I have two groups which are about their departments in college (For example, group1=faculty of x, group2=faculty of y). However, the size of the group1 is 12 and size of the group2 is 99. There is a big gap between the group size. My intention is to test their perceptions based on their departmen. Is it appropriate to compare them? I will use independent t-test.
I have two independent groups: from one I have drawn a sample of size less than 30 due to its small population, while the second group has sample size 100.  Is it appropriate to use an independent samples t-test in this case?
Suppose you have a group isomorphism given by the first isomorphism theorem: $$G/\ker(\phi) \simeq \operatorname{im}(\phi)$$ What can we say about the group $\ker(\phi)\times \operatorname{im}(\phi)$? In particular, when does the following hold: $$G\simeq \ker(\phi)\times \operatorname{im}(\phi)?$$ I ask this question because i want to prove that $GL_n^+(\mathbb{R}) \simeq SL_n(\mathbb{R}) \times \mathbb{R}^*_{&gt;0}$, with $GL_n^+(\mathbb{R})$ the group of matrices with positive determinant. I proved that $SL_n(\mathbb{R})$ is a normal subgroup and that $GL_n^+(\mathbb{R})/  SL_n(\mathbb{R}) \simeq \mathbb{R}^*_{&gt;0}$, using the surjective homomorphism $\det(M)$. I tried something with semidirect products but I got stuck. Show that $\gcd(a^2, b^2) = \gcd(a,b)^2$ \dots versus \ldots - is there a difference? Is there a difference between \dots and \ldots, the ellipsis?  If yes: Which? When to use \dots and when to use \ldots?  If no: Why are there both at all? In how many different ways can you split $10$ people into two groups with the same amount of people?  My attempt: Since the order in which you choose someone doesn't matter, I chose to calculate the combinations: $10 \choose 5$ How to create the matrix attached that has 2x2 blocks in its diagonal and a square outlining them? I don't know about anyone else, but I have a reasonable list of favourites now, and it can be rather difficult to find the relevant ones at any given time. Sometimes, I have a specific one in mind, others I just want to see all favourites relating to a specific topic. It would be nice if there were a way to organise my favourites by tag, or OP, or possibly just search them.  How can we improve the favourites feature?  Please post your ideas on how we might improve the favourites feature of the Stack&nbsp;Exchange sites. Try to keep each answer to one idea so that the votes reflect the popularity of a single idea. Prove that: $ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$ How to prove the following trignometric identity? $$ \cot7\frac12 ^\circ = \sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$$  Using half angle formulas, I am getting a number for $\cot7\frac12 ^\circ $, but I don't know how to show it to equal the number $\sqrt2 + \sqrt3 + \sqrt4 + \sqrt6$.  I would however like to learn the technique of dealing with surds such as these, especially in trignometric problems as I have a lot of similar problems and I don't have a clue as to how to deal with those.  Hints please!  EDIT:  What I have done using half angles is this: (and please note, for convenience, I am dropping the degree symbols. The angles here are in degrees however).  I know that $$ \cos 15 = \dfrac{\sqrt3+1}{2\sqrt2}$$  So,   $$\sin7.5 = \sqrt{\dfrac{1-\cos 15} {2}}$$ $$\cos7.5 = \sqrt{\dfrac{1+\cos 15} {2}} $$  $$\implies \cot 7.5 = \sqrt{\dfrac{2\sqrt2 + \sqrt3 + 1} {2\sqrt2 - \sqrt3 + 1}} $$ I'd like to be able to tail the output of a server log file that has messages like:  INFO SEVERE   etc, and if it's SEVERE, show the line in red; if it's INFO, in green.  What kind of alias can I setup for a tail command that would help me do this? How does SQL Server handle logins when the user is a part of multiple groups with various privileges? I'm trying to wrap my head around how SQL Server decides to grant certain rights based on a users login privileges.  If a user is part of a group "READ ONLY" which has access to DB1, but they are also a part of a group "SysAdmin" which has full db owner rights to the same DB1, which rights are they given when they log in?  On a similar note. What if they are part of a group but also have a standalone account with other middle of the road privileges? Wifi doesnt work on fresh ubuntu 16.04 The set of limit points of the sequence $1,\frac12,\frac14,\frac34,\frac18,\frac38,\frac58,\frac78,\frac1{16},\frac3{16},\ldots$ Stein's lemma condition (Apologies if I break some conventions, this is my first time posting!)  