A bowling ball with a negative initial velocity slows down as it rolls down the lane toward the pins Is the bowling ball's acceleration positive or negative as it rolls toward the pins?

In these situations, you usually define one direction as being positive and the other as negative. In this problem, the question does this for us: it clearly tells us that when the ball is moving down the lane, its velocity is negative. "Down the lane" is the negative direction.

Average acceleration is give by

a_ave = Δv/Δt

where Δv is a change in velocity and Δt is elapsed time.

Δv = v_final - v_initial

In this problem, the ball "slowed down," but did not change direction. This means that the NUMBER associated with "v" got smaller (5 ... 4 ... 3 ...) but the SIGN (-) did not change.

I hope it makes sense, then, that "v_final" was a smaller (slower) number with a negative sign (moving down the lane) while "v_initial" was a larger (faster) number with a negative sign (moving down the lane).

The subtraction of a large negative number from a small one (like, for instance, -4 - [-10]) is a POSITIVE number whose value is given by the difference (+6).

So Δv is positive, and Δt is ALWAYS positive (no matter what).
Therefore a_ave, the quotient of two positive numbers, will be positive.