Answer:
y ≈ 15272.9(1.06306^x) . . . . from regression tool
y = 15300(1.0624^x) . . . . calculated "by hand"
Explanation:
An exponential model is one that uses the exponential function ...
f(x) = a·b^x
to model the data. The value of 'a' is the function value when x=0, the initial value. The value of 'b' is the growth factor, the multiplier between function values for successive values of x.
__
tool
An exponential regression tool makes short work of this. Such tools are available on some calculators, apps, and spreadsheets. All that is required is entering the data into a table and invoking the tool. The one shown in the attachment tells you the model is approximately ...
y ≈ 15272.9(1.06306^x)
__
averaging
We can average the growth rate factors a couple of different ways. When we think of "averaging," we usually think of the arithmetic mean, the sum of values divided by their number. If we average the growth rates in this way, we get ...
average rate of growth =
((161/153) +(173/161) +(184/173) +(196/184) +(207/196) +(220/207))/6 ≈ 1.0624
Using the first table value as the initial value, the model formed this way is ...
y = 15300(1.0624^x)
__
For exponential growth it may make more sense to compute the geometric mean. This is the n-th root of the product of n numbers. In this case, it reduces to the 6-th root of the ratio of the last to the first:
average rate of growth = (220/153)^(1/6) ≈ 1.0624
This gives the same growth factor as above, hence the same model.
The second attachment shows the application of this model to the domain used in the table. We see there are some minor differences when rounding the function value to the nearest hundred.
The differences in our two models seem to arise from rounding the data values in the table to the nearest hundred.
__
summary
The values given in the table can be used in different ways to arrive at two similar, but different exponential models of the table data:
- y = 15272.9(1.06306^x)
- y = 15300(1.0624^x)
Interestingly, the function calculated "by hand" has a very slightly better correlation with the table data. The difference in correlation coefficients is in the 6th decimal place.