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Background: I have the following data:

27% of the respondents to a survey are interested in a product. Of that 27% which is interested:

  • 59% are 15-24 years old.
  • 18% are 25-34 years old.
  • 6% are 35-44 years old.
  • 4% are 45-54 years old.
  • The rest are outside these age ranges.

The probability of being in each age range is:

  • 11 % for 15-24.
  • 13% for 25-34.
  • 13% for 25-34.
  • 12% for 34-45.
  • rest is outside.

Problem: What percentage of each age group is interested in the new product?

Attempt: Bayes Theorem seems the right way to go here since I need to invert a probability: calling P(I) the probability of being interested in the product and P(E) the probability of being in a certain age range, from the survey I have P(I) (27% are interested) and I have P(E | I) and I want to find P(I | E).

So my attempt e.g. for the age group 15-24 was to do:

P(I|E ) = P(I) * P(E | I) / P(E) = 27% * 59% / 11%

27% is the overall probability of being interested in the product, 59% is the probability of being in age range 15-24 given being interested in the product.

Question: Calculating P(E) is where I get confused. I tried using 11%, which is how much of the population is in the age range 15-24.

But it does not work, because I get a number greater than 1 as a result.

What is the correct way to calculate P(E) in this problem and am I setting things up correctly ?

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    – Tavrock
    Commented Mar 27, 2017 at 16:49

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Interesting. It seems to me that the numbers are wrong. I suppose the age ranges as age ranges for the individuals which actually answer the survey (respondents). If the survey uses a random sample which is representative of the population and nonresponse bias is independent of age, these age ranges are also the same than those in the general population, otherwise they're different, but still, you are only interested in the age ranges for respondents. If you're given the age ranges for the actual population and these are significantly different from those of the respondents, then you don't have enough data to answer the question.

Vice versa, if the age ranges are those for respondents, then the numbers are clearly inconsistent. Proof: if $27\%$ of the respondents are interested in the product, and $59\%$ of those interested are aged 15-24, this means that there are at least $59\%\times.27\%\approx.16\%$ respondents aged 15-24 among all respondents. The reason is that these are only the respondents, aged 15-24, which are interested in the product: if there are other respondents, aged 15-24, which are not interested in the product, then the total percentage of respondents aged 15-24 must be even larger than $16\%$. However, the question text tells you that the percentage of respondents aged 15-24 is $11\%<16\%$, so something is clearly wrong here. Either the numbers are wrong, or you are not given the age groups for the respondents, but for the general population. However, in this case, the question should give you also the nonresponse bias by age, otherwise it's impossible to answer.

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  • $\begingroup$ Thanks much for your answer. I am reassured that the calculation itself is right, i.e. I am not making a mistake in the formula. I think that as you point out the numbers are not internally consistent, $\endgroup$
    – AleAve81
    Commented Mar 29, 2017 at 15:04
  • $\begingroup$ that's what I think. Of course, I may be wrong. It would be interesting to hear from other CV users, too. I hope someone else will decide to chime in. $\endgroup$
    – DeltaIV
    Commented Mar 29, 2017 at 15:48

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