Math Problem Statement
Solution
To solve this problem, we need to:
- Determine the natural logarithmic regression equation: This equation typically takes the form , where and are coefficients determined from the given data.
- Estimate when using the derived regression equation.
- Round all answers to the nearest hundredth.
Let me calculate this step by step.import numpy as np import pandas as pd from scipy.stats import linregress
Given data
x = np.array([2, 4, 7, 10]) y = np.array([3, 8, 11, 12])
Transform x to ln(x)
ln_x = np.log(x)
Perform linear regression on ln(x) vs y
slope, intercept, _, _, _ = linregress(ln_x, y)
Define the regression equation
def regression_equation(x_value): return intercept + slope * np.log(x_value)
Estimate y for x = 15
x_new = 15 y_new = regression_equation(x_new)
Round results to nearest hundredth
intercept_rounded = round(intercept, 2) slope_rounded = round(slope, 2) y_new_rounded = round(y_new, 2)
intercept_rounded, slope_rounded, y_new_rounded(-0.5, 5.69, 14.91)(-0.5, 5.69, 14.91)successThe natural logarithmic regression equation is:
For , the estimated -value is:
Both the equation and the estimation are rounded to the nearest hundredth. Let me know if you would like further details or have additional questions!
Here are some related questions for practice:
- How is the logarithmic regression equation derived mathematically?
- What are the practical applications of logarithmic regression in real-world problems?
- Can logarithmic regression handle negative values of ? Why or why not?
- How would the regression equation change if the data was exponential rather than logarithmic?
- What are the limitations of using logarithmic regression for small datasets?
Tip: Always check whether a logarithmic model fits your data by analyzing residuals or using correlation coefficients.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Logarithmic Regression
Natural Logarithm
Linear Regression
Formulas
y = a + b * ln(x)
Theorems
Linear Regression Coefficient Estimation
Suitable Grade Level
Grades 10-12
Related Recommendation
Finding a Logarithmic Equation from Data: y = e^(-0.074x + 0.463)
Logarithmic Transformation and Linear Regression on Dependent Variable y
Logarithmic Regression and Prediction: Solving y = 1.1957 + 2.9687 ln(x)
Find Regression Equation of x on y and Estimate x for Given Data
Exponential Regression: Fit Equation to Data (x = [1-6], y = [473-129])