I am new to statistics and have run into some trouble understanding computing confidence intervals and am seeking some help. I will outline the motivating example in my textbook and hopefully someone can offer some guidance.   Example   There is a population of mean values and your goal is to figure out the true mean (as best you can). In order to accomplish this, a number of samples are taken, each of which has a mean value.   Next, because we know by the central limit theorem that as the number of samples increase, the sampling distribution will be normally distributed, we use the equation $z = \frac{X - \bar{X}}{s}$  (noting that in this case s = standard error) to compute a lower and upper bound taking each sample mean as the mean for the z-score equation and z-scores of -1.96 and +1.96, for example,  to compute a 95% confidence interval.   I’ve included a graph from my textbook in attempt to add clarity.    So I do not understand how it is you can use each sample mean as the mean value in our z equation to compute intervals. We know that the sample distribution is normally distributed so isn’t it the case that only the mean of all the samples can be used? How can we compute an interval around each mean value that contributes to the sampling distribution?  Any help with this would be much appreciated    Note: I'm reading "Discovering Statistics Using IBM SPSS Statistics 3rd Edition" by Andy Field and this example is from pg 43-45
I know roughly and informally what a confidence interval is.  However, I can't seem to wrap my head around one rather important detail:  According to Wikipedia:     A confidence interval does not predict that the true value of the parameter has a particular probability of being in the confidence interval given the data actually obtained.     I've also seen similar points made in several places on this site.  A more correct definition, also from Wikipedia, is:     if confidence intervals are constructed across many separate data analyses of repeated (and possibly different) experiments, the proportion of such intervals that contain the true value of the parameter will approximately match the confidence level   Again, I've seen similar points made in several places on this site.  I don't get it.  If, under repeated experiments, the fraction of computed confidence intervals that contain the true parameter $\theta$ is $(1 - \alpha)$, then how can the probability that $\theta$ is in the confidence interval computed for the actual experiment be anything other than $(1 - \alpha)$?  I'm looking for the following in an answer:   Clarification of the distinction between the incorrect and correct definitions above. A formal, precise definition of a confidence interval that clearly shows why the first definition is wrong. A concrete example of a case where the first definition is spectacularly wrong, even if the underlying model is correct.
I heard Andrew Ng (in a video I unfortunately can't find anymore) talk about how the understanding of local minima in deep learning problems has changed in the sense that they are now regarded as less problematic because in high-dimensional spaces (encountered in deep learning) critical points are more likely to be saddle points or plateaus rather than local minima.   I've seen papers (e.g. ) that discuss assumptions under which "every local minimum is a global minimum". These assumptions are all rather technical, but from what I understand they tend to impose a structure on the neural network that make it somewhat linear.  Is it a valid claim that, in deep learning (incl. nonlinear architectures), plateaus are more likely than local minima? And if so, is there a (possibly mathematical) intuition behind it?   Is there anything particular about deep learning and saddle points? I think this story dates from the 1950s, plus or minus a decade - I read it about 30-35 years ago in a (then) fairly old story collection. It involves explorers on a planet falling into a gelatinous object that dissolves most of their bodies except for their brains and nervous systems. About four of them fall in to the same creature, and they discover that by thinking about it they can grow limbs and other body parts. At first they try to fight each other, by growing arms so they can hit each other's brains - and one of them realises that growing a skull would be a good idea - but later they agree to grow legs and move the creature around co-operatively. I would love to identify this story. Are data members allocated in the same memory space as their objects in C++? I'm not entirely clear on how Counterspell works, when trying to counter higher-level spells: the CS description only talks about disrupting a creature you can see, within 60 feet, casting a spell.   However, when trying to counter a higher level spell:   Do I know what spell is being cast?  Do I at least know (somehow intuit, based on VSM components) the level of the spell? Or can I tell the spell/level of the spell by sensing a disturbance in The Weave? (I.e., if the spell is being cast using Subtle Spell, can I still tell what's going on?)   The point of the question is to determine how fruitful CS can be in combat--if I know what the other guy/gal is casting (or at least the level), I can decide whether I want to spend my L8 slot to counter his/her L8 slot. But if I have no idea, and I end up using a L8 slot to counter a L3 spell... seems a bit painful.  Hope this made sense! Locking an iPhone if button is broken? My friend's iPhone's sleep/wake button no longer works. Is there any way to turn off the screen besides setting up auto-lock and waiting a few minutes? Crossfading Video AND Audio with ffmpeg This Super User answer  has got me so far, but now that I have tried many solutions and seen every SO link out there, it's time to ask you all for help.  I am trying to crossfade 2 videos that are 10 seconds each in ts format with audio and video at the same time.   