What your asking about comes as part of a full collage course in Thermodynamics. (one of the top 10 most difficult courses on the planet).
But you just need the equation.
p2/p1 = (v1/v2)^gama
For air gama is a constant 1.4, just trust me. For a 15:1 air fuel mixture a gama of 1.3 is generally accepted. For the compression test best to shut the fuel off to conduct it, at least if your going to do the science project discussed here.
You can't get the effective compression ratio without knowing the geometric compression ratio, intake valve closing time, bore, stroke and connecting rod length and doing some really elaborate math and trigonometry. rod length and intake valve closing time are difficult to get. Therefore if you can get the published compression test requirements at sea level and multiply that by 0.853 would be your best bet.
But let me explain any way.
v1 is the volume of the cylinder + the volume of the combustion chamber - the volume of the cylinder below valve closing.
v2 is simple its just the volume of the combustion chamber and when I say volume of the combustion chamber I mean everything above the piston when its at TDC.
in your case p1 = 14.7 * 0.853 = 12.54
Solve for p2
p2 = ((v1/v2)^gama)*p1
we dont need to know v1 or v2 just need to know the ratios. v1/v2 = the effective compression ratio. So lets just assume its 7 for the moment.
Plug in ((7)^1.4)/12.54 = 191.2 PSIA
As dugald pointed out that is 191.2 PSIA (A for absolute) in your case subtract 12.54 to get PSIG (G for what the gauge should read)
Let me caution you about one thing. The automotive industry uses a different way to state compression ratio and is you do happen to find a publisher value for effective compression ratio I am not sure how it is calculated.
in automotive terminology CR = Volume of cylinder/volume of combustion chamber. The v1/v2 in the thermodynamics equation if it were to be looked at as geometric compression ratio would = (Volume of cylinder + volume of combustion chamber)/volume of combustion chamber))for example a 9:1 CR engine would be solved as v1/v2 = 10. Just another little thing to piss you off.
And finally the standard disclaimer. There is no such thing as a perfect isotropic or adiabatic process. But since the motion is reasonably fast with reasonably low temperatures involved there is little time for the temperature rise to be dissipated to the block or water jacket, and thereby loosing its stored energy. Therefor the adiabatic process closely approximates the situation.
See this for a brief discussion.
https://www.grc.nasa.gov/www/k-12/airplane/compexp.html