I have difficulty of understanding p-value from An Introduction to Statistical Learning by Gareth James • Daniela Witten • Trevor Hastie Robert Tibshirani (2015) : 67  Consider simple linear regression equation $$Y= β_0+ β_1 X$$  Null hypothesis states Y doesn't relate to X, Mathematically $β_1\approx0$ To reject Null hypothesis, we should prove $β_1 &gt;&gt; 0$ After performing linear regression I calculated coefficient to be $\hat{β_{1}}$ .But since number of samples are limited, There is no confidence to reject null hypothesis.   So I calculate t-statistic as $$t_{statistic}=\frac{\hat{β_{1}} - 0}{SE(\hat{β_{1}})}$$ It is trivial to understand that higher t-statistic implies more confidence in $β_{1}$ since $β_{1}$ is larger than error in calculation of itself, hence we can reject Null hypothesis for high t-statistic values    From:  $$t=\frac{\hat{β_{1}} - {β_1}}{SE(\hat{β_{1}})}=\frac{Error\,in\,calculated\,β_{1}}{(Estimated\,Variance)/(\sqrt{n})}$$    Note here that ${β_1}$ &amp; ${SE(\hat{β_{1}})}$ are random variables and vary with batch of samples observed. Let's denote t-distribution as $T(t,n)$ and ${t\,statistic}$ as $T_{st}$   Probability of observing  ${t_{obs}}\ge{t\,statistic}$ is p-value $${p}{-}{value} = \int_{|t|&gt;|T_{st}|}T(t,n)$$ What is mathematical derivation for ${p}{-}{value}\approx 0 \implies β_1 &gt;&gt; 0$ is large ?    I have a quantitative explanation that t-statistic follows t-distribution, if we assume $H_{0}$ is true, and ${p}{-}{value}$ i.e. $\int_{|t|&gt;|T_{st}|}T(t,n)$ denotes probability of getting a value of "t" $\ge T_{st}$ which being less implies hypothesis is false
After taking a statistics course and then trying to help fellow students, I noticed one subject that inspires much head-desk banging is interpreting the results of statistical hypothesis tests.  It seems that students easily learn how to perform the calculations required by a given test but get hung up on interpreting the results.  Many computerized tools report test results in terms of "p values" or "t values".  How would you explain the following points to college students taking their first course in statistics:   What does a "p-value" mean in relation to the hypothesis being tested?  Are there cases when one should be looking for a high p-value or a low p-value? What is the relationship between a p-value and a t-value?
how to use common modulus attack? I am struck with the following problem:  Let Alice, Bob, Chris and Eve communicate over a public network. They encrypt all messages they send using RSA system. Bob and Chris have the RSA modulus $n_B$ and $n_C$ respectively with $n_B$ = $n_C$ But different public encryption exponents: $e_B \neq e_C$. Suppose $\gcd(e_B, e_C)=1$, and that Alice sends the same secret message to Bob and Chris. Show how Eve can decipher the message. How do I work recursively through a directory tree and execute a specific command on each file, and output the path, filename, extension, filesize and some other specific text to a single file in bash. What is a reliable method to locate floating islands in Terraria? How to sort a ConcurrentSkipListMap by value? Is it true that a subset that is closed in a closed subspace of a topological space is closed in the whole space? I have a non homework related question from a text and require a nice clear proof/disproof please  Is it true that a subset that is closed in a closed subspace of a topological space is closed in the whole space?  my ideas:  if $H$ is the subset of the topological space $X$  if the subset is closed in the closed subspace, the complement is open in the subspace, which means the complement is of form $U\cap H$ for some $U$ open in $X$  if the subspace is closed the complement is open which means complement of $H=U$ for some open $U$ in $X$  kind thanks Method overloading in Objective-C? Show that $\lim_{z \rightarrow \infty} -z^xe^{-z} = 0$ Let $x&gt;0$ be a constant. Show that $\lim_{z \rightarrow \infty} -z^xe^{-z} = 0$.  Do I apply L'hopital's rule here? It doesn't seem to help, for example, after one round of differentiating the numerator and denominator, I get $\frac{d}{dz} z^x = xz^{x-1}$ and $\frac{d}{dz}e^z = e^z$. I am unsure on how to proceed. Materials for learning TikZ I feel intimidated in learning TikZ. I was wondering if there are some nice materials (or just your replies) at different level:   for providing big picture, which is important to beginners to not lose direction during learning. for comprehensive reference.   Examples are important, but only examples are not always clear for understanding the big picture and organization. Suppose $a_n \ge 0$ for all natural numbers $\Bbb{N}$ and $\sum{a_n}$ converges. Is it necessarily true that $\sum{(a_n)^2}$ converges?  I think yes.  Clearly as $a_n \ge 0$ it follows $(a_1+a_2+\dots+a_n)^2 \ge a_1^2+a_2^2+\dots+a_n^2 \ge 0$ and so $(\sum_{n=1}^{\infty}{a_n})^2 \ge \sum_{n=1}^{\infty}{a_n^2} \ge 0$.  As we are told $\sum{a_n}$ converges to say $L$ where $L$ is a real number it follows $L^2 \ge \sum_{n=1}^{\infty}{a_n^2} \ge 0$ and so therefore $\sum{(a_n)^2}$ converges.  