To determine the coffee sales when the temperature is 40°F, we'll analyze the correlation between temperature and sales using a linear model based on the provided data. We can summarize the steps to build our model and predict the sales as follows:
1. Data Collection: We have the temperature and coffee sales data collected as follows:
- 55°F: [tex]$34,000
- 58°F: $[/tex]33,000
- 66°F: [tex]$27,000
- 70°F: $[/tex]25,000
- 75°F: [tex]$22,000
- 80°F: $[/tex]17,000
- 85°F: [tex]$12,000
2. Model Construction: By fitting a linear model to this data, we obtain a linear equation that represents the relationship between temperature (T) and sales (S). The general form of the linear equation is:
\[
S(T) = a \cdot T + b
\]
Here, \(a\) and \(b\) are the coefficients of the line, which were determined to be approximately \(a = -0.7208\) and \(b = 74.6415\).
3. Predicting Sales: Using this linear equation, we can predict the coffee sales at a given temperature. We plug in the temperature of interest (in this case, 40°F) into the equation.
\[
S(40) = -0.7208 \cdot 40 + 74.6415
\]
4. Calculation:
\[
S(40) = -28.832 + 74.6415 = 45.8095
\]
Thus, when the temperature is 40°F, the expected coffee sales are approximately \(45,809.5\) dollars.
So, according to our model, the coffee sales when the temperature is 40°F would be about $[/tex]45,809.50.
