I have a set of 2000 numbers that form an approximately normal distribution, with a mean of about 0.24, and an SD of 0.06.  These form an estimate of the population.  I now have a single sample measurement of 0.38 and would like to determine the chance (p value) that this single sample measurement comes from the same population distribution.  (Unfortunately, in my work the nature of the "test" means that it is unfortunately theoretically impossible to obtain another measurement).    It must be true that the further that the single sample is from the mean of the population, the more unlikely that single event is.  Furthermore, of course one needs to take into account the distribution of the population - had the SD of the population been 6 rather than 0.6 the chance of obtaining the single measurement at 0.38 would be much higher, and hence the p value lower.  I am happy to assume for my purposes that the distribution from which the sample has been obtained would have the same SD as the population.  The question is how I calculate the likelihood (p-value) that the observed single sample comes from a distribution with the same mean as the population?  I have thought about carrying out a one sample t-test, testing whether the observed population is different from the single sample value, but that doesn't seem right to me as that test assumes that the single sample value is the mean of the distribution being tested against, rather than a single sample.  What is the best approach to this analysis?  Thanks in advance for any advice and help.
For an ecology project, my lab group added vinegar to 4 tanks containing equal volumes of pond water, 1 control with no elodea (an aquatic plant) and 3 treatments with the same amount of elodea in each. The purpose of adding the vinegar was to reduce the pH. The hypothesis was that the tanks with elodea would go back to their normal pH quicker. This was indeed the case. We measured the pH of each tank daily for about two weeks. All the tanks eventually returned to their natural pH, but the length of time that this took was much shorter for the tanks with elodea.  When we told our professor about our experimental design, he said that there exists no statistical test that can be performed on the data to compare the control to the treatment. That because there was no replicate for the control (we only used one control tank) we cannot calculate variance and so we can't compare the sample means of the control and the treatment. So my question is, is this true? I definitely understand what he means. For example, if you took the height of one man and one woman, you can't draw conclusions about their respective populations. But we did 3 treatments, and the variance was small. It seems reasonable to assume that the variance would be similar in the control?  Update:  Thank you for the excellent answer. We got more water and elodea from the wetland and decided we would run the experiment again with smaller tanks but this time with 5 controls and 5 treatments. We were going to combine this with our original data but the starting pH of the tanks was different enough that it doesn't seem valid to consider the new experiment to be sampled from the same population as the original experiment.  We considered adding different amounts of elodea and trying to correlate speed of pH remediation (measured as time elapsed until pH returned to its original value) with amount of elodea, but we decided that wasn't necessary. Our objective is only to show that the elodea makes a positive difference, not to construct some kind of predictive model for exactly how the pH responds to differing amounts of elodea. It would be interesting to determine the optimal amount of elodea, but that's probably just the maximum amount that can survive. Trying to fit a regression curve to the data wouldn't be especially illuminating because of the various complicated changes that occur to the community when adding a large amount. The elodea dies, decomposes, new organisms start to dominate, and so on.
Effect of each parameter on a Monte Carlo Simulation I was wondering what is the best way to determine the effect of each random parameter on the result obtained from a Monte Carlo Simulation.  I realise I have asked a similar question , but this time I am making this question in a more general manner as I didn't get any answers previously.   Background: I'm essentially running a Monte Carlo Simulation where I sample randomly from 65 Normal Distributions and 2 Uniform Distributions. Then, i then input these values to a linear equation and run this for 500.000 iterations. I just wanted to determine the best way to establish the relative effect of each of these 67 (in total) random variables on the final outcome from my Monte Carlo Simulation. How to store and preserve film negatives and slides? I bought 35mm dedicated film scanner (Nikon CoolScan III) and I'm planning to digitize old negatives and slides I found at home.  How should I store and preserve these slides and films after I cut them and scan? How to choose best negative pages/sleeves, binders etc. and what to avoid? How does the Comma Operator work Are humans the only animal that requires "clean water"? Package incompatibilites: etoolbox, hyperref, and bm, standalone? I am trying to transition to using the etoolbox package, but seem to have run into some difficulties and they appear related to the hyperref and bm packages which to me seem totally unrelated.  The MWE example compiles since \RemoveEToolbox is defined and shows that there is no apparent problem before etoolbox was used.    Commenting out the \def\RemoveEToolbox{} line produces two problems:   With both the \renewcommand, and the usepackage{hyperref} commented, this results in "Runaway argument? Paragraph ended before \bm@test@token was complete". Including the usepackage{hyperref} (with \renewcommand commented) results in "Undefined control  sequence. \Call@AtVeryEndDocument ...cumentHook \@undefined"   But if uncomment the \renewcommand{\bm}{#1} line (effectively doing without any \bm), then this seems to compile.  I have been using this for a while now and things appeared to have been working fine, but now that I attempt to use etoolbox I seem to run into problems.  \documentclass{standalone} \def\RemoveEToolbox{}% Shows that this works fine without {etoolbox}  \usepackage{amsmath} \usepackage{xcolor} \usepackage{bm}  \ifdefined\RemoveEToolbox     \newcommand*{\iftoggle}[3]{#2}% Default to true value of "if" \else     \usepackage{etoolbox}     \newtoggle{paper}     \toggletrue{paper}      %\renewcommand{\bm}[1]{#1}% Why can't I have a normal \bm with {etoolbox}? \fi  % \usepackage{hyperref}  \begin{document}  This document is intended for  \iftoggle{paper}{paper}{electronic} distribution.  \bm{\textcolor{blue}{Solve $x^2-1=0$}}  \begin{align*}     a &amp;= b\\     \iftoggle{paper}{         c &amp;= d\\     }{}     e &amp;= f\\ \end{align*} \end{document} I want to quantify the relationship between two variables, A and B, using mutual information. The way to compute it is by binning the observations (see example Python code below). However, what factors determines what number of bins is reasonable? I need the computation to be fast so I cannot simply use a lot of bins to be on the safe side.  from sklearn.metrics import mutual_info_score  def calc_MI(x, y, bins):     c_xy = np.histogram2d(x, y, bins)[0]     mi = mutual_info_score(None, None, contingency=c_xy)     return mi Getting equation from differential equations I have:  $\dfrac {dx} {dt}$=$-x+y$  $\dfrac {dy}{dt}$=$-x-y$  and I am trying to find $x(t)$ and $y(t)$ given that $x(0)=0$ and $y(0)=1$  I know to do this I need to decouple the equations so that I only have to deal with one variable but the decoupling is what I am having trouble with  Do I set them equal to each other and then just move like terms to separate sides getting two different equations and then integrate? New to the cloud so pardon this question as I'm certain it's relatively simple!  I've setup my Google Cloud Instance and I'm now trying to connect to it using FileZilla. I have a good understanding of how to do this... but I've been completely unable to download my .ppk file, public key, and private keys. The documentation clearly gives the folder that the files are located in.... but I'm unable to navigate to these folders (it's as if they don't exist) furthermore I'm not really sure how to download something to my computer through the SSH terminal.  Also somewhat infuriating is that I've found the keys in the web portal on Google Cloud however there is no option to download them. I can only view them in clear text.  Any help would be greatly appreciated! Omitting the Milliseconds in a Date Can someone please let me know the minimum system requirements and GPU requirement to run blender 2.80 and eevee smoothly?   Also please let me know the optimum system requirements for the purpose of    environment modeling Game development charachter modelling and animation   Also if someone can suggest me a good desktop/laptop for personal use that is compatible with blender 2.80 in a low budget.  Any help will be much appreciated.  Thanks in advance! jQuery call function from a string Trying to find vehicles which are free between 2 variable dates Can we have some tools to handle link rot? There are many links posted in answers, and some in questions, that are either stale or broken.  An example is the change to  on-line documentation this year to do away with separate 'Optional' and 'Core' tasks areas in the documentation - the old links 404 now.  I've fixed a bunch of 'optionaltasks' broken links, but searching for 'coretasks' in Stack&nbsp;Overflow questions returns over 300 matches...  Are there any tools for us to review and fix these - with an emphasis on the fixing - without sifting through completely manually?  Is it time for some refactoring tools for Stack&nbsp;Overflow content, in particular, for links?  Also, I'd be interested to know if there is an analysis of links (as found in the Stack&nbsp;Overflow data dumps) out there. Answer: there is a  of a sample of links and images, but it's getting a bit out-of-date. $\displaystyle\int_{0}^{2\pi}\frac{1}{a+ \cos\theta}\,d\theta$ Updating a field programmatically, hook_node_update Currently trying to update a field every time a node is created or updated. However the value is not being populated within the node, do I have access to the node object with this particular hook? What could I be missing?    function vbtoken_node_update($node) {         entity_get_controller('node')-&gt;resetCache(array($node-&gt;nid));         $types = node_type_get_types(); //What are the current Node Content Types?       $yes = ($types['volunteer_project']-&gt;type);        if($node-&gt;type === $yes){           $hash = md5($node-&gt;title . $node-&gt;nid . $node-&gt;nid);         $hashed = substr($hash, 0, 6);         $node = node_load($node-&gt;nid);         $node-&gt;tcode[$node-&gt;language][0]['value'] = $hashed;         node_save($node);          watchdog('vbtoken', 'Added a new Token code to %nid', array('%nid' =&gt; $node-&gt;nid));          }         else          {           dpm('not working dude');         }      } Cauchy distribution characteristic function Can sweetened condensed milk be substituted for whole milk and sugar? I am making a fudge recipe that calls for 1cup whole milk and 5 cups sugar. It also uses 2 sticks of butter and 25 marshmallows. I'm wondering if I can omit the milk and cut back on the sugar by using sweetened condensed milk. Let $B$ be a real Banach Space whose dimension is at least $2$, and let $S$ be a subset of $B$ that is an open ball. Is the complement of $S$ (with respect to $B$) always connected?  Idea One could then perhaps make use of the facts that every $2$-dimensional Banach space is homeomorphic to the Euclidean plane and that - for the Euclidean plane - my question has an obvious &quot;YES&quot; answer (since every open ball is the interior of a circle). When Apple was in the high $600 or so range, it split 6 times and made it more affordable to purchase more shares for less, even though the value was technically the same. I see companies like Google are extremely high per share and am wondering why or why not they would or wouldn't split their stock right now. Necromancy In Harry Potter How is Necromancy accomplished in the Harry Potter universe?  I thought that people cannot be raised from the dead in Harry Potter.  Were bodies just use as puppets that can only do simple tasks, for instance when Dumbledore and Harry were getting the locket, the  were there to guard the locket.  Also, besides Inferi were there any Zombies in Harry Potter?