Say I am using a t statistic, the (two-tailed-t test) p-value as i understand is given as pr(t$\geq$|t| |null is true), why cant it be that pr(t=t|null) or at least some small relative frequency in an infinitesimal distance around t, such as f(t)dt ?
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?
I am working in Illustrator (15.0.2 (CS5) or 16.0.0 (CS6)) and when I try to move objects small distances, they snap to the nearest two pixels on some invisible grid. I can't find anything in the preferences. Here are some screen shots: Testing whether two regression coefficients are significantly different (in R ideally) If this is a duplicate question, please point to the right way, but the similar questions I've found here haven't been sufficiently similar. Suppose I estimate the model $$Y=\alpha + \beta X + u$$  and find that $\beta&gt;0$. However, it turns out that $X=X_1+X_2$, and I suspect $\partial Y/\partial X_1 \ne \partial Y / \partial X_2$, and in particular, that $\partial Y/\partial X_1 &gt; \partial Y / \partial X_2$. So I estimate the model $$Y=\alpha + \beta_1 X_1 + \beta_2 X_2 +u$$and find significant evidence for $\beta_1,\beta_2&gt;0$. How can I then test whether $\beta_1&gt; \beta_2$? I considered running another regression $$Y=\alpha +\gamma(X_1 - X_2) + u$$ And testing whether $\gamma&gt;0$. Is this the best way?  Also, I need to generalize the answer to many variables, i.e. suppose we have $$Y=\alpha + \beta^1 X^1 + \beta ^2 X^2 + \dots + \beta^nX^n + u$$ where for each $j=1,\dots,n$, $X^j=X_1^j+X_2^j$, and I would like to test for each $j$ whether $\partial Y/\partial X_1^j \ne \partial Y / \partial X_2^j$.  By the way, I am primarily working in R. Firstly, this is a homework problem so please do not just give an answer away. Hints and suggestions are really all I'm looking for.  I must prove the following using mathematical induction:     For all $n\in\mathbb{Z^+}$, $1+2+2^2+2^3+\cdots+2^n=2^{n+1}-1$.   This is what I have in my proof so far:     Proof: Let $p\left(n\right)$ be $\sum_{j=0}^n 2^j=1+2+2^2+\cdots+2^n=2^{n+1}-1$. Then $P\left(0\right)$ is    \begin{align} \sum\limits_{j=0}^0 2^j=1=2^1-1.\tag{1} \end{align} Hence    $P\left(0\right)$ is true. Next, for some $k\in\mathbb{Z^+}$, assume    $P\left(k\right)$ is true. Then \begin{align}  \sum\limits_{j=0}^{k+1}2^j&amp;=\left(\sum\limits_{j=0}^k  2^j\right)+2^{k+1}\tag{2}\\ &amp;=2^{k+1}-1+2^{k+1}\tag{3} \\ &amp;=2\cdot 2^{k+1}-1\tag{4}\\ &amp;=2^{k+2}-1\tag{5}\\ &amp;=2^{\left(k+1\right)+1}-1.\tag{6} \end{align}   Therefore, $P\left(n\right)$ holds for all $n$.$\;\;\;\;\blacksquare$   Is everything right? My professor went over the methodology in class quite fast (and I had a bit too much coffee that night) and I want to make sure I have it down for tomorrow night's exam. No more disk space: How can I find what is taking up the space? I've run into a problem on one of my servers running 16.04: there is no disk space left.  I have no idea what is taking up the space. Is there a command to list the current directory sizes, so I can traverse and end up in the directory taking up all the space? How can I convert monsters from D&D 3 or 3.5 to 2ed monsters? When a user on our site loses his password and heads off to the Lost Password page we need to give him a new temporary password. I don't really mind how random this is, or if it matches all the "needed" strong password rules, all I want to do is give them a password that they can change later.  The application is a Web application written in C#. so I was thinking of being mean and going for the easy route of using part of a Guid. i.e.  Guid.NewGuid().ToString("d").Substring(1,8)   Suggesstions? thoughts? So I know the basic gist is that fusion power's main issue is sustaining the fusion. I also know that there are two methods. The Torus method and the laser method. The torus magnetically contains plasma and heats it with radiation and accelerates the plasma around to make strong enough collisions that protons fuse. The laser method uses 192 lasers and focuses it on tiny frozen hydrogen pellets and aims to initiate fusion each time pellets are dropped.  The though struck me when we could sorta combine the two designs together. The torus doesn't have to worry about making fusion happen at a specific location but it has issues in that the plasma is unevenly heated and leaks. On the other hand, the laser design is extremely complicated in the level of precision needed and would have to repeat this for every pellet. This lead me to think to make something precise and contained at the same time.  I see that particle colliders are able to direct two beams of protons and have them collide at a specific spot with a very precise energy. Couldn't we tune the energy of the two beams of protons to the energy required for them to fuse? We have the ability to smash them into bits, surely we have the ability to have them fuse. (I'm thinking about the type of collider that circles two beams in opposite directions)  It would be at much lower energies than normal colliders and would be very precise and it would be possible to fuse at a specific location that has greater leeway because for protons that missed collision, they'd just circle around again! Thus protons would efficiently be used and very little would be wasted. There wouldn't be problems of plasma leakage because we are focusing them in a thin tight beam.   