Double counting or double filtering?

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Note: Not sure which subforum this should go in - guess I'll put it here, since although it doesnt require any interest rates, discounted cash flows, etc., it is a business question.


Background:

I'm creating an extremely simplistic rough model of expected customer conversions from introducing widgets into my product assortment, for a trial period.

Assumptions:
  • 1 in 5 people will buy a widget in any given year (20% of total population)
  • I will be running an in-store test program, selling widgets, for 3 months
  • 100,000 people will visit my store and see the widgets during these 3 months
  • I expect 1% of impressions (customers that see widgets for sale) will yield purchases (conversions) from my store

Question:

How many customers convert? I see two possibilities:

Calculation ACalculation B
Impressions during 3 month test100,000100,000
% Total Population that buy widgets20%20%
% Buying widgets during test period25% (3/12 months)
Conversion rate1%1%
Expected Conversions20050


Arguments for Calc A:
  • The 100,000 people already only reflect the customers that will visit during the 3-month test period. Thus, multiplying by 25% again will mean Calc B is double-filtering.
  • If we imagine the test was for an entire year, then we would correctly expect 20% of our 400,000 customers to buy widgets during the year - after the 1% conversion rate, we'd therefore (again, correctly) expect 800 people to buy widgets from my store. Calc A gives us precisely 25% of 800, which is logical for a quarter-year test - it adds up to the yearly expectation if we multiply by 4 quarters!

Arguments for Calc B:
  • While we only have 100,000 customers during those 3 months, this doesn't mean that 20% of them will buy widgets during only these 3 months - it should imply that 20% of the 100,000 customers will buy widgets within a 12 month period. Thus, we should multiply by 25%, despite already only having 1/4 of the yearly customers represented in the model.
  • If we imagine the test lasted for only 1 day (to stress test the model), then it doesn't seem logical to use Calc A - indeed, since 20% of people will buy a widget in any given year, why would we assume 20% of a single day's customers would all buy a widget on that very day?



I'm extremely inclined to believe Calc B is the correct approach. Beyond a simply "A is right / B is right" answer, can you please provide a simple way to explain why? Thanks a bundle!
 
Note: Not sure which subforum this should go in - guess I'll put it here, since although it doesnt require any interest rates, discounted cash flows, etc., it is a business question.


Background:

I'm creating an extremely simplistic rough model of expected customer conversions from introducing widgets into my product assortment, for a trial period.

Assumptions:
  • 1 in 5 people will buy a widget in any given year (20% of total population)
  • I will be running an in-store test program, selling widgets, for 3 months
  • 100,000 people will visit my store and see the widgets during these 3 months
  • I expect 1% of impressions (customers that see widgets for sale) will yield purchases (conversions) from my store

Question:

How many customers convert? I see two possibilities:

Calculation ACalculation B
Impressions during 3 month test100,000100,000
% Total Population that buy widgets20%20%
% Buying widgets during test period25% (3/12 months)
Conversion rate1%1%
Expected Conversions20050


Arguments for Calc A:
  • The 100,000 people already only reflect the customers that will visit during the 3-month test period. Thus, multiplying by 25% again will mean Calc B is double-filtering.
  • If we imagine the test was for an entire year, then we would correctly expect 20% of our 400,000 customers to buy widgets during the year - after the 1% conversion rate, we'd therefore (again, correctly) expect 800 people to buy widgets from my store. Calc A gives us precisely 25% of 800, which is logical for a quarter-year test - it adds up to the yearly expectation if we multiply by 4 quarters!

Arguments for Calc B:
  • While we only have 100,000 customers during those 3 months, this doesn't mean that 20% of them will buy widgets during only these 3 months - it should imply that 20% of the 100,000 customers will buy widgets within a 12 month period. Thus, we should multiply by 25%, despite already only having 1/4 of the yearly customers represented in the model.
  • If we imagine the test lasted for only 1 day (to stress test the model), then it doesn't seem logical to use Calc A - indeed, since 20% of people will buy a widget in any given year, why would we assume 20% of a single day's customers would all buy a widget on that very day?



I'm extremely inclined to believe Calc B is the correct approach. Beyond a simply "A is right / B is right" answer, can you please provide a simple way to explain why? Thanks a bundle!

C=400 000 customers a year, 80 000 impressions, 800 sold or
sold in three months = ((C*0.2)*0.01)(1/4) = 200
customers in three months = C*(1/4)
impressions in three months = (C*(1/4))*0.2
sold in three months = ((C*(1/4))*0.2)*0.01=200

So you did take the 25% already. Your stress test says you would sell ~2.19 widgets per day (on average) which is reasonable. It isn't that everyone decided to buy that day, it is that the percentage who decided to by is the same.
 
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