Questions: Company X tried selling widgets at various prices to see how much profit they would make. The following table shows the widget selling price, x, and the total profit earned at that price, y. Write a quadratic regression equation for this set of data, rounding all coefficients to the nearest tenth. Using this equation, find the profit, to the nearest dollar, for a selling price of 5.25 dollars. Price (x) Profit (y) 6.00 5440 7.00 6582 8.75 7876 11.50 8349 13.75 7111 14.75 5886

Solution Steps

The data provided consists of selling prices \( x \) and corresponding profits \( y \):

\[ \begin{array}{|c|c|} \hline \text{Price } (x) & \text{Profit } (y) \\ \hline 6.00 & 5440 \\ 7.00 & 6582 \\ 8.75 & 7876 \\ 11.50 & 8349 \\ 13.75 & 7111 \\ 14.75 & 5886 \\ \hline \end{array} \]

Using quadratic regression on the data points, we derive the equation of the form:

\[ y = ax^2 + bx + c \]

The calculated coefficients are:

\[ a = -143.4, \quad b = 3039.3, \quad c = -7664.4 \]

Thus, the quadratic regression equation is:

\[ y = -143.4x^2 + 3039.3x - 7664.4 \]

To find the predicted profit for a selling price of \( x = 5.25 \), we substitute \( x \) into the regression equation:

\[ y = -143.4(5.25)^2 + 3039.3(5.25) - 7664.4 \]

Calculating each term:

1. \( (5.25)^2 = 27.5625 \)
2. \( -143.4 \times 27.5625 = -3943.575 \)
3. \( 3039.3 \times 5.25 = 15973.725 \)

Now, substituting these values into the equation:

\[ y = -3943.575 + 15973.725 - 7664.4 \]

Calculating the final profit:

\[ y = 4339.75 \]

Rounding to the nearest dollar gives:

\[ \text{Predicted profit} = 4339 \]

Final Answer