Research Question:  "Why do some months have 30 days and others have 31 days?"

1. http://www.astro.uu.nl/~strous/AA/en/antwoorden/kalenders.html

A month is a period of time that is (historically) tied to the motion of the Moon around the Earth (how long it takes the moon to orbit the earth). Quite a few different kinds of months are in use:  the calendar month: a period of between 28 and 31 days that divides calendars. In some calendars, such as the Gregorian calendar that is used in the West, the months are no longer directly tied to the phases of the Moon. In other calendars, such as the Jewish and Islamic calendars, each month starts with a specific phase of the Moon.

2.  World Book Encyclopedia – states facts about many different culture’s calendars and reveals that, “Most people in the Western world use the Gregorian calendar, designed by Pope Gregory XIII.  It has 12 months, 11 with 30 or 31 days.  February normally has 28 days.” It is so accurate, “the difference between the calendar and solar years is now only about 26 seconds.” 

3.  Database – From HotBot I learned that our calendar comes from the ancient Romans and is based on the sun currently but was once based on the lunar calendar.  During Julius Caesar’s reign, the solar calendar was used but there were extra days which got put into the 1^st, 3rd, 5th, 7th, 9th, and 11^th months and a day was taken away from February.

Some historians think that Augustus Caesar took a day from September and added it to August, the month named after Augustus, and also moved a day from November to December.

That's why August and December now have 31 days, and September and November have 30. But there's no proof that this is the way it really happened.