The correct answers!

[theweaselking]

I'm going to go back and unscreen all the guesses to the earlier questions in a minute, but here are the correct answers:

The first three questions are all about seeing an answer that the question tries to suggest, which is wrong, and rejecting it. MOST people got these right.

1. A bat and a ball cost $1.10 in total. The bat costs $1.00 more than the ball. How much does the ball cost?

Suggested answer: $0.10. Correct answer: $0.05. The ball costs $0.05, thus the bat costs $1 more ($1.05), thus the combination costs $1.10

2. If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?

Suggested answer: 5/5/5 becomes 100/100/100! Correct answer: 5 minutes. It takes ONE machine 5 minutes to make ONE widget. Running 100 machines at once means each takes 5 minutes, so your total time to make 100 widgets is... 5 minutes.

3. In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

Suggested answer: half the time! Correct answer: 47 days. It doubles each day, therefore if it's covered 100% on day 48, it's covered 50% on day 47.



The fourth question is a little trickier, and far fewer people got it right. It's about the problem of small samples.

4. A certain town is served by two hospitals. In the larger hospital about 45 babies are born each day, and in the smaller hospital about 15 babies are born each day. As you know, about 50% of all babies are boys. However, the exact percentage varies from day to day. Sometimes it may be higher than 50%, sometimes lower.
For a period of 1 year, each hospital recorded the days on which more than 60% of the babies born were boys. Which hospital do you think recorded more such days?

a)The larger hospital
b)The smaller hospital
c) About the same (that is, within 5% of each other)

Here, C) is the obvious answer: The same babies are being born! Why would the numbers be different? Well, that's wrong.

I don't think anyone guessed A, but the rest of the guesses were split between B and C. The correct answer is B, because the smaller sample size means that it takes fewer anomalies to deliver an outside-the-normal result.

In the smaller hospital, 15 babies are born each day. Assuming ~50% boys (as the question does), you expect 7.5 boys per day. "60% boys" is only 9/15 - you only need the 0.5 to go boy (which it will do on 50% of days) and ONE other coin flip to go boys for that hospital to hit 60% boys for a day. I'll spare you the math, but that should happen 30.4% of the time. Here's a chart:



In the LARGER hospital, 45 babies are born each day. Assuming ~50% boys, you expect 22.5 boys per day. "60% boys" is 27/45 - you need the swing baby and FOUR more babies for that hospital to hit 60% boys on a day. Again, math skipped for the overview, but that's about 11.6%. Chart:



So, the small hospital, in the course of a year, expects about a hundred and ten "60% or more boys" days. The large hospital only expects 40ish.



For the record, I find it SUPER INTERESTING that on question 4, nobody said A. Not even as a "random guess".

Comments

[ronebofh.livejournal.com]

Hmm, if i'd noticed the earlier post, i would have said A. Also, you need to redo your math; the problem as stated says more than 60%. Your numbers are at 60% exactly.

[falconwarrior.livejournal.com]

well i got #4 wrong, but that just means i learned something new about statistics today!

[lafinjack.livejournal.com]

[theweaselking.livejournal.com]

With discrete trials like this, I would expect the conversational-english phrasing of the question to mean "60% or more" - but if you redo as "strictly greater than but not equal to 60%" the answer is still B, it's just not quite as dominant. It's also no longer measuring the same thing - the small hospital is now measuring days when they pass 66% boys whereas the large hospital is counting days where they pass 62% boys, instead of both measuring days when they reach 60%.

But it's still B. I'll do the math with charts in an hour or so.

[theweaselking.livejournal.com]

"10 or more out of 15", 66%, happens 15% of the time. Hospital B expects 55 days a year to get "greater and not equal to 60%" boys.

"28 or more out of 45", 62%, happens 6.7% of the time. Hospital A expects 24 days a year to get "greater and not equal to 60%" boys.

The answer is still B.

If you hold them both to the same standard again ("66% or more") Hospital A need 30, and is down to meeting that 1.7% of the time.

[ronebofh.livejournal.com]

Yeah, i figured it would still be the same, but precision matters!

[frith]

Tsk. I should have figured out the first one. As for the fourth, I could see that the smaller hospital had a smaller sample size and so is more prone to statistical error than the larger hospital, but I guessed that it wasn't that prone to error. 9_9 I should have gone with the psychology angle -- if you are asking the question, then you are looking for "small sample size really does skew the data that badly" answer.