<p>A statistical manifold is just a manifold equipped with a torsion-free affine connection and a Riemannian metric that satisfies the Codazzi equation. Although the term originates from information geometry, statistical manifolds have long existed in affine differential geometry. The class of statistical manifolds with cubic forms divisible by the metric arises from affine differential geometry. We examine the geodesic connectedness of affine connections on this class of statistical manifolds. In information geometry, the geodesic connectedness of the affine connections are often assumed. In Riemannian geometry, the geodesic connectedness of the Levi-Civita connection follows from its geodesic completeness by the well-known Hopf-Rinow theorem. However, the geodesic connectedness of general affine connections is more challenging to achieve, even for the Levi-Civita connection in pseudo-Riemannian geometry or for affine connections on compact manifolds. Motivated by the Hopf-Rinow theorem in Riemannian geometry, we show conditions under which geodesic completeness implies geodesic connectedness for affine connections on statistical manifolds with cubic forms divisible by the metric. As an application, we establish a Cartan-Hadamard type theorem for statistical manifolds.</p>