Trouble with finding number of people who own a cell phone in certain year

Question

I have a problem where a function is exponential and it comes with a table. I am trying to figure out how to find the number of people who own a smartphone in the Year 2020. I tried to use rate of change and other means, but I was stumped by this one question. Thank you

The value I am trying to estimate is $A_{0}$ and $P$

table image

question ← This is the original problem

Answer 1

Let $\widehat{y_t}$ be our estimate for the number in year t, and $y_t$ be the actual numbers (as per data). We wish to find estimates such that the total error between $y_t$ and $\widehat{y_t}$ is minimized.

One loss function that we could use over the errors is $|\cdot|^2$ (absolute value squared) such that $E = \sum\limits_{t=0}^{7} |\widehat{y_t}-y_t |^2$.

Then $A_0,P$ are minimizers of this loss. $A_0,P = argmin [E] = argmin[(A_0 P^t -y_t)^2]$. You can pass this to some numerical solver, and may find values like $A_0 = 2.122,P=1.1167$.

\begin{array}{|c|c|c|c|} \hline Year & Actual y & y Predicted & Error \\ \hline 1977 & 2.1 & 2.12E+00 & 4.99E-04 \\ \hline 1978 & 2.4 & 2.37E+00 & 8.89E-04 \\ \hline 1979 & 2.6 & 2.65E+00 & 2.21E-03 \\ \hline 1980 & 3 & 2.96E+00 & 1.92E-03 \\ \hline 1981 & 3.3 & 3.30E+00 & 1.92E-06 \\ \hline 1982 & 3.7 & 3.69E+00 & 1.70E-04 \\ \hline 1983 & 4.1 & 4.12E+00 & 3.08E-04 \\ \hline \end{array}

Once you have a model, it's straightforward to predict. You would predict $\widehat{y}_{2020} = A_0P^{2020-1977}$ (and obtain a value around 245 millions)

Answer 2

You should post the full question and your attempt at a solution! As you have not stated the function I will assume it is of the form: $$f(t)=ae^{bt}$$ where $a,b$ are constants. $$f(0)=2.1=ae^{b(0)}\to 2.1=ae^0=a$$ Substituting the new value for a $$f(1)=2.4=2.1e^b\to b=\ln(\frac{2.4}{2.1})$$ Now using both $a,b$ $$f(2020-1977)=2.1e^{(0.134)(43)}$$ $$f(43)=667.77$$