What good is multiple choice?

open question:
what is the expected percentage of examinees to answer a multiple choice question correctly, accounting for random guessing?

thoughts:
if a question has five multiple choice answers, a student has 20% chance to blind guess the correct answer. If the number of examinees to answer the question right is significantly lower than 20%, does that suggest that the question is unfair, intentionally misleading, or incorrect?

for example question 87 on GRE Physics 9677 had 6%! of students answering correctly. They'd have done better not studying ANY physics and blindly picking ABCD or E! Why did this question make more students answer INCORRECTLY than one could reasonably expect for a multiple choice answer!?

I think it shows a certain intelligence that the majority of students put similar thoughts to avoid the correct answer. Like when Muhammad Ali purposefully chose wrong answers on his IQ test to get out of the army, and the testers concluded that to answer that many answers wrong, far more than one would expect from just random guessing, he must be aware of the right answer and purposefully avoiding it. Here of course, the physics students have no reason to avoid answering correctly for a low score, but I believe that they are onto the right idea, but getting tricked by the question.

But were the 6% who answered correctly truly more expert in their knowledge, or were they 'worse thinkers' for not arriving at the same conclusions as their peers!? Or did they simply get lucky with a guess?

What value does a test have when over half the questions are expected to be answered incorrectly by the average test taker, and the majority of questions have a correct answer rate no better than guessing, and some significantly worse!??

OH I forgot there is a 1/4 point penalty for guessing. So that means a hard problem will have fewer people answering. And the percent answering correctly is out of all test takers, including those that didn't answer the question at all. So maybe 6% answered the question and 6% got it right with 0% answering the problem incorrectly, not necessarily that 100% answered and 94% got the question wrong.

I pick (E) Not enough information provided to arrive at a conclusion.

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But even then, a multiple choice test is not a valid reflection of real life problem solving. You get better at these tests by practicing shortcuts to eliminate wrong answers, then quickly selecting the better choice of two or three candidates.

Solving worthwhile problems isn't like picking the most attractive bachelor on a TV dating show. You don't just take what has already been done. You have to create an answer nobody has found before.

GRE 9677 problem 2. Faraday's Law of Inductance

GRE Physics practice exam 9677

Problem:
 

2. The circuit shown above is in a uniform magnetic field that is into the page and is decreasing in magnitude at the rate of 150 tesla/second. The ammeter reads:

(A) 0.15 A
(B) 0.35 A
(C) 0.50 A
(D) 0.65 A
(E) 0.80 A

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Solution:

Correct answer: (B) 0.35 A.

Percent of test takers answered correctly: 29%

By Faraday's Law of Inductance, a circuit in the presence of a changing magnetic field ... induces an emf voltage... with the direction of the magnetic field when the magnetic field is decreasing...

Stated in the Maxwell-Farady equation

In the simple arrangement of the problem, the magnetic field is uniform and the line and surface integrals are trivial so the equation has the simple form

With only one loop in the circuit, we reduce all this down to just:

|V| = (|rate of change in magnetic field|) x (area)

= (150 tesla/s) * (0.1 m x 0.1 m) = 0.15 V

Now we use the "right hand rule" to find that the magnetic field is producing a clockwise current. (Right hand thumb points with the magnetic field into the page, fingers curl clockwise with the direction of induced current.)

The voltage generated by the decrease in magnetic field acts against the direction of change...

"if the magnetic field is increasing, the induced field acts in opposition to it. If it is decreasing, the induced field acts in the direction of the applied field to try to keep it constant." Lenz's Law http://hyperphysics.phy-astr.gsu.edu/hbase/electric/farlaw.html

So in this case the induced current flows with the magnetic field, in the clockwise direction.
 
The current in the circuit diagram is in a counter-clockwise direction. (The little bar on the DC voltage source is the negative terminal and the big bar opposite is positive. By convention, current flows negative to positive.)

We have all then all the components to the circuit equation:

V_dc - V_induced = I x R

(5.0 V) - (0.15 V) =  I x (10 ohms)

gives the solution (B) for the current

I = 0.35 A

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Sidenotes

Tesla is measure of magnetic field strength = Volt * second / meter^2

Wikipedia (tesla unit) gives some reference numbers for real world magnetic fields:

• 31.869 µT (3.2 × 10⁻⁵ T) – strength of Earth's magnetic field at 0° latitude, 0° longitude
• 5 mT – the strength of a typical refrigerator magnet
• 1.25 T – magnetic flux density at the surface of a neodymium magnet
• 1 T to 2.4 T – coil gap of a typical loudspeaker magnet
• 1.5 T to 3 T – strength of medical magnetic resonance imaging systems in practice, experimentally up to 17 T^[12]

So the 150 Tesla /s decrease in this problem is from a HUGE magnetic field holy crap! Any field with THAT much strength beats anything humans can generate. It would more likely come from some massive collapsing star!

• 45 T – the current (2015) world record for continuous field magnets ^[19]
• 10⁸ - 10¹¹ T (100 MT-100 GT) – magnetic strength of the average magnetar

So this problem seems like an academic textbook-ish exercise, not real world applicable!

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Only 29% of test takers got this question right. I don't think people couldn't think of what to do. I think whether to add or subtract the induced voltage is what threw most test takers off.

A temptation is to point the current opposite to the magnetic field. But Lenz's Law states that the direction of the current opposes the change in magnetic field, not the magnetic field itself! In this case, the induced current went in the same direction as the magnetic field.