The wording of this question seems ambiguous, but also, the answer as explained by the college board does not seem correct. Has anyone else seen this question and reviewed the supposed answer?
? 250 / 2000 is 1/8th, so that means you can just scale everything in the table by 8. So (B).
(Note that I believe there were 2 different Oct PSAT dates, so maybe you took a different test?)
They say that 250 were polled, but that is split between Seniors (120) and Freshman (130). Of the 130 Freshman polled, 30 said math was their favorite. That’s 23%. And you still don’t actually know how many of the 2,000 students are Freshman.
@tptutor I don’t have access to the question, but I recall a similar question on the SAT that I took, in which the correct answer was “E) The answer cannot be determined…” simply because you didn’t know the male/female distribution of the population. The “obvious” answer is obtained by simply multiplying a few numbers, but that choice was incorrect.
If that was an answer choice, perhaps it could’ve been correct. However I don’t know the exact question.
The college board has an explanation on their site. They take the 30 freshman and divide by the whole 250 polled, then multiply that by the full population of 2000, to get 240. Makes now sense - ambiguous information and incorrect math.
@tptutor was it an estimation question of any sort (E.g. “Based on the sample data, about how many…”)?
Otherwise you can’t determine the exact number of freshmen.
Because 250 were polled, and 130 were freshman, then 130 / 250 of the students are freshman. It’s reasonable to predict that 13/25th of the (freshman + seniors) are freshman.
So you could do (30 / 130) * 2000 * (13 / 25) = 240.
(No, “Unknown” was not an answer choice.)
Edit: Remember that the question said there were 2,000 freshmen and seniors. (Not 2,000 students.)
Here’s the College Board’s explanation:
“Based on the table, the proportion of freshmen who selected math as their favorite subject is 30 over 250. If the sample is representative of a high school with 2,000 freshmen and seniors, then there will be a total of the 30 over 250 times 2,000 equals 240 freshmen in the high school who would select math as their favorite high school subject.”
This seems pretty sloppy. You know how may Freshman were polled - 130 - and that 30 of them - 23% - prefer math. To then make an assumption that the number polled accurately reflects distribution of students in each class seems unwarranted. Why not assume 1,000 per class? That seems more reasonable.
But I guess their caveat, “…the sample is representative…”, implies they polled a proportional sample.
@tptutor I agree. The question should say something like “about how many…” I’m pretty sure past SAT questions have done this, but I don’t have a specific example.
In statistics, if you randomly sample some students, it is more common to make a confidence interval (e.g. “95% confident that between 225 and 255 students select Math as their favorite subject”) rather than say “240 students select Math as their favorite subject.”
MITer94,
What do you think about the statement, “… the sample is representative…”?
Is there anything in the way that statistic is formally taught (or conducted) that directly implies that if I polled more people from one population, then that population is larger in that exact proportion?
I guess since the sample size is large enough, it makes sense.
I’m not exactly sure what you mean here - could you elaborate?
It’s true that with a larger sample size, the sample means should converge to the actual population mean (with very high probability). But because in many cases this is impractical, we usually pick a small enough sample size that is still representative. But I’m not much of a statistics question.
Basically, my main point was that the question should’ve said, “About how many…”
The questions can only be answered correctly if you assume that since more Freshman were polled, then there are more Freshman in the population of 2,000 students, and that the expected ratio of Freshman to Seniors is the same as the 130 to 120 selected in the poll.
I thought this was a lot to assume, but wondered if the phrase “… the sample is representative…” directly implies that that is how the polling would be done.
Not a big deal… just curious if they were being sloppy, or I just didn’t know accepted conventions in statistical polling.
This is basically what a representative sample is. If you sample a bunch of people randomly from the entire population, then you should have a fairly representative sample.
An example of a “bad” sample is if you selected the school’s multivariable calculus class as your sample. See why?
I think it’s fine, but a little sloppy on CB’s part.
First of all, I don’t think you can argue that any of the other answers are better than 240. What answer do you think is correct?
Also, I think the question carefully uses the correct statistical terms. It says “the results of a survey of a random sample of…” and then later says “If the sample is representative…, what is the predicted number …?”
If a random sample of 100 people out of 1,000 show that 25% are X, and that sample is representative, than it is correct to predict that 250 in the full population of 1,000 will be X.
From an MITer90… 
I don’t think there is a correct answer… but am ok with the idea that a properly constructed sample would choose Freshman and Seniors in numbers relative to the actual proportion that they exist in the school - making 240 correct.
This is fine. I would certainly pick 240 over other choices.
Quite simply, this is a losing test taking strategy / attitude. There is always (at least) one answer which will be marked correct. Your goal when taking a test is to find that answer. Not to be smarter than the test and figure out why all answers are wrong.
And for this specific question, I think the question was fine, and the answer was right. The only reason you were confused is because you might not be fully familiar with some of those statistics phrases.