I'm not going to do your homework for you. Instead of trying to plug numbers into some formula, go back to what Nyquist and Shannon were really saying.
You want to send data encoded with 16 different analog levels. Divide the voltage range of your signal up so that 16 levels are maximally spaced apart. That means each level will be 1/15 of the range from the next. This means each level has 1/32 noise margin. In other words, you can add up to 1/32 noise to each signal and still be able to distinguish which level it is at the other end.
The question now becomes, what is the time for the worst case step to settle to within 1/32 of its final value. That is the absolute minimum time the transmitter has to dwell on each level for it to be distinguishable at the other end. I'll let you work out the 1/32 settling time of a step that is limited to 3 kHz. In the end, you have to remember you are sending Log2(16) = 4 bits at a time.
That was the case without noise. What noise does is add a certain amount of error that never settles away. Convert the noise level to the fraction of full scale. The signal needs to settle to within the 1/32 level minus the noise. Without noise, your minimum settling level is 1/32 = .03125. If you think the maximum noise is .01, for example, then you have to wait for settling to within .03125 - .01 = .02125. Conversely, you can do this calculation in logarithmic scale, typically in units of dB.
There is more you can do if you are allowed to make some assumptions about the noise. Actually, we already made one assumption above, which is that the noise has a maximum voltage excursion. That may not be true from just a dB figure. If, for example, you know the noise will average out over time (doesn't cover the full 3 kHz bandwidth, in this case not including low frequencies), then you can eventually recover a level even if the noise amplitude is larger than the error band around each level. Effectively though, you are reducing the bandwidth of your channel because you are adding low pass filtering at the receiver. By the way, this is actually how GPS receivers decode some of the signals. The satellite signals are so weak that they are something like 20 dB below the noise floor. They are recovered by some fancy math, with one way to look at it being that a lot of filtering is applied at the receiver, effectively reducing the frequency range of the channel to where the signal does exceed the noise.