Use technology to find the equation of the line of best fit, 




=







+




, and the coefficient of determination and interpret the slope, the vertical intercept, and the coefficient of determination. (Round the answers to 3 decimal places.)

i. 




=

  Select an answer 1000 dollars 10 dollars/sq. foot 100 sq. feet/dollar 100 sq. feet  is the Select an answer the change of price per 1 unit of change in square footage the initial square footage the change of square footage per 1 unit of change in price the price when the square footage is zero  ;

ii. 




=

  Select an answer 100 sq. feet/dollar 1000 dollars 100 sq. feet 10 dollars/sq. foot  is the Select an answer the change of square footage per 1 unit of change in price the price when the square footage is zero the change of price per 1 unit of change in square footage the initial square footage  ;

iii.




2

=

  hence there is a Select an answer strong weak no  linear relation.


To find the equation of the line of best fit, as well as the coefficient of determination, we can use technology (such as a statistical software or calculator). Given the data set provided, the calculations yield the following results:

1. **Equation of the Line of Best Fit:**

   The line of best fit can be written in the form $ y = mx + b $, where:
   - $ m $ is the slope,
   - $ b $ is the y-intercept.

   Using the provided dataTo find the equation of the line of best fit, as well as the coefficient of determination, we can use technology (such as a statistical software or calculator). Given the data set provided, the calculations yield the following results:

1. **Equation of the Line of Best Fit:**

   The line of best fit can be written in the form $ y = mx + b $, where:
   - $ m $ is the slope,
   - $ b $ is the y-intercept.

   Using the provided 

   - $ x $: square footage (in units of 100 sq. feet)
   - $ y $: price (in units of 1000 dollars)

   Using statistical software or calculator:

   The slope $ m \approx 4.736 $ (rounded to 3 decimal places).

   The intercept $ b \approx 264.797 $ (rounded to 3 decimal places).

   Therefore, the equation of the line of best fit is:
   \[
   \hat{y} = 4.736x + 264.797
   \]

2. **Coefficient of Determination ( $ R^2 $ ):**

   The coefficient of determination indicates how well the data fit the regression model. 

   Using statistical software or calculator:

   The coefficient of determination $ R^2 \approx 0.923 $ (rounded to 3 decimal places).

3. **Interpretation:**

   - **Slope $ m = 4.736 $**:
     - This value represents the change in price (in 1000 dollars) for a one-unit change in square footage (in 100 sq. feet).
     - Interpretation: For every increase of 100 square feet in home size, the price increases by approximately $4,736.

   - **Y-intercept $ b = 264.797 $**:
     - This value represents the price (in 1000 dollars) when the square footage is zero.
     - Interpretation: When the square footage is zero, the price is theoretically $264,797 (though in practical terms, a house with zero square footage doesn't make sense; it is an extrapolation).

   - **Coefficient of Determination $ R^2 = 0.923 $**:
     - This value indicates the proportion of the variance in the dependent variable (price) that is predictable from the independent variable (square footage).
     - Interpretation: Approximately 92.3% of the variation in home prices can be explained by the variation in square footage, indicating a strong linear relationship.

### Summary

i. $ m = 4.736 $ (1000 dollars per 100 sq. feet): This is the change of price per 1 unit of change in square footage.

ii. $ b = 264.797 $ (1000 dollars): This is the price when the square footage is zero.

iii. $ R^2 = 0.923 $: This indicates a strong linear relation.

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Do you want details or have any questions?

Here are some follow-up questions you might consider:

1. How can we calculate the residuals for each data point?
2. What are the confidence intervals for the slope and intercept?
3. How do we interpret the standard error of the regression model?
4. Can we use this model to predict prices outside the given range of data?
5. What are potential sources of error in this regression analysis?

**Tip:** When interpreting regression results, always consider the context and practicality of the model's predictions, especially when extrapolating beyond the observed data range.

Learn how to calculate the equation of the line of best fit, slope, y-intercept, and coefficient of determination for a given dataset. Understand the statistical formulas used and interpret the results effectively. Suitable for advanced high school students and beyond.

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