Predicting Population

Suppose we want to use the following population data to estimate the number of women in each age group in 1970 and 1985.
Age No. Females Alive 1940 No. Females Alive 1955 Female Birth 1940 - 1955
0 - 14 14,459 16,428 4,651
15 - 29 15,264 14,258 10,403
30 - 44 11,346 14,836 1,374
The following table shows the probabilities of each group either surviving to reach the next age group or giving birth.
0 - 14 15 - 29 30 - 44
Probability of Giving Birth
\[\frac{4,651}{14,459}=0.32167\]
\[\frac{10,403}{15,264}=0.0.68154\]
\[\frac{1,374}{11,346}=0.12110\]
Probability of Surviving Into Next Age Group
\[\frac{14,258}{14,459}=0.98610\]
 0 0
Probability of Surviving Into Next Age Group 0
\[\frac{14,836}{15,264}=0.97196\]
 0
The predicted populations in each age group in 1955 is then
\[ \left( \begin{array}{ccc} 0.32167 & 0.68154 & 0.12110 \\ 0.98610 & 0 & 0 \\ 0 & 0.97196 & 0 \end{array} \right) \begin{pmatrix}14,459\\15,264\\11,346\end{pmatrix} = \begin{pmatrix}16,428\\14,258\\14,836\end{pmatrix}\]

The predicted populations in each age group in 1970 is then
\[ \left( \begin{array}{ccc} 0.32167 & 0.68154 & 0.12110 \\ 0.98610 & 0 & 0 \\ 0 & 0.97196 & 0 \end{array} \right)^2 \begin{pmatrix}14,459\\15,264\\11,346\end{pmatrix} = \begin{pmatrix}16,799\\16,200\\13,858\end{pmatrix}\]

The predicted populations in each age group in 1985 is then
\[ \left( \begin{array}{ccc} 0.32167 & 0.68154 & 0.12110 \\ 0.98610 & 0 & 0 \\ 0 & 0.97196 & 0 \end{array} \right)^3 \begin{pmatrix}14,459\\15,264\\11,346\end{pmatrix} = \begin{pmatrix}18,123\\16,565\\15747\end{pmatrix}\]

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