Air is composed of gas molecules, which are pieces of matter and therefore have mass. Here are 2 ways to answer the question, hinging upon what one includes when referring to "the basketball".
First, if the ball is taken to be the rubber "skin" as well as anything enclosed by that skin, then filling it with air will increase the mass of the ball. This is because the ball now has the skin, which has some mass and the amount of that mass has not changed, as well as some amount of enclosed air and that air does have mass.
On the other hand, if the ball is said to be only the skin, then filling it with air will not increase the mass. This is because the mass of the skin is constant regardless of what it contains and the air, although it does have mass, is not considered to be part of the ball.
In the first case, the amount by which the mass of the ball (=skin + air) increases is impossible to determine with only the given information. This is because gases always fill their volume and hence the density (i.e., mass per volume) of gases depends on their pressure Increasing the pressure essentially means . More gas molecules means more mass in the same volume, so density increases. One way to estimate the mass of dry air is with the equation:
M = V*P/(R*T)
where M = mass, V = volume, P = pressure, R = specific gas constant of dry air ~= 287 J/kg-K, and T = absolute temperature (measured from absolute 0, not from 0 C or 0 F).
The diameter of a standard basketball (converted using diameter * pi = circumference) is:
29.5/pi inch = 0.239 cm,
so the volume (when inflated) is:
V = 4/3*pi*(0.239 / 2)3 = 0.00714 m3.
The air pressure of a properly inflated basketball is ~8 lb/in2 gauge pressure, meaning pressure relative to the surrounding atmospheric pressure. The absolute pressure of the air (assuming it is inflated near sea level where ambient pressure is:
1 atmosphere = 14.5 lb/in2, or = 101325 Pa in SI units) is therefore ~22.5 atm ~= 152000 Pa.
Then taking a temperature of ~20 C = 293 K, one finds a mass of air enclosed by the basketball skin as:
M ~= 0.013 kg = 13 g ~= 0.03 lbs.
In this response to an academic journal article, the mass in a basketball was both calculated and measured, with a mass of air in the inflated basketball of ~11.7 g.
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