A small company is selling a new board game, and they need to know how many to produce in the future, After 12 months, they sold 4 thousand games; after 18 months, they sold 7 thousand games; and after 36 months, they sold 15 thousand games. 1) Could this information be reasonably estimated using a single linear model?

Yes, the information could be estimated using a single linear model.

So remember that if a line passes through two points (x₁, y₁) and (x₂, y₂) the slope of that line can be written as:

Here we have 3 points:

(12, 4)

(18, 7)

(36, 15)

Where the first value represents the number of months and the second the number of games sold in thousands.

From these we can form 3 pairs and make 3 slopes, then we can take the mean slope:

So the mean slope is:

s = (0.5 + 0.44 + 0.46)/3 = 0.467

y = 0.467*x + b

To find the mean value of b, we evaluate the function in the 3 points, and then we take the mean for b.

y  - 0.467*x = b

for each point we have:

4  - 0.467*12 = b = -1.6

7 - 0.467*18 = b = -1.4

15 - 0.467*36 =  -1.8

Taking the mean we get:

(-1.6 - 1.4 - 1.8)/3 = -1.6

So a line that models this can be:

y = 0.467*x - 1.6

to test the line, we can evaluate it in x = 36, we will get:

y = 0.467*36 - 1.6 = 15.212

Comparing, with the actual value, which is 15, we get a percentual error of:

100%*(15.212 - 15)/15 = 1.4%

As you can see, the percentual error with this line is really small, which means that the line is a really good estimation for the relation between the number of months and the number of games sold.

If you want to learn more about estimations, you can read:

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