Answer
$1.667$
Work Step by Step
When we put the plastic piece over the black dot, the microscope will be focused on the virtual image of the black dot, not the black dot itself.
We know that the image distance is given by
$$s'=\dfrac{n_2}{n_1}s$$
where here $n_1$ is for the plastic index of refraction while $n_2$ is for air index of refraction.
Solving for $n_1$,
$$n_1 =\dfrac{n_2}{s'}s$$
Plugging the known (see the figure below),
$$n_{\rm plastic}=n_1 =\dfrac{1.0}{0.6}\cdot 1=\color{red}{\bf 1.667}$$
