Two offices in different towns of the same life insurance company write 25-year term assurance policies. Below are data from these two offices relating to policyholders of the same age. Both deaths and policies in force are on an age last birthday basis. Gasperton Great Hawking Policies in force on 1 January 2009 2,000 1,770 Policies in force on 1 January 2010 2,100 1,674 Deaths in calendar year 2009 25 21 (ii) Calculate the central death rate for the calendar year 2009 at this age for the offices in Gasperton and Great Hawking. A detailed examination of the records shows that 50% of the policyholders in Gasperton at both censuses were smokers, and 20% of policyholders in Great Hawking at both censuses were smokers. National death rates at this age for smokers in 2009 were 40% higher than those for non-smokers. (iii) Estimate the central death rates for smokers and non-smokers in Gasperton and Great Hawking. The first part is simple and straight. For the second part the solution says Rate_Gasperton = 0.5 Rate of Smokers in Gasperton + 0.5 Rate of Non-Smokers in Gasperton. It is not clear how we get this equation. I understand that when population is split 50-50 in smokers and non smokers the exposed to risk will be divided by a factor of 2 but we don't know the split for the deaths so how can we split the rates like in the above equation. Any ideas ? Thanks
We know that in Gasperton 25 of the central exposed to risk of 2050 died, giving us a central death rate of 25/2050 = 0.0122. But we're then told that half of this population are smokers and hence have a higher mortality rate (by a factor of 1.4) The info we have leads us to the equation: ((0.5 * 2050) * Mns) + ((0.5 * 2050) * Ms) = 25 Where Mns is the central death rate for non smokers, Ms for smokers. Since we know that Ms = 1.4Mns so we can rewrite the equation: 1025 Mns + 1025 (1.4Mns) = 25 So 1025 Mns + 1435 Mns = 25 and Mns = 25 / 2460 = 0.0102 Ms = 1.4 * 0.0102 = 0.01423 Similarly for Great Hawking: (0.8 * 1722 * Mns) + (0.2 * 1722 * 1.4 * Mns) = 21 Mns (1377.6 + 482.16) = 21 --> Mns = 0.01129 Ms = 0.01581
Now if we add the central rates for Gasperton and Hawking then we would get = 2*(0.01222) because both regions have the same central rates. However, if we calculate from first principles that is add the deaths in both regions = 46 and then also add the census information for both regions for 2009 and 2010. Subsequently calculate the exposed to risk and then the rate of mortality the answer is = 0.01222. So it appears that adding the rates for two regions is not equal to the rate of the combined regions. This is the example I was alluding to in the other post.
OK, I agree that if you have calculated the crude central death rates seprately for two populations, you can't just add these two rates together to get the central death rate for the populations combined. But your question in the other post was: The answer to that question is yes. In this case we can find the central exposed to risk for Gasperton and Great Hawking combined, simply by adding the Central Exposed to Risk for Gasperton to the Central Exposed to Risk for Great Hawking (I.E. 2050 + 1722 = 3772). And the central death rate is simply the combined deaths of 46 / 3772 = 0.0122 for the combined population. In part one of the exam q I think you were supposed to notice this when you calculated them separately. But nowhere in this exam question were we asked to actually calculate a death rate for the combined groups. In particular you cannot do so in the second part of the question anyway, because each population contains a different proportion of smokers.
