research is done on the behavior of materials under stress, and I think the way materials fracture/crack is a useful diagnostic tool.
As a starting point, waves travel at the speed of sound in a material, which is to say they are mechanical waves. Do cracks also propagate at the speed of sound?
Another related question is, do the cracks in a solid move at a constant speed? Obviously it is possible for a crack to form and stop propagating, so this might imply a dissipation of some vibrational amplitude until the energy is below the point of being able to break bonds/fragment atoms (sheets for some solids). It also might imply that the crack does not behave like a wave and spreads at a non-constant speed. Which is right?
Lastly, do all solids follow the same rules for the speed at which they fracture? I'm thinking of glass, which probably is not a good place to start on this question because it is an amorphous solid (some people say incredibly viscous liquid but this feels like semantics). Obviously some glass will crack into large pieces and some will fracture like a spiderweb. Are these processes fundamentally the same but with different microscopic details? And how does fracturing in glass relate to something simpler like the breaking of a single atom solid? I haven't seen this happen, but surely it is possible.
Obviously there were quite a few question there, but they were only meant to give a general idea of the types of questions would be nice to have answered. Maybe the question is actually quite simple and one answer will suffice, or maybe it's complicated and there are multiple good answers.

Random sampling isthe best sampling technique because it guarentees that our sample will be represantive of the population.
True or false?