Math Problem Statement

Find the regression​ equation, letting the first variable be the predictor​ (x) variable. Using the listed​ lemon/crash data, where lemon imports are in metric tons and the fatality rates are per​ 100,000 people, find the best predicted crash fatality rate for a year in which there are 400 metric tons of lemon imports. Is the prediction​ worthwhile? Use a significance level of 0.05. Lemon Imports 231 263 353 475 524

Crash Fatality Rate 16 15.8 15.6 15.5 15 Question content area bottom Part 1 Find the equation of the regression line. ModifyingAbove y with caretequals    enter your response hereplus​(    enter your response here​)x ​(Round the​ y-intercept to three decimal places as needed. Round the slope to four decimal places as​ needed.)

Solution

numpy as np import pandas as pd from scipy.stats import linregress

Lemon imports and crash fatality rates data

lemon_imports = np.array([231, 263, 353, 475, 524]) crash_fatality_rates = np.array([16, 15.8, 15.6, 15.5, 15])

Perform linear regression

slope, intercept, r_value, p_value, std_err = linregress(lemon_imports, crash_fatality_rates)

Calculate regression equation components

regression_equation = f"y = {intercept:.3f} + {slope:.4f}x" intercept, slope, regression_equation, r_value**2, p_valueThe regression equation for the given data is:

y=16.5900.0027xy = 16.590 - 0.0027x

Where:

  • yy is the crash fatality rate.
  • xx is the lemon imports in metric tons.

Key findings:

  • R2=0.867R^2 = 0.867: This indicates that approximately 86.7% of the variation in crash fatality rates can be explained by lemon imports.
  • p=0.0215p = 0.0215: The p-value is less than the significance level of 0.05, so the relationship is statistically significant.

Next, I will calculate the predicted crash fatality rate for 400 metric tons of lemon imports. Predicted crash fatality rate for 400 metric tons of lemon imports predicted_rate = intercept + slope * 400 predicted_rateThe predicted crash fatality rate for a year in which there are 400 metric tons of lemon imports is approximately 15.496 per 100,000 people.

Conclusion:

Since the relationship is statistically significant (p<0.05p < 0.05) and the R2R^2 value shows a strong explanatory power, the prediction is considered worthwhile for this dataset.

Would you like a deeper dive into the calculations or analysis?

Related Questions:

  1. How can you calculate the residual for a specific data point in the regression?
  2. What does the R2R^2 value signify in this context?
  3. How do we assess if the linear model is appropriate for this data?
  4. Could non-linear regression provide a better fit for this data?
  5. What are potential limitations or misinterpretations of this analysis?

Tip:

Always evaluate the p-value and R2R^2 together to judge both statistical significance and practical relevance.

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Math Problem Analysis

Mathematical Concepts

Linear Regression
Statistical Analysis
Correlation

Formulas

y = mx + b (Equation of a line)
R^2 = 1 - (SS_residual / SS_total) (Coefficient of Determination)
p-value calculation for significance

Theorems

Least Squares Regression Theorem
Statistical Significance Testing (p-value)

Suitable Grade Level

Grades 11-12 or Undergraduate Statistics