Improve Your Understanding of College Level Statistics Problems

I have the following 2 samples
data 1 : 59.09 59.17 59.27 59.13 59.10 59.14 59.54 59.90
data 2: 59.06 59.40 59.00 59.12 59.01 59.25 59.23 59.564
And I need to check whether there is differentiation regarding the mean and the variance between the 2 data samples at significance level a=0.05
I think that the first thing I need to do is to check whether the samples come from a normal distribution in order to infer whether i should proceed using parametric or non marametric tests...
However using lillietest in matlab returned that both samples do not follow the normal distribution...
Any ideas on how should I proceed with checking the differentiation tests ? Should I perform ttest ? Or should I proceed by using something like Wilcoxon ? (p.s please confirm that that both data samples do not follow normal distribution...)

Why does one counterexample disprove a conjecture?
Can't a conjecture be correct about most solutions except maybe a family of solutions?
For example, a few centuries ago it was widely believed that $2^{2^n}+1$ is a prime number for any n . For n=0 we get 3 , for n=1 we get 5 , for n=2 we get 17 , for n=3 we get 257 , but for n=4 it was too difficult to find if this was a prime, until Euler was able to find a factor of it. It seems like this conjecture stopped after that.
But what if this conjecture isn't true only when n satisfies a certain equation, or when n is a power of 2$\ge <!--\; \ge \; -->$ 4 , or something like that? Did anybody bother to check? I am not asking about this conjecture specifically, but as to why we consider one counterexample as proof that a conjecture is totally wrong.P.S. Andre Nicolas pointed out that Euler found a factor when n=5, not 4 .

Let $X_i$ be Gaussian random variables with $\mathrm{\mu <!--\; \mu \; -->}$ = $10$ and ${\mathrm{\sigma <!--\; \sigma \; -->}}^2$ = $1$. We decide to use the test statistic $\stackrel{^<!--\; ^\; -->}{\mathrm{\mu <!--\; \mu \; -->}}$ = $\frac{1}{20}\underset{i=1}{\overset{20}{\sum <!--\; \sum \; -->}}{X}_i$ and following tests:
$|\stackrel{^<!--\; ^\; -->}{\mathrm{\mu <!--\; \mu \; -->}}-<!--\; -\; -->10|>\mathrm{\tau <!--\; \tau \; -->}$: Reject $H_0$
$|\stackrel{^<!--\; ^\; -->}{\mathrm{\mu <!--\; \mu \; -->}}-<!--\; -\; -->10|\le <!--\; \le \; -->\mathrm{\tau <!--\; \tau \; -->}$: Cannot reject $H_0$
1) Find $\mathrm{\tau <!--\; \tau \; -->}$ if you want to have $95\mathrm{\%<!--\; \%\; -->}$ confidence in the test.
2) You find that $\stackrel{^<!--\; ^\; -->}{\mathrm{\mu <!--\; \mu \; -->}}=10.588$. Do you reject $H_0$? If so, what is the p-value?
Using textbook equations, I found out that for
1) $\mathrm{\tau <!--\; \tau \; -->}$ = ${\mathrm{\sigma <!--\; \sigma \; -->}}_{\stackrel{^<!--\; ^\; -->}{\mathrm{\mu <!--\; \mu \; -->}}}$ = $1.96\sqrt{\frac{1}{20}}$ $=0.392$
2) $|10.588-<!--\; -\; -->10|=0.588>0.392\to$ Reject $H_0$
p-value $=0.003$

My problem is the following. I like to know if there exist a sentence true in complex a field but false in a field of positive characteristic.

In orthodox probability theory, only certain trivial Boolean algebras with very few elements contain, for any given A and B, an event X which satisfies P(X) = P(B|A). Later extended by others, this result stands as a major obstacle to any talk about logical objects that can be the bearers of conditional probabilities.
I'm aware that in Bayesian analysis, the conditional likelihood P(B|A) is not a probability distribution. There must be a connection here right? What are some simple examples of conditional events that are not in the event algebra?

Outline the constraints which could be involved in a business risk analysis.

p-values behave uniformly. Now as p(np) is fixed and n goes to infinity, binomial converges to normal(Poisson). Now suppose I take random binomial samplings and fir normal(Poisson) to it, for say n = 1000. Will my p-value still be uniformly distributed or as binomial converges to normal(Poisson), p-values mostly will be in 0.8-1?

Extending the Concepts and Skills
In sampling from a population, state which type of sampling design corresponds to each of the following experimentaldesigns:
a. Completely randomized design
b. Randomized block design

You run a significance test to determine if the average salary of Americans is $35,000. You end up rejecting the null hypothesis. You then find that the actual average salary is somewhere between $25000 and $2700o. This is significantly lower than expected!
Which of the following best describe the analysis above?
- Both statistically significant and practically significant.
- Statistically significant, but not practically significant.
- Practically significant, but not statistically significant.
- Neither statistically significant nor practically significant.

How the false positive value affects accuracy?
$TP(truepositive)=2739$
$TN(truenegative)=103217$
$FP(falsepositive)=43423$
$FN(falsenegative)=5022$
$accuracy=\frac{TP+TN}{TP+TN+FP+FN}$
In this case the accuracy is $0.68$. Can I say that I have low accuracy because the value false positive is high? There is any relathion between false positive and the parameters true positive or true negative?

