First, you want a deep-cycle battery, not a car starting battery. The latter is designed to deliver high-current for short periods of time and not be discharged very deeply at all. If you do, it'll have a very short life.
A Group 27 size deep-cycle battery typically has a capacity of 100 amp-hours at a 5A (20 hour) discharge rate. However, if you want reasonable life out of it, you won't discharge it below 50%.
An 85% efficient inverter (converts 12VDC to 120VAC) typically draws 10% of the appliance's 120VAC wattage as amps from the battery. So a 120VAC appliance that really uses 50 watts will cause the inverter to draw about 5 amps from the battery. In theory, that appliance will discharge a fully charged 100 amp-hour battery from 100% charge down to 50% charge in 50amp-hour/5amps=10hours.
On a notebook power supply, you have to consider whether the power rating is on the input or the output. If it's output, you need to multiply that by about 1.15 to find the wattage it inputs because the power supply is also less than 100% efficient. If you only know the output voltage and maximum output current, you can multiply those together and then by 1.15 to estimate the input wattage. In your case, that's 19 V x 3.42 A = 65 output watts x 1.15 = an estimated 75 input watts.
Keep in mind that the notebook power supply's rating is its maximum capacity, i.e. what it could deliver when operating the notebook while charging its battery if heavily discharged. If you're constantly operating the notebook off the deep-cycle battery, it's like being plugged into residential power all the time, so the notebook battery won't discharge, and the notebook power supply will never be drawing near it's maximum.
If we estimate the power supply to use 80 watts maximum, we might guess it uses 40 watts or less when the notebook battery is fully charged, causing the inveter to draw 4 amps from the deep cycle battery. In that case, 50 amp-hours divided by 4 amps will give just over 12 hours of use on 50% of the Group 27 battery's amp-hour capacity.
