2003 Saturn LW wont start?

My 2003 Saturn (65K mi.) wont start. When I turn the key it attempts but doesnt catch and start. After 5 minutes of attempting and waiting, then it starts and runs great. The starter doesnt sound weak and i have a new battery. THe temp outside is around 22 degrees. And the engine light comes on sometimes but other times it doesnt. After an episode its fine for a while - starts right up. I went to AutoZone and they said they recvd bad responses about the transmission - modules inside (??) anyway, why would the transmission impact the ability to start the car? are those diagnostics always accurate? ideas please....