Math Problem Statement

What is the natural logarithmic regression equation for the following data? Estimate the y-value for x = 15. Express answers to the nearest hundredth.

Solution

To solve this problem, we need to:

  1. Determine the natural logarithmic regression equation: This equation typically takes the form y=a+bln(x)y = a + b \ln(x), where aa and bb are coefficients determined from the given data.
  2. Estimate yy when x=15x = 15 using the derived regression equation.
  3. 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:

y=0.50+5.69ln(x)y = -0.50 + 5.69 \ln(x)

For x=15x = 15, the estimated yy-value is:

y=14.91y = 14.91

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:

  1. How is the logarithmic regression equation derived mathematically?
  2. What are the practical applications of logarithmic regression in real-world problems?
  3. Can logarithmic regression handle negative values of xx? Why or why not?
  4. How would the regression equation change if the data was exponential rather than logarithmic?
  5. 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.

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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