Usually, the prediction interval has this shape in the image.  I don't know why the end of the interval is wider than the center.
I have noticed that the confidence interval for predicted values in an linear regression tends to be narrow around the mean of the predictor and fat around the minimum and maximum values of the predictor. This can be seen in plots of these 4 linear regressions:    I initially thought this was because most values of the predictors were concentrated around the mean of the predictor. However, I then noticed that the narrow middle of the confidence interval would occur even if many values of were concentrated around the extremes of the predictor, as in the bottom left linear regression, which lots of values of the predictor are concentrated around the minimum of the predictor.  is anyone able to explain why confidence intervals for the predicted values in an linear regression tend to be narrow in the middle and fat at the extremes?
How are "needs to be washed," "needs washing," and the regional variant "needs washed" to be distinguished"? If I leave fully-cooked food (particularly meat) out at warm temperature - say on the counter or in a crock pot that's been turned off -  how long will it stay safe to eat?  Does it make any difference if I re-cook the food afterward? It seems to me that tuples are just as elementary as sets. Can tuples be reduced to sets? I can't seem to get this hat on straight! I carefully positioned my hat to fit my head just right:     But it seems to be shifted to the left a few pixels elsewhere (same in Firefox and Chrome):   Stack Overfow   Skeptics   The top bar is also affected on some sites:   Stack Overfow   Meta SE   InfoSec   However, it is in the correct position in the top bar on some sites somehow:   Skeptics   Bricks   It's not a huge deal of course, it just bothers perfectionists like me. How do you concatenate characters in java? Concatenating strings would only require a + between the strings, but concatenating chars using + will change the value of the char into ascii and hence giving a numerical output. I want to do System.out.println(char1+char2+char3... and create a String word like this.  I could do   System.out.print(char1); System.out.print(char2); System.out.print(char3);   But, this will only get me the characters in 1 line. I need it as a string. Any help would be appreciated.  Thanks I'm including \usepackage{hyperref} so that each instance of \ref (as well as each page number in the index and the table of contents) automatically links to that page.  When I'm viewing the document (in ) and I click a link, I jump to the linked page.  Is there any easy way to go "back" to where I was before I clicked the link?  (And is the answer any different when using Adobe Acrobat reader?) Nautilus not generating thumbnails in 13.10 Like the title says, Nautilus is not generating thumbnails for any file types (pictures, documents, pdf, etc) after a clean install of 13.10.  I have checked the preferences and it is set to always generate thumbnails.  Help please! Iterating through columns in table using ModelBuilder? I am unable to find any info on how to use ModelBuilder to run a certain spatial analysis function on each column in a table.  I have a table to sample locations and at each location there are a series of measures (e.g. temperature, flow rate, number of certain species).  For each of those attributes (columns in the table), I was to run the IDW function from the surface tools.  This would be easy if I had each attribute in a separate shp/feature class, but not very efficient.  I tried Iterate Feature Selection, but was unable to pass the field name as the Z value in the IDW function.  This is for ArcGIS 10.1 (Basic and Spatial Analyst license) Show $\frac{1}{4}\leq \mu(A\cap B \cap C)$ Let $\{X,\mathcal{A},\mu\}$ be a probability space. Let $A$, $B$, $C$ be sets in $\mathcal{A}$, which satisfy $$\mu (A\cap B)\geq \frac{1}{2}, \quad \mu (A\cap C)\geq \frac{1}{2}, \quad \mu (B\cap C)\geq \frac{1}{2}.$$ Show that $\frac{1}{4}\leq \mu(A\cap B \cap C)$.  I've managed to show in a previous assignment that $$\mu(A\cap B \cap C)\leq \min\{\mu (A\cap B), \mu (A\cap C), \mu (B\cap C)\},$$ but now I'm stuck with the above inequality. I've tried to draw it, but still no luck. Any hints? I completed the two contracts I was given (Scout and Heavy), and earned two of the 'Teufort' weapons for turning them in.  I expected to get some new contracts so I could continue to earn some more of the schmancy new weapons, but I can't seem to find out how to get more.  Do I need to buy the ticket again to unlock more contracts? If not, how are contracts assigned? Where can I find every gnome? I shifted to Ubuntu 10.10 now I am facing problem to have C/C++ compiler. What are the possible ways so that I can use C/C++ compiler using all the libraries (like graphic, math, conio, stdlib, etc) as in Microsoft Windows creating executable file? One of the ways I quickly rename files in Windows is   F2 &gt; Rename &gt; Tab (to next file) &gt; Rename ...   But in Ubuntu/Nautilus, I can't tab to next file. But being on Linux, I think there must be a command line alternative.  However, sometimes, I may want more control over how to rename specific files. In that case, perhaps its better to be able to tab to the next file Does the potential energy of a single object exist? I know that potential energy exists for a system of objects. But why? Is there no potential energy for a single object system? I assume that there is.  Consider the earth-object system. Suppose an external force lifts the object above the earth. Then the work done by it gets stored in the earth-object system as potential energy.Right?  Now, my question is that is the potential energy always zero for a single object system.   MY REASONING  If we consider the object only as the system and lift it above the Earth, we are doing positive work on it and filling it with energy and the gravitational field of the Earth is doing negative work on it and draining it of energy. Thus whenever the object is moving it has kinetic energy which is fine. But when it is not moving,i.e. when its weight is equal to the force applied by us, the net work done is zero and so the object should not have any kinetic energy. Does it have potential energy? Variance of sum of $k$ randomly drawn numbers from $1,...,n$. How to get less spacing in math mode LaTeX commands like this:  Here again, four different simulations are run on networks of size  $C^{G} = 25 \times\ 25 = 625$, $C^{G} = 50 \times\ 50 = 2500$, $C^{G} = 250 \times\ 250 = 62500$, and $C^{G} = 500 \times\ 500 = 250000$...   When compiled generate the following:    The spacing in math mode really irritates me. There's too much white space around the equal sign, and the multiplication sign has more right padding than left padding. Is there any way to reduce this spacing? How to export SVG without lines, polylines and polygons? Can I travel to the UK unaccompanied on an accompanied child visa now that I am over 18? Why many abstract nouns get an indefinite article while it's taught that abstract nouns shouldn't get it? $K \subset \mathbb{R}^n$ is compact iff it is closed and bounded I want to prove $\Leftarrow$.  I just need one more step, now. If I find a compact set that contains K, then it will finish the proof. Rudin take a k-cell to do that, but before it he proves that every k-cell is compact, and I think I can take a more simple compact to do this step. I has thinked in the closed ball, but don't know how to prove that it is compact. Can anyone help me to find this set, if it really exists? Thanks.