It seems that this idea has girth, or I feel this way at least, can someone back me up by offering some calculations on how to calculate the efficiency? How would I go about calculating the two circling beams of protons and at what specific velocity would be needed? etc. How do I change my login provider(s)? what is the total number of distinct arrangements of $n$ oranges and $n$ apples around a round table? I have no idea how to go about. Mathematical operators, such as function names, should be set in roman type, not italics. Latex already has commands for some operators, including \max, \min, and \log.  How can I define additional such commands? Uninstall, Activate, Deactivate a plugin: typical features & how-to I'm making a WordPress plugin. What are typical things I should include in the uninstall feature?  For example, should I delete any tables I created in the install function?  Do I clean up my option entries?  Anything else? Proving this can be done as follows: consider a finite group G and elements $g_i \in G$ for some integer $i$. Now consider $\langle g_i \rangle = \{g_i^n: n\geq 0\}$, a generator. It can be proved that $\langle g_i \rangle \leq G$ and that the order of $g_i$ is equal to the order of $\langle g_i \rangle$, so $|\langle g_i \rangle| \leq |G|$.  We can now use Lagrange's theorem which states that if $H \leq G$ then $|H|$ divides $|G|$ and we're done. But...  Is there a simpler way to prove this fact? Encode URL in JavaScript? How do you safely encode a URL using JavaScript such that it can be put into a GET string?  var myUrl = "http://example.com/index.html?param=1&amp;anotherParam=2"; var myOtherUrl = "http://example.com/index.html?url=" + myUrl;   I assume that you need to encode the myUrl variable on that second line? Find the inverse of a Trig Matrix I don't have a clue of what's going on. We haven't learn this in class so I need all the help possible. The more detailed of an explanation, the better. Thanks in advance. The only info I have is that this matrix is Orthogonal. Which means I know the answer, just don't know how to get it.  \begin{bmatrix}        \cos\theta &amp; -\sin\theta \\        \sin\theta &amp; \cos\theta       \end{bmatrix} euler's theorem of homogeneuos function suppose that $f$ is a mapping from $\mathbb R^n \to \mathbb R$.  a positively homogeneous of degree $n$ and suppose $f_1, f_2, .... f_N$ are continuous for $a \neq 0$. then   $$\sum_{i=1}^n a_if_i(a)= n*f(a)$$   now, they asked me to prove this. (Which is a total no no to me)  I write my proof as;  $f$ is a mapping from $\mathbb R^n \to \mathbb R$  A homogeneous function of degree $n$ if for all $a \in \mathbb R^n$ and all $L&gt;0$, $f(La)=L^n\,f(a)$. let $a \in \mathbb R^n$ and define function $g$ be a mapping from $[0,\infty) \to \mathbb R$. so I write $g(L) = f(La)-L^nf(a)$. then $L≥0$ let $L=0$ then, $g(L)=0$ and therefore $g'(x) =0$?  then i don't know what's goes next. can anyone help me about this? Finding roots/zeros for collision detection? For the most simple of 2D games, I have implemented a posteriori collision detection (overlapping rectangles) on the x/y Cartesian plane, but am now interested in understanding the basics of a priori collision detection...  In the link , I noticed the reference to Newton’s Method.  So, I want to better understand the link b/w finding zeros of an equation and detecting a posteriori collision.    Let's start with the most obvious case of finding roots of an equation:  If you have a projectile's trajectory modeled as a quadratic function, you can find the roots to predict at what time the height will equal zero (or any other constant height, actually, by just solving f(x)=C to be f(x)-C=0 and then finding roots of that expression).    So, as I understand, finding the root can tell me the time at which the object will be at any chosen height.  More broadly, we basically know the (x,y) location of the object for any given time.  But how does root finding, in particular, translate to detecting collisions with another object?  (The other object may be stationary, or be moving.  If the latter, do you also determine roots/location of this moving object as well? Calculate the (x,y) trajectory of both objects and determine where they will intersect?)  Again, the basic question is where does root finding come into play?  Any links to a basic primer on this topic, with a simple example would also be greatly appreciated. What is the formula for armor/magic resist reduction? As far as I can tell, it's something like this:  damage reduction = armor / (armor + 100)   Is this accurate? Is there any way to hide taskbar labels and not combine taskbar buttons? There are three options for customizing taskbar buttons in Windows 7:   Always combine, hide labels Combine when taskbar is full Never combine   Is there any way to hide labels and not combine taskbar buttons? Or even better, to exclude only certain applications from combining?  My main problem is that I switch a lot between two instances of a single application, and having to hover over combined icon and then choosing the correct one. Alt-Tabbing is not a good alternative when I'm switching between more than three applications; clicking on taskbar only once is what I'm looking for. Management of fruit bowls on the Enterprise D I have noticed that guest quarters on the Enterprise D have fruit bowls on their tables.  A prominent instance is the fruit bowl in Berlinghoff Rasmussen's quarters in "".  The first thing that Rasmussen does when Data takes him to his guest quarters is take a piece of fruit and shove it in his pocket.  Who replicates fruit for these fruit bowls before guests arrive?  More generally, who removes unused fruit and tidies quarters after guests leave?  Is there a ship hospitality staff, perhaps including Guinan's wait staff in Ten Forward? What is the best way to dump entire objects to a log in C#?