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?
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.
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.