A carpenter uses a hammer to build wooden objects, so we need to find a similar relationship in the analogies. A "metalworker" uses a "welding torch" to build metal objects. "Metalworker" and "welding torch" have the same relationship as carpenter and hammer.

A captain uses a boat, but the boat is not simply a tool for the captain, it is the captain's home. 

Birds hatch from eggs, there is no user or tool in this analogy.

Chefs make food, but food is not the tool of the chef.

The policeman is not the tool of the baton, but vice versa. Analogies must match conceptually, so if you have a user paired with a tool, you need to find an answer that also has a user and a tool.

Mechanics repair cars. Similarly, "plumbers" fix "toilets."

None of the other analogies have this relationship:

Drivers drive limos but typically do not repair them. Filmmakers make movies. Cowboys ride horses. A salesman sells cars.

"Chauffeurs" drive "cars" as their occupation. Similarly, "pilots" fly "planes" as their occupation. 

None of the other analogies have this relationship:

A lawyer works in the court, but does not operate the court in the same way that a pilot or a chauffeur operate a plane or a car. Painters paints pictures, while photography is the art of taking pictures with a camera. Runners run races; there are no tools in this analogy. An oarsman rows a boat, but a steamship has no oarsmen as it is steam-powered, thus the oarsman does not operate the steamship as his profession.

The correct answer is "chalk is to teacher."

This analogy expresses a user and tool relationship. 

A pen is a tool that an author uses to do his or her job. Therefore, we need to find an identical relationship for the second half of the analogy. 

A teacher uses chalk to perform (specific tasks) his or her job. 

A "geologist" is someone who studies rocks, someone who examines rocks. So, to solve this analogy you need to determine which of these answer choices is studied or examined by a "statistician." A "statistician" is primarily concerned with statistics, or "data." To provide extra help, "data" is facts collected and analyzed.

A "grocer" is someone who sells food, and other household products for a living. So, to solve this analogy you need to determine which of these is a product made or sold by a "mason." The correct answer is "stone." A "mason" is someone who cuts and shapes stone and marble for a living.

An "acupuncturist" is someone who uses "needles" to treat a patient. So, to solve this analogy you need to determine which of these is something at a "chiropractor" might use. Well, a "chiropractor" is someone who treats back problems. So, the correct answer is "massage." Additionally, a "surgeon" is a doctor operates on people; "sorcery" is magic; "vaccination" is inoculation, treating someone with a small amount of a disease to prevent them from contracting a more serious version of the disease.

The correct answer is "actor is to script."

This analogy expresses a user/tool relationship. A doctor uses a thermometer to perform his or her job. An actor uses a script to perform his or her job.

The correct answer is "paintbrush is to artist."

This analogy explores the relationship of a tool and the user of the tool. Clippers are used by a barber to perform the task of cutting hair; likewise, a paintbrush is used by an artist to paint.

"Trash" is a noun, and "discard" (which means to throw away) is an action that one typically does with that noun. To complete this analogy, you have to determine which object matches with the action "consume." To "consume" means take in or eat, so the correct answer is “food," because just as one typically discards trash, one typically consumes food.