$15/hr * 40 hr/wk * 50 wk/yr (2 weeks vacation) = $30,000/yr. Most people would consider that a living wage. Federal poverty level for a family of 4 is just $24,250/yr.
So the $15/hr target is too high. If you target the poverty level for a family of 4 (assuming it's a single income family), the target is $12.12/hr. Poverty level for a single person is $11,770/yr, which translates into $5.89/hr, which is actually below the current minimum wage of $7.25/hr. So the current minimum wage is in the right ballpark of a compromise between singles and single-income families.
Yes this assumes full employment throughout the year. The minimum wage has to be tied to productivity because wages are tied to productivity. If you try to set the minimum wage based on poverty levels for people not being productive the full year, you end up eliminating jobs of people who are fully employed and productive the full year. Inability to find full employment is an employment problem (number of jobs available), not a wage problem (how much you're paid for a job).
IMHO the problem isn't the minimum wage, it's the capital gains tax is way too high for lower income people. People always complain the 15% capital gains tax is too low without really researching who actually pays a 15% income tax. The tax rate is graduated meaning just because you're in the 25% tax bracket doesn't mean you pay a 15% income tax. The threshold where you actually pay a 15% income tax (single, standard deduction) is about $58,500. The threshold where the average American pays 15% income tax (after credits, exemptions, and itemized deductions) is closer to $90,000 (you can figure this out from the IRS tax stats). So it makes little sense for people making less than this to invest their money when it's going to be taxed more than if they just spent it and increased their income via raises (e.g. raising the minimum wage) rather than investments/savings.
The economy rewards you with income for two things - generating productivity (working), and deciding where productivity is needed (managing/investing). The current flat 15% capital gains tax effectively discourages lower income people from participating in the latter. It needs to be graduated like income tax so lower income people have more incentive to save and invest. (The rationale for the capital gains tax rate being lower than income tax rate at higher incomes is the same. It encourages rich people to invest their money thus re-injecting it into the economy, instead of wasting it on gold toilet seats. Same logic applies to lower income people, except some of them "waste" their money on big screen TVs, iPhones, car leases, etc.)