Predict The Cost Of Premium Gasoline At A Station Where The Price Of Regular Gas Is $3.50. Show Your Work Or Explain How You Got The Answer. Regular Premium 3.54 4.29 3.39 4.09 3.69 4.39 3.27 3.87 3.49 3.99

Introduction

In the world of mathematics, predicting the cost of premium gasoline is a simple yet intriguing problem. Given the price of regular gas, we can use mathematical techniques to estimate the cost of premium gas. In this article, we will explore this problem and provide a step-by-step solution.

Understanding the Problem

The problem states that the price of regular gas is $3.50, and we need to predict the cost of premium gas. We are given a set of data points that represent the prices of regular and premium gas at different stations. The data points are:

Analyzing the Data

To predict the cost of premium gas, we need to analyze the relationship between the prices of regular and premium gas. We can use a simple linear regression model to estimate the relationship between the two variables.

Step 1: Calculate the Mean of Regular Gas Prices

To calculate the mean of regular gas prices, we need to add up all the prices and divide by the number of data points.

import numpy as np
regular_gas_prices = np.array([3.54, 3.39, 3.69, 3.27, 3.49])

mean_regular_gas = np.mean(regular_gas_prices)
print("Mean of Regular Gas Prices: {{content}}quot;, mean_regular_gas)

Step 2: Calculate the Mean of Premium Gas Prices

To calculate the mean of premium gas prices, we need to add up all the prices and divide by the number of data points.

# Define the prices of premium gas
premium_gas_prices = np.array([4.29, 4.09, 4.39, 3.87, 3.99])

mean_premium_gas = np.mean(premium_gas_prices)
print("Mean of Premium Gas Prices: {{content}}quot;, mean_premium_gas)

Step 3: Calculate the Slope of the Linear Regression Line

To calculate the slope of the linear regression line, we need to use the following formula:

m = (n * Σ(x_i * y_i) - Σx_i * Σy_i) / (n * Σx_i^2 - (Σx_i)^2)

where m is the slope, n is the number of data points, x_i is the price of regular gas, y_i is the price of premium gas, and Σ denotes the sum.

# Define the prices of regular and premium gas
regular_gas_prices = np.array([3.54, 3.39, 3.69, 3.27, 3.49])
premium_gas_prices = np.array([4.29, 4.09, 4.39, 3.87, 3.99])

sum_xy = np.sum(regular_gas_prices * premium_gas_prices)

sum_x = np.sum(regular_gas_prices)

sum_y = np.sum(premium_gas_prices)

sum_x_squared = np.sum(regular_gas_prices ** 2)

slope = (len(regular_gas_prices) * sum_xy - sum_x * sum_y) / (len(regular_gas_prices) * sum_x_squared - sum_x ** 2)
print("Slope of the Linear Regression Line: ", slope)

Step 4: Calculate the Intercept of the Linear Regression Line

To calculate the intercept of the linear regression line, we need to use the following formula:

b = (Σy_i - m * Σx_i) / n

where b is the intercept, m is the slope, n is the number of data points, x_i is the price of regular gas, and y_i is the price of premium gas.

# Calculate the intercept of the linear regression line
intercept = (sum_y - slope * sum_x) / len(regular_gas_prices)
print("Intercept of the Linear Regression Line: ", intercept)

Step 5: Predict the Cost of Premium Gas

To predict the cost of premium gas, we can use the linear regression equation:

y = m * x + b

where y is the price of premium gas, m is the slope, x is the price of regular gas, and b is the intercept.

# Predict the cost of premium gas
predicted_premium_gas = slope * 3.50 + intercept
print("Predicted Cost of Premium Gas: {{content}}quot;, predicted_premium_gas)

Conclusion

In this article, we used a simple linear regression model to predict the cost of premium gas given the price of regular gas. We calculated the mean of regular and premium gas prices, the slope and intercept of the linear regression line, and finally predicted the cost of premium gas. The predicted cost of premium gas was $4.14.

Discussion

The linear regression model assumes a linear relationship between the prices of regular and premium gas. However, in reality, the relationship may be non-linear. To improve the accuracy of the prediction, we can use more advanced models such as polynomial regression or decision trees.

Limitations

The linear regression model has several limitations. It assumes a linear relationship between the variables, which may not be true in reality. Additionally, the model is sensitive to outliers and may not perform well with noisy data.

Future Work

In future work, we can explore more advanced models such as polynomial regression or decision trees to improve the accuracy of the prediction. We can also collect more data points to increase the accuracy of the model.

References

• [1] "Linear Regression" by Wikipedia
• [2] "Polynomial Regression" by Wikipedia
• [3] "Decision Trees" by Wikipedia