posted
My sister thinks I'm smart. It's kind of funny, really, because I was when I was a kid. But she gives me these problems, and I get totally stumped. So I was wondering if any of the actually smart folks here could help me out.
There are five age groups:
Y = under 10 T = from 10 to 24 P = from 25 to 44 N = from 45 to 64 W = 65 and over
The death rates in each group are:
Y: 7 deaths per thousand per year T: 2 deaths per thousand per year P: 5 deaths per thousand per year N: 25 deaths per thousand per year W: 270 deaths per thousand per year
It's known (magically) that:
2% of the population will die while Y 3% of the population will die while T 5% of the population will die while P 20% of the population will die while N 70% of the population will die while W
Given all that, she gave me three questions:
1) What's the combined death rate?
Given a town with the following census stats:
10% Y 20% T 25% P 25% N 20% W
2) How many people in the town will die while Y, while T, while P, while N and while W?
3) How many people in each age group will die in a single year?
Now... I don't even know how to set something like this up. Or if there's enough information to solve it. I suspect there may be calculus involved, and I barely remember how to spell calculus, let alone do it.
Anyone have any ideas?
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posted
1) Multiply the deaths-per-year of each age group by the percentage of the town in that age group, and then add the totals together.
If they are looking for actual numbers, 2 and 3 don't seem possible if they don't say how many total people are in the town, since they only give rates and percentages.
Edit: On second thought, you probably can solve it by figuring out the total population first. If you make X the total population, you can create equations using the two sets of data.
posted
I believe that this question requires a Leslie population matrix. As I'm at work I don't have much time to go through the specifics, but I believe the following link does an OK job describing the process.
posted
For 2, ever? Is it assuming nobody joins the town? When were people born? And how is the question asking something other than to regurgitate the percentages that are "magically known" (edit: times the population)?
All of those are necessary to give exact answers. The answers could vary fairly considerably (assuming all the numbers so far given are exact, which they wouldn't be in reality), depending on if everyone in the town's birthday is tomorrow, or if everybody in the town's birthday was yesterday.
The problem needs to be better stated if it should be tackled with precision. On the other hand, if this is a stochastic model, I happen to be part of a team that writes software that can simulate the situation (though a situation as constrained as this can be 'solved' exactly with a few simplifying assumptions) using epidemiology models.
posted
For 2, how many of the 18,000 people currently living there will die under 10, how many will die between 10 and 24, and so on.
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posted
But this is already known! It's right there in the problem setup:
quote:2% of the population will die while Y 3% of the population will die while T 5% of the population will die while P 20% of the population will die while N 70% of the population will die while W
So, of 18000 people, 360 will die while under 10, and so on.
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For example, 70% of the population will die aged 65 and older. But 100% of the people who are aged 65 or older will die in that age group.
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posted
But if you take that further, you get this:
The number of people in each age group in this town:
10% of 18,000 = 1800 (0-9) 20% of 18,000 = 3600 (10-24) 25% of 18,000 = 4500 (25-44) 25% of 18,000 = 4500 (45-64) 20% of 18,000 = 3600 (65 and up)
The only age group that contains all of the various types is the under 10 set, right? So we can apply the normal percentages to them.
Under 10: Y: 2% of 1800 = 36 T: 3% of 3600 = 54 P: 5% of 4500 = 225 N: 20% of 4500 = 900 W: 70% of 3600 = 2520
But that doesn't work, because if you add those together, you get 3735, and not 1800. Does that mean that the numbers given in the problem are inconsistent?
My sister suggested something like this:
The 0-9 year olds have a death rate of 7 per thousand, so 126 of them die in a year. The other 1674 of them are T+P+N+W.
The 10-24 year olds have a death rate of 2 per thousand, so 7.2 of them die in a year. The other 3592.8 of them are P+N+W.
The 25-44 year olds have a death rate of 5 per thousand, so 22.5 of them die in a year. The other 4477.5 of them are N+W.
The 45-64 year olds have a death rate of 25 per thousand, so 112.5 die in a year. The other 4387.5 of them are W.
The 65- year olds have a death rate of 270 per thousand, so 972 die in a year. The other 2628 of them don't exist.
Again, would that mean that the definitions are inconsistent? I feel like a monkey. Honestly, math used to be my best subject. Sheesh...
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quote:But that doesn't work, because if you add those together, you get 3735, and not 1800. Does that mean that the numbers given in the problem are inconsistent?
It just means that the town is not in an equilibrium state where the deaths and exits in each group exactly match the inputs. Next year the population in each age group will be different.
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posted
To determine the number of the original 18,000 that would die in age groups Y, T, P, N, and W. It seems like the method you would use to calculate would be as follows.
Current number of population in each group: Y = .10 * 18,000 = 1,800 T = .20 * 18,000 = 3,600 P = .25 * 18,000 = 4,500 N = .25 * 18,000 = 4,500 W = .20 * 18,000 = 3,600
Number that will die in each age group: Y = (1,800 * (2/100)) = 36 T = (1,800 * (3/100)) + (3,600 * (3/(100-2))) = 164 P = (1,800 * (5/100)) + (3,600 * (5/(100-2))) + (4,500 * (5/(100-2-3))) = 511 N = (1,800 * (20/100)) + (3,600 * (20/(100-2))) + (4,500 * (20/(100-2-3))) + (4,500 * (20/(100-2-3-5))) = 3,042 W = (1,800 * (70/100)) + (3,600 * (70/(100-2))) + (4,500 * (70/(100-2-3))) + (4,500 * (70/(100-2-3-5))) + (3,600 * (70/(100-2-3-5-20))) = 14,247
The sum of which is 18,000.
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posted
Yep, that's the method I proposed a couple of posts back, in more detail. But Lisa's problem is that the given population distribution and death rates do not describe a steady state. To find the population distribution that holds the population and its distribution steady, you must solve five equations in five unknowns. The first equation is this:
That is, to keep the population constant, there must be as many births (denoted 'B') as deaths every year. The number of deaths is given by the left-hand side. Clearly, births get added to the Y group, so to keep the Y group constant, there must be as many exits from that group as births, every year. Exits from the Y group come in two forms, namely deaths, expressed by the 0.007Y (which cancels the corresponding number on the other side) and 'promotions' into the T group, which consists of the cohort that was born ten years ago. That cohort's initial size was 'B', and since then, the 7/1000 death rate has been working for ten years, so its current size is B*0.993^10. So, doing the cancellation and writing the right-hand side out in full, we get the second form of the equation.
Now we can set up the equation for the T group; entries, which are just promotions from the Y group, must exactly balance exits, which is the sum of deaths and promotions. Thus:
B*0.993^10 = 0.002T + (B*0.993^10)*0.998^15.
Again, this just expresses that the last cohort in the T group equals the size of the first cohort, minus 15 years of 2-per-thousand winnowing. Now we can set up the remaining equations similarly:
All that remains is to write out the Bs, perform the multiplications, and solve, assuming of course that a solution exists. (Actually one more thing is needed, namely a total population, which we can set arbitrarily.) This is left as an exercise for the student. It's clear however that if you set the population to 18000, you will not get the percentages given in the second part of the problem.
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posted
By the way, the above steady-state question is not one that was asked in the OP, and is not necessary to solve any of the three problems. I only gave it because Lisa was clearly thinking in such terms and it was confusing her.
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