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The formula for the volume of a sphere is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( V \) is the volume, and \( r \) is the radius of the sphere.
Given that the diameter of the basketball is 9 inches, the radius \( r \) will be half of that:
\[ r = \frac{diameter}{2} = \frac{9\ inches}{2} = 4.5\ inches \]
Now we can use the formula to calculate the volume, using 3.14 as an approximation for pi:
\[ V = \frac{4}{3} \cdot 3.14 \cdot (4.5\ inches)^3 \]
\[ V = \frac{4}{3} \cdot 3.14 \cdot 91.125\ in^3 \]
\[ V = 4.18667 \cdot 91.125\ in^3 \]
\[ V = 381.50625\ in^3 \]
Rounded to the nearest tenth, the volume of the basketball would be:
\[ V \approx 381.5\ in^3 \]
Thus, the correct option is \( 381.5\ inches \) cubed.