Why is P value interpreted under null hypothesis? is explained very nicely in the link . But still i have not understood  Why is P value interpreted under null hypothesis?
If we disbelieve $H_0$, why quote a p value calculated assuming $H_0$ was true? Hypothesis testing seeks to reject a null hypothesis ($H_0$) on the basis of an assumption made about the sample following a certain distribution. This assumption is conditional on $H_0$ being true. So what is the flaw with the following reasoning?     If I reject $H_0$ in favor of the alternative hypothesis ($H_1$), I   must then assume that (1) $H_1$ is true, and (2) that at the   same time the variable under examination follows the distribution   used to reject $H_0$. Since we also stated that the assumed   distribution is conditional on $H_0$ being true, it appears that $H_0$   has got to be true in order to justify the assumption made about the   distribution. Logically, it would therefore seem that both $H_0$ and   $H_1$ have to be true, which is a paradox, because they are mutually   exclusive!
Prove the definitions of $e$ to be equivalent How to prove that the following definitions of $e^x$ are equivalent, with as simple tools as possible and without any knowledge of $e$ or logarithms ?  $$\sum_{n=0}^{\infty} \frac{x^n}{n!}=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n$$  Also preferably, prove that this is $a^x$ for some real number $a&gt;0$. How to add tags to files in Windows 7 so that they appear in search results In Windows 7, I want to add tags to my files (PDF, Word, Images etc) so that they appear directly in the search results, regardless of their content, title etc.    shows how to search by tag, but it does not tell how to add a tag in any file.  In the long term, I'm planning to add such tags programmatically as well. Creating polygons from points in Mapinfo Ordinal dependent variable with continuous independent variables What is a plain English explanation of "Big O" notation? I'd prefer as little formal definition as possible and simple mathematics. What are the integer solutions to $a^{b^2} = b^a$ with $a, b \ge 2$ I saw this in quora.  What are all the integer solutions to $a^{b^2} = b^a$ with $a, b \ge 2$?  Solutions I have found so far:  $a = 2^4 = 16, b = 2, a^{b^2}  = 2^{4\cdot 4} =2^{16}, b^a = 2^{16} $.  $a = 3^3, b = 3, a^{b^2} = 3^{3\cdot 9} =3^{27}, b^a = 3^{3^3} =3^{27} $.  In the general case, $a$ and $b$ have the same set of prime divisors, so let $a =\prod_P p_i^{a_i}$, $b =\prod_P p_i^{b_i} $ with each $a_i \ge 1, b_i \ge 1$.  $b^a =b^{\prod p_i^{a_i}} =(\prod p_j^{b_j})^{\prod p_i^{a_i}} =\prod p_j^{b_j\prod p_i^{a_i}} $  $a^{b^2} =a^{\prod p_i^{2b_i}} =(\prod p_j^{a_j})^{\prod p_i^{2b_i}} =\prod p_j^{a_j\prod p_i^{2b_i}} $  Therefore, for each $p_j$, $b_j\prod p_i^{a_i} =a_j\prod p_i^{2b_i} $.  I haven't gotten any further than this.  I conjecture that there are no other solutions. How to access java-classes in the default-package? I'm working now together with others in a grails project. I have to write some Java-classes. But I need access to an searchable object created with groovy. It seems, that this object has to be placed in the default-package.   My question is: Is there a way to access this object in the default-package from a Java-class in a named package? Does anyone know the location of this login hotmail background? How to change the base of logarithmic-scale plots? How can aperture be f/11 on a lens with an aperture range designation of 3.5-5.6? Traditionally  at the end of a filename after uploading it via the image uploader. That doesn't seem to work any more. I've tested it on Q&amp;A as well as chat.        (Yes, I know big M doesn't work)  Could we have this fixed please? Difference between shutdown and shutdownNow of Executor Service I'm using Fedora and wonder if there is some way I can exclude kernel updates when I run yum update, until I specifically want to update the kernel.    The reason why I am concerned about this is because sometimes it requires a lot of extra research finding patches for some programs and extra time spent recompiling them to work on the new kernel.  I am thinking VMWare specifically as one example.   I wonder if there is some way for yum update to NOT update the kernel and only notify me that a new one is available when it goes through its routine so that I can decide to put it off until a more convenient time.  Or is this thinking likely to cause other issues? Using expect for inputting commands Heres the thread I'm bouncing off of:   So I'm trying to run this command: cd /etc/openvpn/easy-rsa; . ./vars; ./build-key username  However, after running that command, I need to input nothing [just press enter] 10 times, and press y twice. The user on the last thread suggested I use except, but I couldn't get it working. Here are the questions it asks:  root@suffice-vpn:/etc/openvpn/easy-rsa# ./build-key usernamtest Generating a 2048 bit RSA private key ..................+++ ........