Yeah, the wording is quite poor.

The second question about the distribution of random answers to the test is literally incoherent. They probably meant to ask a question about the expected number of random answers on a randomly completed test, but they worded it such that they are asking about the distribution of answers on a single test.

>Assuming that every question in this 20-question test has only 1 right answer and you choose to answer each question randomly, what is the chance that you're so bad that you only get 4 questions right or less?

They probably mean for you to assume that 1) each question has 5 answer choices, and 2) you select one of them with equal probability. But given that this is question #2, assumption (1) is kind of a guess, and (2) should be explicitly stated... (the answer btw has to do with the binomial distribution - and you definitely need some heavy arithmetic, so are they testing your gut instinct?)

The messed up question I was referring to, and I think bloaf was referring to, is a later one, the second one involving random answers:

> Assuming that every question in this 20-question test has only 1 right answer and you choose to guess the answer of each question randomly, then in terms of the number of correctly-guessed answers, approximately 68% of them will fall between:

> A. 2.2 and 5.8

> B. 0.8 and 7.2

> C. 6.8 and 12.5

> D. 7 and 13

> E. None of the above

Another messed up question was the one early on asking whether the dot product operator is "associative/commutative on addition" or not. That's not a thing that actually means something.

I answered none of the above precisely because associativity doesn't apply to an operator that maps 2 vectors to a scalar. But it still felt wrong.

Oh, I was parsing the responses like "(Non-associative, commutative and distributive) on addition". And no, the dot product is not commutative on addition. But it's meant to be "Non-associative, commutative and (distributive on addition)".

I think the following is poorly worded and possibly incorrect:

>What is the area enclosed by the function f(x) = sin²x - x²(sin³x)(cos²x) and the x-axis — between -π/4 and π/4?

Well, the second part is an odd function so we know its areas are 0 over symmetric regions, assuming "area" means "signed area". So we're left with sin²x which has a trick: the integral of sin²x and cos²x are equal over any of the quarter cycles of the graph -- same shape and everything. So we have that over our domain, pi/2 = Int(1) = Int(cos²x+sin²x) = Int(sin²x+sin²x) = 2Int(sin²x)... so the answer would be pi/4 (which doesn't appear).

I also suspect that some of the answers must be wrong, possibly in the counting or probability questions where it is extremely easy to make a logical error.

In my opinion the author should have a third party scrutinize his/her answers. I'm quite certain I didn't get 40% of the questions wrong. (I have a math PhD, taught calculus for 4 years, scored 170 on the quant section of the GRE, and went through this whole "test" very carefully).

The function is non-negative on [-pi/4, pi/4], so you don't need any assumption about area meaning "signed area."

Your math is wrong on that problem. The integral of sin^2 over [-pi/4, pi/4] is (pi - 2)/4, which was one of the options. The functions sin^2 and cos^2 only have equal integrals over properly aligned quarter-cycles like [0, pi/2].

The second summand due to the sin^3 is not nonnegative (so you do need to use oddness to avoid the integral)

Right on the second part, my bad.

I read that part of your comment as re-interpreting the question, instead of just reinterpreting the prior part of the sentence it's in, never mind.

Regardless -- I think it's a valid reading to say the "area between" say, g(x) = x and the x-axis from -1 to 1 equals 1. Only people who are trained would say it's 0. So I don't think something like that really measures "extended problem solving and deductive thinking skills" which is a pretty lofty goal for a test.

> Some of the questions are quite poorly worded, and I'm highly suspicious about one question with non-integral boundaries on a probability distribution defined over the integers.

Yup. It sounds like they were assuming that you would use some specific simplifying approximation.

It looks like the "correct" answer was supposed to be "2.2 and 5.8" -- another poster mentioned the server sends answers to the client. But P(X=3 or X=4 or X=5) is .598.

It looks like they used the normal cdf approximation of the Bernoulli distribution, which would give .6857.

That is some horrible approximation then! The closest range sum to 68% is for {3,4,5,6} summing to 70%, but even if available as an answer, the only correct answer remains "none of the above".