Is Edwin Jaynes correct about statistics?
I've recently been reading Edwin Jaynes's book, Probability Theory: The Logic of Science, and I was struck by Jaynes's hostile view of what he dubs "orthodox statistics." He repeatedly claims that much of statistics is a giant mess, and argues that many commonplace statistical techniques are ad hoc devices which lack a strong theoretical footing. He blames historical giants like Karl Pearson and Ronald Fisher for the field's sorry state, and champions Bayesian methods as a healthier alternative.
From my personal perspective, his arguments make a lot of sense. However, there are a number of factors that prevent me from taking his criticism of statistics at face value. Despite the book's being published in 2003, the majority of its contents were written in the mid 20th century, making it a little dated. The field of statistics is reasonably young, and I'm willing to bet it's changed significantly since he levied his critiques.
Furthermore, I'm skeptical of how he paints statistics as having these giant methodological rifts between "frequentists" and "Bayesians." From my experience with other fields, serious methodological disagreement between professionals is almost nonexistent, and where it does exist it is often exaggerated. I'm also always skeptical of anyone who asserts that an entire field is corrupt--scientists and mathematicians are pretty intelligent people, and it's hard to believe that they were as clueless during Jaynes's lifetime as he claims.
Questions:
1. Can anyone tell me if Jaynes's criticisms of statistics were valid in the mid 20th century, and furthermore whether they are applicable to statistics in the present day? For example, do serious statisticians still refuse to assign probabilities to different hypotheses, merely because those probabilities don't correspond to any actual "frequencies?"
2. Are "frequentists" and "Bayesians" actual factions with strong disagreements about statistics, or is the conflict exaggerated?

For $H_0:\mathrm{\pi <!--\; \pi \; -->}=0.20$, for instance, the score test statistic is $Z(s)=-<!--\; -\; -->2.50$, which has two-sided P-value $0.012<0.05$, so $0.20$ does not fall in the interval.
It is understood that the score test statistic is $-<!--\; -\; -->2.5$ using $n=25$ and $\stackrel{^<!--\; ^\; -->}{\mathrm{\pi <!--\; \pi \; -->}}=0$ (in the previous description), but the part where the P-value is $0.012$ is not understood.

How can a median be greater than the mean?

Lack of unique factorization of ideals
I'm aware of the result that integral domains admit unique factorization of ideals iff they are Dedekind domains.
It's clear that $\mathbb{Z}[\sqrt{-<!--\; -\; -->3}]$ is not a Dedekind domain, as it is not integrally closed. I'm having difficulty demonstrating directly that $\mathbb{Z}[\sqrt{-<!--\; -\; -->3}]$ does not admit unique factorization of ideals.
Obviously we only need one example of an ideal that doesn't have a unique factorization into prime ideals, but a good answer would provide either a process or some intuition in finding such an example.

I'm confused about the interpretation of P value in hypothesis testing. I know that we set significance level as 0.05 which is the threshold we set for this test so that it won't suffer from Type I error by 5%.
And we are comparing P to significance level, does it mean P is the probability of making type I error based on the sample?

I was working on the following problem:
Consider two probability density functions on $[0,1]:f_0(x)=1$ and $f_1(x)=2x$. Among all tests of the null hypothesis $H_0:X\sim <!--\; \sim \; -->f_0(x)$ versus the alternative $X\sim <!--\; \sim \; -->f_1(x)$, with significance level $\mathrm{\alpha <!--\; \alpha \; -->}=0.1$, how large can the power possibly be?
I think we need to begin by looking at an arbitrary test with significance level$\mathrm{\alpha <!--\; \alpha \; -->}=0.1$ but I am having a hard time doing this. I am not even sure this is the direction we want to head in so I was hoping to get some hints regarding this problem.

I'm trying to solve following type of combined date given question. Imagined that there's two factories. They are X and Y. "X" factory has 10 workers and "Y" has 20 workers. These two factory's production respectively given as Σx^2=2950 and ΣY^2=5000. Moreover, "X" factory mean is 18 and "Y" factory mean is 15.
If the whole population is considered as a one. How to calculate the mean and standard deviation in a situation like this one? I'm not expecting the final answer, I rather would like to know the procedure that how to solve such a question.

Why do electrons in a current spread apart, from their point of view?
There are many videos explaining how special relativity causes magnetism, with the most relevant part being that from the reference frame of the electrons, the positively charged ions in the wire have gotten closer together, and the negatively charged electrons have spread apart, a non-zero electric charge in a wire. We can determine that this must happen using the fact that in a lab reference frame, the wire is electrically neutral, and applying a Lorentz transformation to the electrons and ions. However, the electrons must see a different mechanism causing the space between them to increase. They need to see something pushing them apart. They don't know that the wire needs be electrically neutral in another reference frame. What's this mechanism that they see? Does it have anything to do with the period of time when they're accelerating?
I saw a bunch of questions that seemed similar to this one, but I didn't find any of the answers to be satisfactory. They all seemed to just explain how we know that the wire must be charged in different reference frames, using the Lorentz tranformation.