To determine the quadratic regression equation and predict the profit at a selling price of $x = 20.75$, let's proceed step by step:

1. Construct a quadratic regression equation: This involves finding the best-fit equation in the form: $y = ax^2 + bx + c$ where $a$, $b$, and $c$ are constants.

2. Use the data points:

   • $(13.25, 553553)$
   • $(15.75, 666666)$
   • $(20.25, 856856)$
   • $(29.00, 778778)$
   • $(34.00, 595595)$

3. Fit the quadratic regression model using these points.

4. Predict the profit when $x = 20.75$ by substituting $x$ into the equation.

Let me calculate this.import numpy as np from numpy.polynomial.polynomial import Polynomial import pandas as pd

Data points (price, profit)

prices = np.array([13.25, 15.75, 20.25, 29.00, 34.00]) profits = np.array([553553, 666666, 856856, 778778, 595595])

Fit a quadratic regression model

coefficients = np.polyfit(prices, profits, 2) a, b, c = coefficients # Coefficients for y = ax^2 + bx + c

Predict profit at x = 20.75

predicted_price = 20.75 predicted_profit = np.polyval(coefficients, predicted_price)

Round coefficients and prediction to the nearest hundredth or whole number

coefficients_rounded = [round(coef, 2) for coef in coefficients] predicted_profit_rounded = round(predicted_profit)

coefficients_rounded, predicted_profit_rounded([-2806.57, 134349.53, -736950.95], 842397)([-2806.57, 134349.53, -736950.95], 842397)successThe quadratic regression equation is:

$y = -2806.57x^2 + 134349.53x - 736950.95$

Using this equation, the profit for a selling price of $x = 20.75$ dollars is approximately $842,397.

Would you like a detailed explanation of the process?

Related Questions:

1. How can we interpret the coefficients of this quadratic equation in a business context?
2. What is the maximum profit predicted by this model, and at what price does it occur?
3. How can the goodness of fit for this quadratic model be evaluated?
4. What are some alternative models if the data doesn't fit a quadratic regression well?
5. How does the predicted profit compare to the actual data points?

Tip:

When analyzing regression results, always check the residuals or fit errors to ensure the model's reliability for predictions.