Estimating number of slaves imported before 1790 I have a statistics problem I am having a hard time figuring out how to model mathematically. The 1790 US Census counted 697,681 slaves and 59,196 free Africans in the United States. (A) assume importation began in 1620 and increased according to some unknown exponential function (B) assume a rate of natural increase at 2.5% per annum (C) assume all free Africans are manumitted slaves or their descendants and the rate of manumission is constant during the entire period Given these assumptions, what is the function that would describe the annual number of slaves imported during that time (1620-1790)?

Tyson Atkins 2022-10-24 Answered
Estimating number of slaves imported before 1790
I have a statistics problem I am having a hard time figuring out how to model mathematically.
The 1790 US Census counted 697,681 slaves and 59,196 free Africans in the United States.
(A) assume importation began in 1620 and increased according to some unknown exponential function
(B) assume a rate of natural increase at 2.5% per annum
(C) assume all free Africans are manumitted slaves or their descendants and the rate of manumission is constant during the entire period
Given these assumptions, what is the function that would describe the annual number of slaves imported during that time (1620-1790)?
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Answers (1)

ehedem26
Answered 2022-10-25 Author has 13 answers
If S ( t ) and F ( t ) are the populations of slaves and free Africans at time t, I would interpret your assumptions as saying
S = A e k t + r S F = r F + m S S ( 0 ) = F ( 0 ) = 0
where r = 0.025 is the rate of natural increase (assumed here to apply to both slaves and free Africans, though I doubt that is realistic), m the per capita rate of manumission, A the initial importation rate, k the rate of increase of the importation rate, and t is measured in years after 1620.
However, note that there are three unknown parameters A, k and m and only two items of data ( S ( 170 ) and F ( 170 )), so you're not going to be able to determine those parameters.
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