Credit Card scam telephone call

After getting a bunch of calls from The CI Institute, or whatever they go by, I stopped picking up the phone when the number did not display. Usually, these calls are identified by "Unknown Caller." However, for some reason, I picked one up today. Here was the conversation (one-sided, a recording....)

voice: Hello. This is your credit card company calling to let you know you can reduce your interest rate. Press 1 to speak with an operator....
me:

Maybe if I had more guts (or nothing better to do) I would have hit 1, and tried to deduce who it was. I mean, really. If they couldn't tell me my company's name or my account number, then you'd know, right? And if they did, then I would have had a problem. Anyway, I didn't want to go through with it. But, I wonder how many people do?