I am working on proving Stein's characterization of the Normal distribution: for Z $\sim N(0,1)$ and some differentiable function $f$ with $E[|f'(Z)|] &lt; \infty$, $$E[Zf(Z)] = E[f'(Z)]$$ Writing the LHS expression in integral form and integrating by parts, I eventually obtain: $$E[Zf(Z)] = \frac{1}{\sqrt{2\pi}} \left[ -f(z) \cdot \exp \left\{ \frac{-z^2}{2} \right\} \right] \Bigg|_{-\infty}^{\infty} + E[f'(Z)]$$ Now I need to show that the first expression on the right hand size is zero. Intuitively, this seems clear because of the exponential term, but I am having trouble explicitly applying the condition on $f'$ to prove this rigorously. Any ideas? Converse to Chinese Remainder Theorem So as seen on this question , we know that if $(n,m) \neq 1$, then $\mathbb{Z} /mn \mathbb{Z} \ncong \mathbb{Z}/n\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z}$, because the right hand side does not have an element of order $nm$.  But take a more general setting. Let $R$ be a commutative ring, and let $A,B$ be ideals in $R$. $A$ and $B$ are said to be comaximal if $A+B=R$. If $A, B$ are comaximal, then we have: $R/AB \cong R/A \times R/B$. In this setup, is the converse true?      If we have $A, B$ ideals in $R$ such that $R/AB \cong R/A \times R/B$, do we always have that $A$ and $B$ are comaximal? Persistent live drive with a partition for persistence There are some web sites describing  new tools new features in [new versions of] old tools  that can create a persistent live drive from an iso file with Ubuntu 19.10 and the corresponding Ubuntu family flavours.  Why and how does it work? How is it different from earlier methods for persistent live drives? Isn't live the default for a cloned drive? Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$ Method defined as testMethod do not support web service callouts, test skipped I have been struggling with this error for a few days now. I get this error every time I try to run a test for a method that has an httpCallout. I followed the architecture from  , but to no avail. I thought it could be something I was messing up in my code, so I copied all three of the classes from the documentation, and I am still getting the error when I try to run the test. All of the classes are on Salesforce API 29. Has anyone run into this before? how to add a custom lock for the recovery and/or bootloader? Unlocking the bootloader and/or the recovery is very popular discussion across many devices. But doing so exposes a very obvious and inevitable issue with security!  An unlocked recovery means that no matter how secure you make your android lock screen, someone can easily reset phone to restore the phone to a fully functional (and lockless/insecure) state and the worst part, they don't have to erase your data(meaning they can access all your personal files and folders)!  It is almost hilarious! In fact if the phone supports insertion of external SD cards, you can even flash your own software into the phone and maybe bypass the lock screen without wiping any data.  For example: One possible loophole is using the 'Tasker' app which has a feature that can temporarily disable the keyguard. And the best part is that Tasker can be flashed using the recovery!  So is there any method by which I could add a security measure that prevents complete access to my recovery and/or boot loader that will make my phone less insecure?!  Expected answer: Method by which a custom pin/password must be entered before entering the bootloader/recovery. So that my bootloader/recovery is still technically unlocked but simply requires authentication to use (this also ensures that conventional unlocking methods won't work as the phone is not really locked!). This also means that one does not have to lose root access to ensure security! How to get pixel data from a UIImage (Cocoa Touch) or CGImage (Core Graphics)? When updating, I get the following error message:   W: The repository 'http://ppa.launchpad.net/mc3man/trusty-media/ubuntu xenial Release' does not have a Release file.   Here, I find another statement on this error:   This recommends removing certain PPAs; and, I'm not sure if I should do that since it might mean not getting the updates that I need.    Is this what I should do? top x% in [tag] : rules for the tag to be taken into account?