I am 72 years old receive a pension and social security, do I have to continue to pay Federal Income Taxes? We live in Michigan.I was told that after a person reaches 70 years old you are exempt from paying Federal Income tax, is this correct?

No one is ever exempt at any age.
If your SS and pension aren't very much, it's possible that you won't owe tax, or very little. Make sure that you are not counting any SS over the threshold (and for a couple if married). Maybe have a tax preparer go over your 2010 form just to be sure. But the pension is fully taxable, as is any interest and dividends on your savings. If you have paid off a mortgage, you have lost that deduction.