To find the value of n, we need to solve the inequality 1881 < 5n - 2 < 1999.

First, let's solve the left inequality: 1881 < 5n - 2.

Adding 2 to both sides, we get: 1883 < 5n.

Dividing both sides by 5, we get: 376.6 < n.

Since n is an integer, the smallest possible value for n is 377.

Now, let's solve the right inequality: 5n - 2 < 1999.

Adding 2 to both sides, we get: 5n < 2001.

Dividing both sides by 5, we get: n < 400.2.

Since n is an integer, the largest possible value for n is 400.

Therefore, the value of n is any integer between 377 and 400, inclusive.