Statement 1: The pattern and position of fringes always remain same even after the introduction of transparent medium in a path of one of the slit.

Statement 2: The central fringe is bright or dark is depend upon the initial phase difference between the two coherence sources.

If a transparent medium of thickness t and refractive index $\mu$ is introduced in the path of one of the slits, then effective path in air is increased by an amount $(\mu -1)$ due to introduction of plate.

Therefore, the zeroth fringe shifts to a new position where the two optical paths are equal. In such case fringe width remains unchanged.

The central fringe is bright or dark depends upon the initial phase difference between the two coherent sources.