Literature

Fact: human males have a voice pitch in the range 90-160 Hz, with a mean of 120 Hz. The range for women is 180-280 Hz. There is a significant difference - this is vocal sexual dimorphism.
Hypothesis: a Darwinian explanation suggests that there is a link between voice pitch and reproductive success.
Testing the hypothesis: an association has been found, in the Hazda tribe of Tanzania, between the depth of pitch of male voices and the number of children the subjects father.
Conclusion: "These findings suggest that the association between voice pitch and reproductive success in men is mediated by differential access to fecund women. Furthermore, they show that there is currently selection pressure for low-pitch voices in men."

Whilst this sounds like the authors are using the scientific method, the details are not so clear. The authors admit: "we don't know the exact reason that these men with deeper voices have fathered more children". They draw from the work of others to suggest possible mechanisms: "Previous studies have also shown a relationship between testosterone and deeper vocal pitch, and so increased testosterone may contribute to the male's ability to hunt." Successful hunters are likely to be recognised by the tribe, and these men may then be regarded as desirable husbands, perhaps marry earlier and thereby have greater opportunity to father children. None of these steps of the explanation appear to have been validated.
There appears to be no questions rising in the minds of researchers about extrapolating from the Hadza, a Tanzanian hunter-gatherer tribe, to humans in general. The press release comments:

Because of their similarity to the hunter-gatherer lifestyle of our ancestors, the reproductive success of the Hadza could be indicative the way that human beings evolved. "It's possible that vocal dimorphism has evolved over thousands of years, partly due to mate selection," says Apicella. "Perhaps at one time, men and women's voices were closer in pitch than they are today."

These comments deserve to be handled more critically, because although the Hazda are hunter-gatherers, they are not characterised as carnivores. They eat a starchy diet heavy in roots and tubers. How do we know they are not descendants of farming ancestors and no more representative of our ancestors than any other human community?
Can a design framework for research help us here? This framework would posit advantages for sexual dimorphism (including vocal characteristics) that would go beyond reproductive success by predicting a variety of other benefits for individuals and for society. The design framework is not inconsistent with reproductive success, but it would suggest that focusing on this one consequence is a case of trivialising the phenomenon.

Voice pitch predicts reproductive success in male hunter-gatherers
C.L. Apicella, D.R. Feinberg, F.W. Marlowe
Biology Letters, FirstCite, September 25 2007 | doi 10.1098/rsbl.2007.0410

Abstract: The validity of evolutionary explanations of vocal sexual dimorphism hinges upon whether or not individuals with more sexually dimorphic voices have higher reproductive success than individuals with less dimorphic voices. However, due to modern birth control methods, these data are rarely described, and mating success is often used as a second-rate proxy. Here, we test whether voice pitch predicts reproductive success, number of children born and child mortality in an evolutionarily relevant population of hunter-gatherers. While we find that voice pitch is not related to reproductive outcomes in women, we find that men with low voice pitch have higher reproductive success and more children born to them. However, voice pitch in men does not predict child mortality. These findings suggest that the association between voice pitch and reproductive success in men is mediated by differential access to fecund women. Furthermore, they show that there is currently selection pressure for low-pitch voices in men.

See also:

Male voice pitch predicts reproductive success in hunter-gatherers, EurekAlert, 25 September 2007.