The steps are:   Add complex filter black screen Fade out video 1 after 9 seconds, with 1 second fade duration Fade in video 2 at 9 seconds, with 1 second fade duration Trim black filter output to 19 seconds   This works fine for just video alone, but when I attempt to add audio, I can't get past this error message:     Output pad "default" with type audio of the filter instance "Parsed_asetpts_4" of asetpts not connected to any destination   It appears I'm not connecting the audio output properly. I think I need to alter the [over] lines to include audio. But how do I do this?  `ffmpeg -i vid1.ts -i vid2.ts -f lavfi -i color=black -filter_complex \ "[0:v]format=pix_fmts=yuva420p,\ fade=t=out:st=9:d=1:alpha=1,setpts=PTS-STARTPTS[v0];\ [0:a]\ afade=t=out:st=9:d=1,asetpts=PTS-STARTPTS[a0];\ [1:v]format=pix_fmts=yuva420p,\ fade=t=in:st=0:d=1:alpha=1,setpts=PTS-STARTPTS+9/TB[v1];\ [1:a]\ afade=t=in:st=0:d=1,asetpts=PTS-STARTPTS+9/TB[a1];\ [a0][a1]amix=inputs=2;\  [2:v]scale=720x406,trim=duration=19[over];\ [over][v0]overlay[over1];\ [over1][v1]overlay=format=yuv420[outv]" -vcodec libx264 -y -map [outv] final.ts` Travelling on Business visa to UK for tourism purposes Am I Running as a Service "elements required by agent" is "required" here an adjective or verb? "elements required by agent" is "required" here an adjective or verb?  another example "elements expressed by person". How do Horcruxes work? A few questions regarding the Horcruxes.   Had Voldemort actually succeeded in using a Horcrux, would he come back the same age (biologically and/or aesthetically) he was when he made the Horcrux, or the age he would biologically be if he had never died (half died)? If Harry was a Horcrux, then in the graveyard, when Wormtail added Harry's blood to the potion to bring Voldemort back, didn't that technically use up the Horcrux inside of Harry? So in Deathly Hallows when Voldemort kills Harry, shouldn't Harry have actually died rather than just have the Horcrux inside him destroyed, seeing as it was technically used in the Goblet of Fire? Evaluating the limit of a sequence given by recurrence relation $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$. Is my solution correct? Problem  The sequence $(a_n)_{n=1}^\infty$ is given by recurrence relation:   $a_1=\sqrt2$, $a_{n+1}=\sqrt{2+a_n}$.   Evaluate the limit $\lim_{n\to\infty} a_n$.  Solution   Show that the sequence $(a_n)_{n=1}^\infty$ is monotonic. The statement $$V(n): a_n &lt; a_{n+1}$$ holds for $n = 1$, that is $\sqrt2 &lt; \sqrt{2+\sqrt2}$. Let us assume the statement holds for $n$ and show that $V(n) \implies V(n+1)$. We have that $$a_n &lt; a_{n+1}.$$ Adding 2 to both sides and taking square roots, we have that $$\sqrt{2+a_n} &lt; \sqrt{2+a_{n+1}},$$ that is $a_{n+1} &lt; a_{n+2}$ by definition. Find bounds for $a_n$. The statement $$W(n): 0 &lt; a_n &lt; 2$$ holds for $n=1$, that is $0 &lt; \sqrt2 &lt; 2$. Let us assume the statement holds for $n$ and show that $W(n) \implies W(n+1)$. We have that $$0 &lt; a_n &lt; 2.$$ Adding two and taking square roots, we have that $$0 &lt; \sqrt2 &lt; \sqrt{2+a_n} &lt; \sqrt4 = 2.$$ The limit $\lim_{n\to\infty} a_n$ exists, because $(a_n)_{n=1}^\infty$ is a bounded monotonic sequence. Let $A = \lim_{n\to\infty} a_n$. Therefore the limit $\lim_{n \to\infty} a_{n+1}$ exists as well and $\lim_{n \to\infty} a_{n+1} = A$. (For $(n_k)_{k=1}^\infty = (2,3,4, \dots)$, we have that $(a_{n_k})_{k=1}^\infty$ is a subsequence of $(a_n)_{n=1}^\infty$, from which the statement follows.) We have that $a_{n+1} = f(a_n)$. That means that $A = \lim_{n\to\infty} a_n = \lim_{n \to\infty} {f(a_n)} = f(\lim_{n \to\infty} a_n) = f(A) = \sqrt{2 + A}$. Solving the equation $A = \sqrt{2 + A}$, we get $A = -1 \lor A = 2$. Putting it all together, we get that $A = 2$, because the terms of the sequence are increasing and $a_1 &gt; 0$.   Is my solution correct? Site specific magic comment links What's the best way of handling permissions for Apache 2's user www-data in /var/www? Has anyone got a nice solution for handling files in /var/www? We're running Name Based Virtual Hosts and the Apache 2 user is www-data.  We've got two regular users &amp; root. So when messing with files in /var/www, rather than having to...  chown -R www-data:www-data   ...all the time, what's a good way of handling this?  Supplementary question: How hardcore do you then go on permissions?  This one has always been a problem in collaborative development environments. Installing Elementary OS - will it automatically be added to the Grub menu? I currently have Windows 7 and Linux Mint installed.  When my PC starts up, it shows me the Grub menu, it shows an entry for Mint Linux and the Windows boot menu.  I am wondering, if I was to install Elementary OS, would it be automatically added to grub or do I have to put some command in the terminal?  I also want to ask, when I boot from my Elementary OS cd, and install it, just before it finishes, it shows some kind of error saying some error about cannot install to grub or something, I've tried installing to a separate partition and alongside "Them", but had no luck.  I'm really new to Linux, and want to keep my Mint Linux and Windows 7 because I have files on them. natbib: combining chicago style and sorting by appearance How to evaluate environment variables into a string in Python? Difference between "speed up something" and "speed something up" Both seem to be in common usage, but is there a nuanced difference about when one is more appropriate than the other? Thanks. How come unpolarized light does not undergo destructive interference? (Ubuntu/Linux)"find" command gives different results based on quotation marks I was using   find -iname *prib*   and getting only one result "./prib09jack.pdf" with   find -iname "*prib*"    I get many more results that I was looking for such as  ./Dir1/PRIB09/prib09jackTFF1.pdf   and many more, but why did it matter that the quotation marks were around "prib"?   Best. Replacing the core of a solenoid to increase the pulling force