+++ writing new private key to 'usernamtest.key' ----- You are about to be asked to enter information that will be incorporated into your certificate request. What you are about to enter is what is called a Distinguished Name or a DN. There are quite a few fields but you can leave some blank For some fields there will be a default value, If you enter '.', the field will be left blank. ----- Country Name (2 letter code) [US]: State or Province Name (full name) [NY]: Locality Name (eg, city) [Merrick]: Organization Name (eg, company) [IceWare]: Organizational Unit Name (eg, section) [IceWare]: Common Name (eg, your name or your server's hostname) [usernamtest]: Name [server]: Email Address [iceware@programmer.net]:  Please enter the following 'extra' attributes to be sent with your certificate request A challenge password []: An optional company name []: Using configuration from /etc/openvpn/easy-rsa/openssl-1.0.0.cnf Check that the request matches the signature Signature ok The Subject's Distinguished Name is as follows countryName           :PRINTABLE:'US' stateOrProvinceName   :PRINTABLE:'NY' localityName          :PRINTABLE:'Merrick' organizationName      :PRINTABLE:'IceWare' organizationalUnitName:PRINTABLE:'IceWare' commonName            :PRINTABLE:'usernamtest' name                  :PRINTABLE:'server' emailAddress          :IA5STRING:'iceware@programmer.net' Certificate is to be certified until Jan 14 18:27:48 2026 GMT (3650 days) Sign the certificate? [y/n]:y   1 out of 1 certificate requests certified, commit? [y/n]y   As you can see, I press enter 8 times initially [to keep the default input] and then press enter another 2 times to skip the optional commands. Then, I need to press y twice. How can I automate this? I need to have this all happen in one command. Any suggestions? Thanks in advance. $n \mid (a^{n}-b^{n}) \ \Longrightarrow$ $n \mid \frac{a^{n}-b^{n}}{a-b}$ How does one prove that if $n \mid (a^{n}-b^{n}) \ \Longrightarrow$ $ \displaystyle n \mid \frac{a^{n}-b^{n}}{a-b}$ where $a,b, n \in \mathbb{N}$.  What i thought of is to consider $$(a-b)^{n} \equiv a^{n} + (-1)^{n}b^{n} \ (\text{mod} \ n)$$ and if we suppose that $n$ is odd then we have, $$(a-b)^{n} \equiv a^{n} -b^{n} \ (\text{mod} \ n)$$ and since $n \mid (a^{n} - b^{n})$ we have $$(a-b)^{n} \equiv 0 \ (\text{mod} \ n) $$  I think i am far away from the conclusion of the problem, but this is what i could work on regarding the problem. I'm trying to install a dual boot (with Window 10) on my new XPS-15 (with a SSD 256Go).  Everything works well (following  until I go on step "Installation type".  Where I can't select "Something else", and I already have the partition table but completely empty. I can't see any existing partition, or add new one. Prevent questions with movie/tv shows/game spoilers from appearing in the hot questions list I've noticed that occasionally "hot questions" from SciFi and the Movies &amp; TV network contain major spoilers about the movie/show discussed.  The Matrix spoilers:      (just now there's something about Cypher betraying Morpheus in the Matrix, which is a big deal for people who haven't seen the movie).   I like browsing through those but fear that one day one of these questions will spoil a show I'm excited about and yet to finish   Breaking Bad spoilers:      (like the finale of Breaking Bad, imagine a question asking why did one character betray another at the end or something).   Can users posting such questions be asked to label those with a tag (such as 'spoilers') and then somehow have the network hide those from the "hot questions" list? Or possibly format them differently?  Similarly, this could apply to the Arcade network too.  P.S. Love how this question spikes in popularity around the time GoT starts running again. I swear it should be fairly simple technically to ban questions that have a tag that says 'spoilers' from appearing in the Hot Network Questions.  P.P.S. Oh, and with every fresh release of Star Wars (The Last Jedi being the most recent one)!  P.P.P.S. Another year, another Game of Thrones, another spike in popularity! A 64-bit double can represent integer +/- 253 exactly. Given this fact, I choose to use a double type as a single type for all my types, since my largest integer is an unsigned 32-bit number. But now I have to print these pseudo integers, but the problem is they are also mixed in with actual doubles. So how do I print these doubles nicely in Java? I have tried String.format(&quot;%f&quot;, value), which is close, except I get a lot of trailing zeros for small values. Here's an example output of of %f  232.00000000 0.18000000000 1237875192.0 4.5800000000 0.00000000 1.23450000  What I want is:  232 0.18 1237875192 4.58 0 1.2345  Sure I can write a function to trim those zeros, but that's lot of performance loss due to string manipulation. Can I do better with other format code?  The answers by Tom E. and Jeremy S. are unacceptable as they both arbitrarily rounds to two decimal places. Please understand the problem before answering.  Please note that String.format(format, args...) is locale-dependent (see answers below). Reset Search Index in Exchange 2013 Why does Earth behave like Natural Bar Magnet? What is the reason for the Earth to behave like a bar magnet and have poles (North and South poles)?