If the petition contains more than 500 signatures, the secretary shall, within 30 days from the date such petition is filed, verify such signatures by means of a random sampling. The random sample of signatures shall be drawn in such a manner that every signature filed with the secretary is given an equal opportunity to be included in the sample. Such a random sampling shall include an examination of at least 500 or 5 percent of the signatures, whichever is greater.
If the projection made from the random sampling as to signature validity shows the number of valid signatures as between 90 and 110 percent of the signatures needed to declare the petition sufficient, the secretary shall examine and verify each signature filed.