Let \(X\) be a finite set with \(n\) elements. A function \(f: X\to\mathbb{R}\) such that \(\sum_{x\in X} f(x)\geq 0\) is called an \(n\)-weight function. \textit{N. Manickam} and \textit{N. M. Singhi} [J. Comb. Theory, Ser. A 46, 91-103 (1988; Zbl 0645.05023)] conjectured the following: if \(d\) is a positive integer and \(f\) is an \(n\)-weight function with \(n\geq 4d\), then there exist at least \({n-1\choose d-1}\) subsets \(Y\) of \(X\) with \(|Y|= d\), for which \(\sum_{y\in Y}f(y)\geq 0\). \textit{T. Bier} and \textit{N. Manickam} [Southeast Asian Bull. Math. 11, 61-68 (1987; Zbl 0723.05120)] showed that the conjecture is not true for all values of \(n\) and \(d\). The present paper proves the conjecture under the condition \(|\{x\in X: f(x)\geq 0\}|\leq d\leq n/2\).