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:
Where:
- is the crash fatality rate.
- is the lemon imports in metric tons.
Key findings:
- : This indicates that approximately 86.7% of the variation in crash fatality rates can be explained by lemon imports.
- : 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 () and the 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:
- How can you calculate the residual for a specific data point in the regression?
- What does the value signify in this context?
- How do we assess if the linear model is appropriate for this data?
- Could non-linear regression provide a better fit for this data?
- What are potential limitations or misinterpretations of this analysis?
Tip:
Always evaluate the p-value and 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
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