Question: Market researchers were interested in the relationship between the price of bobbleheads and the…

Market researchers were interested in the relationship between the price of bobbleheads and the demand of bobbleheads. Information was collected from a survey and used to obtain the regression equation \(\hat{y} = -0.227x + 50.455\), where \(x\) represents the price of bobbleheads (measured in dollars) and \(\hat{y}\) is the predicted demand of bobbleheads (in units). What is the predicted demand of a bobblehead that has a price of $6.00?

-1.362 units 49.093 units 51.817 units 195.837 units

Solution

The problem provides a regression equation that models the demand for bobbleheads based on their price: \[ \hat{y} = -0.227x + 50,455 \] where \( x \) is the price in dollars, and \(\hat{y}\) is the predicted demand in units. 1. Substitute \( x = 8 \) into the equation to find the predicted demand: \[ \hat{y} = -0.227(8) + 50,455 \] 2. Calculate \( -0.227 \times 8 \): \[ -0.227 \times 8 = -1.816 \] 3. Add this result to 50,455 to find \(\hat{y}\): \[ \hat{y} = -1.816 + 50,455 = 50,453.184 \] Thus, the predicted demand of a bobblehead that has a price of $8.00 is 50,453.184 units.