150 spherical marbles, each of diameter 1.4 cm

We have,

the radius of spherical marble, $r=\frac{1.4}{2}=0.7 \mathrm{~cm}$ and

the radius of the cylindrical vessel, $R=\frac{7}{2} \mathrm{~cm}=3.5 \mathrm{~cm}$

Let the rise in the level of water in the vessel be $H$.

Now,

Volume of water rised in the cylindrical vessel $=$ Volume of 150 spherical marbles

$\Rightarrow \pi R^{2} H=150 \times \frac{4}{3} \pi r^{3}$

$\Rightarrow R^{2} H=200 r^{3}$

$\Rightarrow 3.5 \times 3.5 \times H=200 \times 0.7 \times 0.7 \times 0.7$

$\Rightarrow H=\frac{200 \times 0.7 \times 0.7 \times 0.7}{3.5 \times 3.5}$

$\therefore H=5.6 \mathrm{~cm}$

So, the rise in the level of water in the vessel is 5.6 cm.

Disclaimer: The diameter of the spherical marbles should be 1.4 cm instead 14 cm. The has been corrected above.