A physical quantity P is related to three observables X , Y , and Z...

The relation for the physical quantity is given by $P = \dfrac{X^3 Y^2}{Z^{1/2}}$.

The maximum percentage error in $P$ is given by the sum of the percentage errors of the individual quantities multiplied by their respective powers:
$\dfrac{\Delta P}{P} \times 100 = 3 \left(\dfrac{\Delta X}{X} \times 100\right) + 2 \left(\dfrac{\Delta Y}{Y} \times 100\right) + \dfrac{1}{2} \left(\dfrac{\Delta Z}{Z} \times 100\right)$

Given that the percentage errors in $P$, $X$, and $Z$ are $15\%$, $1\%$, and $4\%$ respectively, we can substitute these values into the equation:
$15 = 3(1) + 2 \left(\dfrac{\Delta Y}{Y} \times 100\right) + \dfrac{1}{2}(4)$

$15 = 3 + 2 \left(\dfrac{\Delta Y}{Y} \times 100\right) + 2$

$15 = 5 + 2 \left(\dfrac{\Delta Y}{Y} \times 100\right)$

$2 \left(\dfrac{\Delta Y}{Y} \times 100\right) = 10$

$\dfrac{\Delta Y}{Y} \times 100 = 5\%$

The percentage error in the measurement of $Y$ is $5\%$.

Answer: $5\%$