My wife was given this problem on an exam:
A town is served by two hospitals. In the larger hospital, about 45 babies are born each day and in the smaller hospital about 15 a day. About 50% of all babies are boys but the exact percentage varies from day to day, sometimes being higher than 50% and sometimes lower than 50%. For a period of one 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?
1. The larger hospital
2. The smaller hospital
3. About the same
I approached it using an extreme example. Suppose the larger hospital had 200 births each day. The odds of all being boys is near zero. Suppose the smaller hospital had 2 births each day. The odds of all being boys is one-third. Therefore it is more likely that the smaller hospital, with a smaller sample size, would have a blip of over 60%, so the answer is the smaller hospital.
The instructor said the correct answer was the larger hospital.
What say my peers?
I would have real similar reasoning as you and say the smaller hospital as well: off the top of my head. But sometimes statistics surprises me, until I go through a lot more rigorous thought and/or read the solution.
The smaller sample would vary from the mean more. I agree with you.
I find some sort of experiment a useful check on my intuition. Making the simplification that there are always exactly 45 births at the large hospital and 15 at the small hospital and setting up a simulation for 45 days in Excel results in 22 days with >60% boys for the big hospital, compared to 52 days for the small hospital.
I agree. I wrote a simulation for both hospitals for a hundred years using a good randomizer, and tabulated the results, allowing for a tie.
Larger 36, Tie 12, Smaller 52.
Second run:
Larger 42, Tie 7, Smaller 51.
> .. setting up a simulation for 45 days in Excel results in .. >60% boys ... 52 days for the small hospital.
Huh?
I just googled the question...google agrees with you , Bruce.
I would be interested in the methodology as to why it is the larger hospital. I would also think it would be the smaller one.
This kinds of reminds me of another somewhat statistical problem. Three women sitting on a park bench eating an ice cream cone. First woman is calmly licking the ice cream cone. Second woman is biting into the ice cream. The third woman is holding on to the cone with both hands and literally enhaling the ice cream. Which one is married? o.O
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The one wearing the wedding ring!!! :-$
You always try to take a sampling of larger numbers in order to more closely estimate the "mean". To me a sampling of a 50-50 chance is going to hit more often in a 45-count sampling than in a 15-count sampling. I know I'm not applying scientific formulas here, just sayin'.....
I would say the larger hospital had more days on which more than 60% of the births were boys, AND more days on which less than 60% of the births were boys.
There are, after all, 16,425 births at the larger hospital compared to 5,475 at the smaller hospital. That's 10,950 more chances to vary from the mean and still come out with a valid mean.
I should have written that my simulation was done for 365 days, not 45 days.
My logic.
60% of 45 = 27
60% of 15 = 9
You need 1 more in each hospital to go over 60%
28/45 = 62%
10/15 = 67%
So the chances of going over 60% are greater in the large hospital.
> My logic.
>
> 60% of 45 = 27
> 60% of 15 = 9
>
> You need 1 more in each hospital to go over 60%
>
> 28/45 = 62%
> 10/15 = 67%
>
> So the chances of going over 60% are greater in the large hospital.
Good point. I changed my mind.
60% OF THE TIME, IT WORKS EVERY TIME....;-)
> My logic....
Further proof that the human mind can devise a logic for anything it wants to believe.
A more analytical approach would be to use the binomial distribution (specifically, as provided by Excel). Still make the approximation that the number of births at the big hospital is 45 every day, and 15 every day for the small hospital. To satisfy the >60% criteria, there must be at least 28 boys born at the big hospital or 10 at the small hospital. This is equivalent to no more than 17 girls or 5 girls respectively. Name this event B, for lots of boys.
Using the binomial distribution indicates the probability of B on any given day is 0.068 for the large hospital and 0.151 for the small hospital.
The exact Excel formulas I used were
=BINOM.DIST(17,45,0.5,TRUE) for the large hospital and
=BINOM.DIST(5,15,0.5,TRUE) for the small hospital
I hadn't worked through it yet, but was thinking along the same lines. The key is what does it take to get past 60 percent as opposed to just below 60.
My thought was along the lines of saying there were three babies born each day at the smaller hospital and 9 born each day at the bigger hospital. Say the requirement had been set at more than 67 percent. In the case of the smaller hospital that would necessitate all three be the same gender while in the case of the larger hospital only 7 out of 9 would need to be the same gender.
That instructor should not be an instructor. Here is a reference that tells how you would calculate the probability and has a similar example:
http://www.fourmilab.ch/rpkp/experiments/statistics.html
As you can see from their graphs as the number of events (n) increases the probability curve becomes a narrower and narrower bell curve centered around the mean.
My thoughts were that it would be easier to deviate from the mean at the smaller hospital based on either the:
Central Limit Theorem or
The Law of Large Numbers....
But it has been so long since stats I could be wrong...
I agree that, as most have reasoned out, the smaller hospital would have more percentage variation.
It's a little harsh to say the instructor shouldn't be teaching, but yes an instructor should understand their class material better than this. What are the odds the teacher has a degree in Education and not in Math/Stats?
