  We give an estimate for Manolescu's $\kappa$-invariant of a rational homology
3-sphere $Y$ by the data of a spin 4-orbifold bounded by $Y$. By an appropriate
choice of a 4-orbifold, sometimes we can restrict and determine the value of
$\kappa$. We give such examples in case of Dehn surgeries along knots in $S^3$.
