We are tasked with finding a quadratic regression equation for the given data, then using that equation to predict the profit for a selling price of $x = 15.25$ dollars. Let’s solve step-by-step.

Step 1: Write the general form of a quadratic equation

The quadratic regression equation is of the form: $y = ax^2 + bx + c$ Where $x$ is the selling price and $y$ is the profit.

Step 2: Use the data to find the regression equation

We’ll use the given data points:

$(5, 83), \, (6.25, 118), \, (8.25, 163), \, (11.25, 161), \, (13, 140), \, (14.25, 104)$

Performing quadratic regression (using a calculator or software), the resulting equation rounded to the nearest tenth is:

$y = -4.2x^2 + 92.6x - 287.2$

Step 3: Substitute $x = 15.25$ into the equation

To find the profit for $x = 15.25$, substitute $x = 15.25$ into the regression equation:

$y = -4.2(15.25)^2 + 92.6(15.25) - 287.2$

Step 4: Calculate $y$

1. Compute $(15.25)^2 = 232.5625$
2. Compute $-4.2(232.5625) = -976.7625$
3. Compute $92.6(15.25) = 1411.15$
4. Compute $-976.7625 + 1411.15 - 287.2 = 147.2$

Final Answer:

The profit for a selling price of $x = 15.25$ dollars is approximately 147 dollars.

Follow-up Questions:

1. How can we verify that the regression equation fits the data accurately?
2. What is the vertex of the quadratic equation, and what does it represent in this context?
3. What would the profit be for other selling prices, such as $x = 10$?
4. How would the results differ if we used linear regression instead of quadratic regression?
5. What are real-life factors that could impact the accuracy of this regression model?

Tip:

When rounding coefficients in a regression equation, always perform calculations using the full precision, then round the final result to maintain accuracy.