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 500 metric tons of lemon imports. Is the prediction worthwhile? Use a significance level of 0.05. Lemon Imports 233 262 356 476 529
Crash Fatality Rate 16 15.8 15.4 15.4 14.9 Question content area bottom Part 1 Find the equation of the regression line. ModifyingAbove y with caretequals 16.636plus( negative 0.0031)x (Round the y-intercept to three decimal places as needed. Round the slope to four decimal places as needed.) Part 2 The best predicted crash fatality rate for a year in which there are 500 metric tons of lemon imports is enter your response here fatalities per 100,000 population. (Round to one decimal place as needed.)
Solution
Part 1: Find the regression equation
We use the given equation for the regression line:
Where:
- is the lemon imports (in metric tons)
- is the predicted crash fatality rate.
Part 2: Predicted crash fatality rate for 500 metric tons
To predict the crash fatality rate when , substitute into the regression equation:
The best-predicted crash fatality rate is 15.1 fatalities per 100,000 population.
Is the prediction worthwhile?
To assess whether the prediction is worthwhile, we must evaluate the significance of the regression line:
- Check the -value for the slope in a hypothesis test for significance ( vs. ) and ensure -value < 0.05.
- Evaluate the coefficient of determination () to measure how well the regression line explains variability in the data.
Without additional data on the -value or , it is hard to definitively say if the prediction is statistically worthwhile, though a weak slope () suggests limited practical significance.
Would you like details on hypothesis testing for regression significance or calculating ? Here are 5 related questions:
- How is the -value for a regression slope calculated?
- What does tell us about a regression model?
- Can correlation alone indicate causation in regression models?
- How do we interpret a negative slope in a regression equation?
- What alternative models could we use if linear regression isn’t a good fit?
Tip: Always visually plot your data and regression line to ensure the model aligns with observed trends!
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Math Problem Analysis
Mathematical Concepts
Linear Regression
Prediction in Statistics
Hypothesis Testing
Formulas
Regression Line: ŷ = b0 + b1x
Slope (b1): Σ[(xi - x̄)(yi - ȳ)] / Σ(xi - x̄)^2
Intercept (b0): ȳ - b1x̄
Theorems
Least Squares Method
Significance Testing for Regression Coefficients
Suitable Grade Level
Grades 10-12
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