Learn Significance Test with Examples & Questions

Probability of determinants being coprime
I have a question that is not of particular significance, but I would love to understand the underlying principles.
Suppose we have two square $3\times <!--\; \times \; -->3$ matrices, $M_1$ and $M_2$ with
$M_1=\left(\begin{array}{ccc}{a}_1& a_2& a_3\\ a_4& a_5& a_6\\ a_7& a_8& a_9\end{array}\right)\text{and}{M}_2=\left(\begin{array}{ccc}{b}_1& b_2& b_3\\ b_4& b_5& b_6\\ b_7& b_8& b_9\end{array}\right)$
with the coefficients $a_n,b_n\in <!--\; \in \; -->\mathbb{Z}$ and $1\le <!--\; \le \; -->a_n,b_n\le <!--\; \le \; -->9$What is the probability that the matrices' determinants are coprime, when uniformly random coefficients satisfying the conditions are chosen.
I am familiar with the Riemann's $\mathrm{\zeta <!--\; \zeta \; -->}$ function way to find out the probability of two random integers being coprime, but I have no clue how to apply that here with additional conditions on the numbers given.
I did test it mechanically, using Mathematica and the result is around 30%, but I would like to see a proper way to do it.
I would love to at least get a few pointers as what to research to tackle this problem.
Thank you very much!

Z-Test
$$\begin{gathered} \mathrm{\mu <!--\; \mu \; -->}<40 \\ z=-<!--\; -\; -->2.105263158 \\ p=.017634141 \\ \stackrel{¯<!--\; ¯\; -->}{x}=38 \\ n=100 \end{gathered}$$
(c) Based on your selection from part (b), what is the value of the standardized test statistic for this significance test?
(d) Based on your selection from part (b), what is the P-value for this significance test?
(e) What is the correct decision for this test, using a 0.05 level of signifcance?
A) Reject the null hypothesis
B) Do not reject the null hypothesis

Does anyone know what the properties of t-values are?
As I know it's just the ration
$t=\frac{\stackrel{¯<!--\; ¯\; -->}{x}-<!--\; -\; -->\mathrm{\mu <!--\; \mu \; -->}}{s/\sqrt{n}}$
which follows a t-distribution.
Used in mean-comparison tests and as a consequence also to check the significance of single independent variables in regression analysis. To generalize, t-values are used for hypothesis testing. Am I right with these properties?
Thanks!

Before using these variance estimates, what must we calculate?

I have "solved" a question in mathematical statistics, but unfortunately received an incorrect answer and I would appreciate any help in understanding why my method did not work in this case. The question is as follows:
In city A, 500 randomly selected people were tested for antibodies to covid-19, which resulted in 110 giving positive results. In city B, a random sample of 500 people was also drawn, and here 91 had positive results. Can it be said that it is statistically certain that city A has a larger population share with antibodies than city B? (Justify your answer with an appropriate test. Use 5% significance level.)
I started by producing $P^{\ast <!--\; \ast \; -->}(A)=\frac{110}{500}=0.22$ and respectively $P^{\ast <!--\; \ast \; -->}(B)=\frac{91}{500}=0.182$.
Then I chose to continued with a one-sided hypothesis testing where I assumed that the data was binomially distributed according to $Bin(n=500,p)$, with$H_0:P=0.182\text{ and }{H}_1:>\mathrm{0.182.}$
And finally brought out the one-sided interval according to:$I_p={\mathrm{\lambda <!--\; \lambda \; -->}}_{\mathrm{\alpha <!--\; \alpha \; -->}}\sqrt{\frac{P^{\ast <!--\; \ast \; -->}(A)(1-<!--\; -\; -->P^{\ast <!--\; \ast \; -->}(A))}{n}}\Rightarrow <!--\; \Rightarrow \; -->[0.189,\mathrm{\infty <!--\; \infty \; -->}].$.And concluded that $H_0$ should be rejected since 0.182 is not included in the interval. But according to the results, $H_0$ should not be rejected and they have also used a different method than I did. Why is this method not applicable in my case?
Note: At first I thought that errors were because I only used α and not $\frac{\mathrm{\alpha <!--\; \alpha \; -->}}{2}$ in the calculation, however, even if I were to replace this, 0.182 is not in the interval.

