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The question refers to a regression analysis between distance and time based on the formula for free fall, where the distance an object falls is related to the square of the time.

Given:
- The model is \( \text{distance} = a \times (\text{time})^2 \).
- The regression output shows that:
  - The constant term is 0.007,
  - The coefficient for \( (\text{time})^2 \) is 2.511.

From the options provided, we are looking for the equation that follows the least-squares regression result. The correct regression equation should be in the form:
\[ \text{distance} = 0.007 + 2.511 \times (\text{time})^2. \]

This matches the first option:
\[ \text{distance} = 0.007 + 2.511(\text{time})^2. \]

Thus, the correct answer is the first option.

Let me know if you'd like more details or have any other questions.

Here are 5 related questions to deepen your understanding:
1. How does the least-squares method work in regression analysis?
2. Why is time squared in the equation for distance fallen under gravity?
3. How can you calculate the error (residual) in regression analysis?
4. What does the R-squared value (95.1%) indicate about the regression fit?
5. How do you interpret the p-values (0.000) shown in the regression output?

**Tip:** Always check the units and exponents in regression models to match them with the physical relationships they represent.

The distance that an object dropped from a given height falls is related to time because release by the model distance = a (time)^2. An engineer collected data on the distance (in feet) for various amounts of time (in seconds) for a random sample of rubber balls. The scatterplot of (time)^2 and distance is roughly linear. The regression analysis produced the following output. Which of the following is the correct equation of the least-squares regression line for this data?

Understand how to determine the least-squares regression equation for data involving the distance fallen by an object over time. This problem covers key concepts like regression analysis, quadratic relationships, and the least-squares method, suitable for students in Grades 10-12.

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