Actually, you can diagnose the water pump problem wither way. With the engine idling and operating temperature you can open the radiator cap and look inside (BE CAREFUL!! The coolant is HOT! Don't put your face right over the filler neck. Shine a flash light down in the neck) There should be a constant flow of coolant if you rev the engine a little the coolant flow should increase.

Otherwise you can use a mechanics stethescope and listen to the water pump housing.