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: y^=16.6360.0031x\hat{y} = 16.636 - 0.0031x

Where:

  • xx is the lemon imports (in metric tons)
  • y^\hat{y} is the predicted crash fatality rate.

Part 2: Predicted crash fatality rate for 500 metric tons

To predict the crash fatality rate when x=500x = 500, substitute x=500x = 500 into the regression equation:

y^=16.6360.0031(500)\hat{y} = 16.636 - 0.0031(500) y^=16.6361.55\hat{y} = 16.636 - 1.55 y^=15.1\hat{y} = 15.1

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:

  1. Check the pp-value for the slope in a hypothesis test for significance (H0:β1=0H_0: \beta_1 = 0 vs. Ha:β10H_a: \beta_1 \neq 0) and ensure pp-value < 0.05.
  2. Evaluate the coefficient of determination (R2R^2) to measure how well the regression line explains variability in the data.

Without additional data on the pp-value or R2R^2, it is hard to definitively say if the prediction is statistically worthwhile, though a weak slope (0.0031-0.0031) suggests limited practical significance.


Would you like details on hypothesis testing for regression significance or calculating R2R^2? Here are 5 related questions:

  1. How is the pp-value for a regression slope calculated?
  2. What does R2R^2 tell us about a regression model?
  3. Can correlation alone indicate causation in regression models?
  4. How do we interpret a negative slope in a regression equation?
  5. 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