Improve Your Understanding of College Level Statistics Problems

A statistics teacher has 4 periods of introductory statistics. She wants to get students’ opinions on a new homework policy. To get a sample, the teacher groups the students by their class performance (A students, B students, etc.). Then she randomly selects 3 students from each class performance group to survey. Which sampling method was used?
- cluster sampling
- simple random sampling
- stratified random sampling
- systematic random sampling

How can the energy of a magnetic field spread at speed of light?
I would like to solve one question. Suppose that we have a point charge. Then we apply a force on it, and as it accelerates, it emits electromagnetic waves and is pushed back due to Abraham-Lorentz force. But then, when we drop it and it moves at constant speed, it starts to create a magnetic field that spreads at speed of light. So as the field occupies more volume, it increases its energy but we are not giving any energy as we have dropped it. So, where does this energy come from?

How do you write down 5 numbers where the mean, median, and mode are all 6?

What do measures of central tendency indicate?

Explain Misconceptions about significance testing?

If $m$ is the number of bits in the array, $k$ is the number of hash functions and we have $n$ entries, then the false positive probability upper bound can be approximated as:
$(1-<!--\; -\; -->e^{\frac{-<!--\; -\; -->kn}{m}}{)}^k$
It further states in the wiki article that:
"The number of hash functions, $k$, must be a positive integer. Putting this constraint aside, for a given $m$ and $n$, the value of $k$ that minimizes the false positive probability is:"
$k=\frac{m}{n}\ln <!--\; \; -->2$
How can I find the $k$ value that minimizes the false positive function ?

When does the wave function spread over the volume of a box?
I have heard colloquially that for any initial state, a particle enclosed in some volume $V$ will spread itself relatively evenly over that volume after large time, so that $|\mathrm{\psi <!--\; \psi \; -->}(\stackrel{\to <!--\; \to \; -->}{x}){|}^2\approx <!--\; \approx \; -->1/V$ may be used as the position probability density in the box.
This seems reasonable but of course it is not true, for example, of energy eigenstates. A sine wave with a wavelength fitting into the box will remain a sine wave for all times; this is the case for any energy eigenstate. However one might argue that an exact energy eigenstate is an idealization.
Does anyone know of a condition on the initial wave function ${\mathrm{\psi <!--\; \psi \; -->}}_0(\stackrel{\to <!--\; \to \; -->}{x})$ which is sufficient to guarantee that for large times,
$|\mathrm{\psi <!--\; \psi \; -->}(\stackrel{\to <!--\; \to \; -->}{x}){|}^2\approx <!--\; \approx \; -->1/V?$
Or else, is there a counterexample of a more realistic state which does not fit the above approximation? My hunch is that any initial state whose fourier transform is supported on all of ${\mathbb{R}}^3$ might give the approximation above, but I don't know of any such result. Energy eigenstates, for example, have a delta-function fourier transform.

What are the outputs for a business risk analysis?

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

If the mean of five numbers is 25, what is their sum?

this problem is from the wackerly, but i have a particular question, in this problem we have to estimate the variance with $S_p^2$ or can we use the information standard deviations that they gave us.
Here is the problem:
Two new drugs were given to patients with hypertension. The first drug lowered the blood pressure of 16 patients an average of 11 points, with a standard deviation of 6 points. The second drug lowered the blood pressure of 20 other patients an average of 12 points, with a standard deviation of 8 points. Determine a 95% confidence interval for the difference in the mean reductions in blood pressure, assuming that the measurements are normally distributed with equal variances.

Tyler played 5 games of basketball. The mean was 10 points. The median was 12 points. What might each of his scores be?

Please, identify the type of data in the following datasets:
1) World’s Highest Temperatures. Information provided by the World Meteorological Association revealed the following data on the highest recorded temperature for each continent.

Consider a disease that is on average caught by $1$ in $N$ people. Let's say that a disease test gives a false positive for $p\mathrm{\%<!--\; \%\; -->}$ of healthy people. For diseased people, it always gives a correct, positive answer.
What's the chance that a person who got a positive result is healthy?

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

A poll from a previous year showed that $10\mathrm{\%<!--\; \%\; -->}$ of smartphone owners relied on their data plan as their primary form of internet access. Researchers were curious if that had changed, so they tested $H_0:p=10\mathrm{\%<!--\; \%\; -->}$ versus $H_a:p\ne <!--\; \ne \; -->10\mathrm{\%<!--\; \%\; -->}$ where p is the proportion of smartphone owners who rely on their data plan as their primary form of internet access. They surveyed a random sample of $500$ smartphone owners and found that $13\mathrm{\%<!--\; \%\; -->}$ of them relied on their data plan.
The test statistic for these results was $z\approx <!--\; \approx \; -->2.236$, and the corresponding $P$-value was approximately $0.025$.
Assuming the conditions for inference were met, which of these is an appropriate conclusion?
a) At the $\mathrm{\alpha <!--\; \alpha \; -->}=0.01$ significance level, they should conclude that the proportion has changed from $10\mathrm{\%<!--\; \%\; -->}$.
b) At the $\mathrm{\alpha <!--\; \alpha \; -->}=0.01$ significance level, they should conclude that the proportion is still $10\mathrm{\%<!--\; \%\; -->}$.
c) At the $\mathrm{\alpha <!--\; \alpha \; -->}=0.05$ significance level, they should conclude that the proportion has changed from $10\mathrm{\%<!--\; \%\; -->}$.
d) At the $\mathrm{\alpha <!--\; \alpha \; -->}=0.05$ significance level, they should conclude that the proportion is still $10\mathrm{\%<!--\; \%\; -->}$.
The correct answer is $c$ but why could it not have been $b$? Why is it $c$?

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'.

The median is called a resistant measure, whereas the mean is a non-resistant measure. What's a resistant measure?

I have 2 groups of people. I'm working with the data about their age. I know the means, the standard deviations and the number of people. I don't know the data of each person in the groups.
Group 1 :
Mean = 35 years old; SD = 14; n = 137 people
Group 2 :
Mean = 31 years old; SD = 11; n = 112 people
I want to combine those 2 groups to obtain a new mean and SD. It's easy for the mean, but is it possible for the SD? I do not know the distribution of those samples, and I can't assume those are normal distributions. Is there a formula for distributions that aren't necessarily normal?