Given a control chart that shows the mean and upper/lower control limits, how do I tell if the cause of out of control points is assignable or not? 
I am given 15 points. The control limits are at +/- 3 $\sigma$. Points 1, 4, 5, 6, 7, 8, 9, 10, 11, 13, and 15 fall within the control limits. Points 2, 3, 12, and 14 are outside of the control limits, with 2 being below the lower control limit and 3, 12, and 14 being above the upper control limit.
How do I know if points 2, 3, 12, and 14 are out of control caused by chance causes or caused by assignable causes?
 A: Yes, you should find and assignable cause for every point that's outside the limits. But things are a little more complicated.
First you have to determine if the process is in control, since a control chart is meaningless when the process is out of control. Nearly 1/4 of your observations falling outside the limits is a strong sign that the process may be out of control. Looking at the chart would be useful to determine whether the process is under control or not.
Besides falling outside the control limits, there are other potential reasons for needing to look for assignable causes for certain observations. For example, if you have several observations in a row falling on the same side of the mean -- especially if they're near the control limit -- they may need to assigned a special cause.
I might be able to be more specific if you'd post the chart itself.
If you want to learn more about control charts, SPC Press has a number of useful free resources. You might also want to look at this book: it's short, concise and very informative.
(Edit:)
I assumed we were talking about real-world data, not an exam question. In this case, the correct answer really is the first one: the points outside the control limits are (probably) caused by assignable causes.
The exam is a little sloppy in its terminology, though: you can't actually tell with 100% certainty that the points outside the control limits are not caused by chance. You can only say that there is a 99.7% probability that a particular point outside the limits is not caused by chance.
A: My understanding of control charts is a little bit different... After the first signal at observation 2, wouldn't the process would be stopped and checked for problems, and then restarted?
In any case, you could use a p-value argument. The probability of observing 4 or more observations (out of 15) beyond their control limits is VERY tiny if the process is actually in control. Let's say the probability of an observation going outside of the control limits while the process is actually in control is about 0.01 (this exact probability depends on the in control distribution of the data), so if the process is in control, we expect a false alarm (ie out of control signal caused by random chance) every 100 observations or so. The probability of observing 4 or more out of control signals (out of 15) while the process is in control is about 0.000012, so it's very unlikely that the signals are due to random chance.
While an actual diagnosis would require you to look at the chart and possibly actually investigate the physical process, because the out of control points are both below and above the control limits, I'm betting there was a scale shift (i.e. increase in variance.)  
A: (Sorry for posting a new answer, I can't reply to comments directly yet)
I don't really agree with the statement:
"Apparently, if you cross either the UCL or LCL, there has to be an assignable cause" 
To keep things simple, if your in control distribution is N(0,1), then you will still obtain false alarms once every 370 observations, on average, using a UCL of 3 and LCL of -3. When the chart signals, the process needs to be investigated. Only then can a reason for the signal be assigned (ie process change or random error.) Setting the UCL and LCL requires the user to balance the desired false alarm/missed detection rate (analogous to the Type I/Type II error trade off in hypothesis testing.) 
You can also wait until a few signals to actually stop and investigate the process, but in that case, you may detect the shift too late if it really occurred at the first signal. Again, you can't have something for nothing and the user must use their judgment to decide on how to set up the control chart and monitor the process.
A: I found something interesting tucked away in a study document from the IEEE geared toward this exam:

  
*
  
*Data points falling within the UCL and LCL range are considered to be in control and caused by chance causes.
  
*Outliers falling above the UCL or below the LCL are considered to be out of control and caused by assignable causes.
  
*If a number of points fall systematically above or below the mean (but are within the UCL and LCL) this may indicate a nonrandom out-of-control state.
  
*The goal of a control chart is to detect out-of-control states quickly.
  
*The chart, alone, will not indicate the root causes of the event, but it will provide investigative leads.
  

Apparently, if you cross either the UCL or LCL, there has to be an assignable cause.
This makes sense, given the Wikipedia definition of characteristics of assignable (special) cause:

  
*
  
*New, unanticipated, emergent or previously neglected phenomena within the system;
  
*Variation inherently unpredictable, even probabilistically;
  
*Variation outside the historical experience base; and
  
*Evidence of some inherent change in the system or our knowledge of it.
  

