Watch out - under full vacuum you will get a bigger deflection than usual formulas can handle accurately. You need a copy of Roark's Formulas for Stress & Strain. For a simply supported rectangular plate under uniform pressure, it gives for the maximum deflection (at the center, of course),
ymax = -(c p b^4)/(E t^3)
where, for your dimensions,
c ~ 0.060
b = 38 in
E = 30 x 10^6 psi (Young's modulus)
t = 0.25 in
p = 14.7 psi
This formula is only suitable for small deflections, i.e., on the order of the half thickness of the plate or less, and the formula gives something like 4 inches deflection under full vacuum. The formula is therefore close to useless as far as accuracy is concerned.
The minus sign is because the deflection is in the inward direction for positive pressure on the outside. It's just the chose sign convention.
A simply supported plate is one whose edges are free to rotate along the line of the edge but cannot move out of plane. If you clamp the plate with fasteners, you won't have simple support, and your deflection will be smaller than given by the formula. It will be about one third as big for a plate perfectly rigidly clamped along all four edges. But that answer is only valid for small deflections, i.e., small pressure differentials, such that the maximum deflection remains within one half of a plate thickness.