Is this correct? Reading order for Cosmere series by Brandon Sanderson My entry into Brandon Sanderson's works is Mistborn Trilogy, I finished and loved it. Then I got to know that many of his series are connected and happen in the same universe - Cosmere. Next I started with the Way of Kings, but am now kind of worried if I might read the series in the wrong order. So what is the right order to read them? is it the publication order..?   Elantris (2005)  Mistborn Trilogy Warbreaker (2009) The Stormlight Archive: The Way of Kings (2010) Mistborn: The Alloy of Law (2011) The Emperor's Soul (2012, novella) Shadows for Silence in the Forests of Hell (2013, short story published in Dangerous Women) The Stormlight Archive: Words of Radiance (2014)    or I have seen people suggest the following order too,   Mistborn Trilogy Elantris and the Emperor's Soul The Way of Kings Warbreaker Words of Radiance The Alloy of Law Shadows of Self   I have also read posts where they say it doesn't matter which order you read it in. So am confused, since I have already started the Way Of Kings (Only a few chapters in), should I stop it and go back to Elantris, and follow an order, or should I just continue finish this and pick other books in an order. Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $ What is difference between: "I have been married with (was married with, married) her for three years." What is the difference between the three sentences below?   I have been married with her for three years. I was married with her for three years. I married her for three years. How to set a shortcut for "Always on Top" for window? In 12.04 (Unity), I would like to assign a shortcut to the "always on top option" for a window. Probably something like Ctrl+Shift+Home to turn on and Ctrl+Shift+End to turn off, or whatever. Is there an option to do this somewhere in the Compiz Settings Manager?  This (old)  claimed that      For compiz, Enable Extra WM Actions. Check the properties, and insert ALT+A to Key field for Toggle Always-On-Top action.   But I couldn't find that. Allowed Wave Functions of System Given a single-particle system with Hamiltonian $H$, what constraints can be put on the wave function at a particular point in time $\psi(x)$? Of course $\psi(x)$ must obey boundary conditions given by $H$. However, in situations where $H$ does not yield strict boundary conditions (e.g. the harmonic oscillator) can $\psi(x)$ be any normalised function? My intuition says no for the following reason: QM textbooks say that any valid wave function can be represented as a linear combination of the eigenstates of $H$. In the case of the harmonic oscillator, the eigenstates of $H$ are a countably infinite set. However the set of all normalised functions is uncountably infinite. It seems to me that one cannot change between one complete set of basis functions to another and reduce the "dimensionality" somehow. This makes me think that the eigenstates of $H$ are not complete, that they imply some constraints on $\psi(x)$. This is not an area of mathematics that I understand well. Can somebody help me out? systemd-resolve high cpu usage after update to 17.04 How can I emulate 1.7.10's adventure mode in 1.8? Understanding the -exec option of `find` I find myself constantly looking up the syntax of  find . -name "FILENAME"  -exec rm {} \;   mainly because I don't see how exactly the -exec part works. What is the meaning of the braces, the backslash and the semicolon? Are there other use cases for that syntax? Is it possible to construct an (asymmetric) encryption scheme from a signature scheme?  If the signature scheme is deterministic and allows  (e.g. ), then the answer is yes (create a forgery (m, sig); encrypt plaintexts using  with sig as the key; send m in clear to the signer along with ciphertexts, who can then compute sig and decrypt these ciphertexts) .  I am interested in similar reductions for probabilistic signature schemes.  What should the properties of a probabilistic signature scheme be so that it can be used for encryption? I have sucessfully installed AniMove by activating experimental plugins (yay! thanks gis.stackexchange). But I can't find it in Qgis! Yes I have activated it. Where is it? Is this a common bug? I searched for it the toolbox, I looked up the names of the several single tools based on this  and only found "Kernel Density", which is filed under SAGA. But I don't have SAGA installed (and don't know if I need to?) Is there any way to automatically unwrap UV map proportionally to object or "world" size?  For example:  I've made two cubes, one is two times bigger than other (scale is applied). Then I unwrapped texture with Smart UV project which will try to span all faces on whole texture space(0 to 1) so bricks on left cube will be bigger than bricks on right cube.   But what I want to achieve is this:     In broader context:  When I import model from Archicad (in form of OBJ) all objects have UV's in some scale (so all bricks have the same size on every imported object). And when I want to create custom object in scene I have to manually scale UV's to adjust bricks size for every new object (instead of scaling to one constant parameter).