Consider the following:
1. Test for significance of the coefficient of determination
2. F-test for overall model significance
3. Test for significance of the slope coefficient
4. Test for significance of the correlation coefficient
Assuming you have a 5% level of significance, which of the above give equivalent conclusions a simple linear regression?
Select one:
a. 1, 2 and 3
b. 2 and 3
c. 1, 2, 3, and 4
d. 2, 3, and 4
e. 2 and 4

Psychologists who study perceptual/motor tasks often refer to something called the “Speed/Accuracy Tradeoff” or SAT. In many human responses we can respond very quickly and sacrifice response accuracy, or we can respond with precision and accuracy and sacrifice speed. Sports psychologists have found that both novice and skilled golfers demonstrate a SAT in various aspects of the game.
Ask any golfer about driving the ball and on top of getting a discussion about club head speed and fairways hit, you’ll likely hear some version of the old saying “there’s a lot of big hitters in the woods.” Hitting it long is great, but lots of long hitters wind up way off line. Some of these big hitters are well known because they’ve won, but lots of the names on the list of the top 25 drivers of the golf ball are unknowns—they can drive the ball a country mile, but they don’t win the big tournaments, perhaps because they have to hit their second shot from the deep woods.
Suppose we select a random sample of 30 golfers and measure both the distance they can drive the ball (a variable related to speed, in particular to club head speed at the point of impact with the ball) and average accuracy (measured as the ratio of fairways hit divided by total number of attempts).
Is there a relationship between speed at impact and accuracy?
What test should you use and why?

How one derives significance test for pearsons correlation coefficient?
I am exploring statistics and probability. What upsets me, only ready to use algorithms are present in the books. But no example how one derives a significance test, where from the test statistics come from etc. At the moment I am interesten in Pearsons coefficient significance test, but really derivation of any popular test will be helpful for me.

The test statistic is z = 2.75, the critical value is z = 2.326. The probability value is_________.
Select one:
a. not measurable
b. greater than the significant level
c. equal to the significant level
d. less than the significant level

A new cream that advertises 52% of women over the age of 50 will experience skin improvement from using their product. A study was conducted in which a sample of 170 women over the age of 50 used the new cream for 6 months. Of those 170 women, 76 of them reported skin improvement (as judged by a dermatologist). Is this evidence that the cream will improve the skin of less than 52 % of women over the age of 50? Perform a hypothesis test using the 0.05 level of significance.
Set up a null and alternate hypothesis.

Mean vs. Median: When to Use?
I know the difference between the mean and the median.
- The mean of a set of numbers is the sum of all the numbers divided by the cardinality.
- The median of a set of numbers is the middle number, when the set is organized in ascending or descending order (and, when the set has an even cardinality, the mean of the middle two numbers).
It seems to me that they're often used interchangeably, both to give a sense of what's going on in same data.
Do they mean (pun intended) different things? When should one be used over the other?

The value $\mathrm{\alpha <!--\; \alpha \; -->}$ that represents the probability of type I error is often referred to as the __________________________ of the test.
The vertical deviationsy1-y^1,y2-y^2,...,yn-y^nfrom the estimated regression line are referred t as the________.$\text{The vertical deviations}{y}_1-<!--\; -\; -->{\stackrel{^<!--\; ^\; -->}{y}}_1,y_2-<!--\; -\; -->{\stackrel{^<!--\; ^\; -->}{y}}_2,...,y_n-<!--\; -\; -->{\stackrel{^<!--\; ^\; -->}{y}}_n\text{from the estimated regression line are referred t as the\_\_\_\_\_\_\_\_.}$
$\text{When the estimated regression line is obtained via the principle of least squares, the sum of the residuals}y;-<!--\; -\; -->\stackrel{^<!--\; ^\; -->}{y};(i=1,2,3,...,n)\text{should in theory be}$

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 .

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.

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?

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 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.

Show the difference between two approaches to significance testing?