Math Problem Statement
The average price per m^2 across different years is given by table. Find the slope of the linear regression equation for Year in terms of price. 1996 - $105000. 2000 - $140000. 2003 - 200000. 2006 - 270000. 2007 - 290000.
Solution
To find the slope of the linear regression equation for the year in terms of price, we need to compute the slope using the formula for a linear regression line:
Where:
- represents the years,
- represents the prices,
- is the number of data points.
First, let's list the year and price dataTo find the slope of the linear regression equation for the year in terms of price, we need to compute the slope using the formula for a linear regression line:
Where:
- represents the years,
- represents the prices,
- is the number of data points.
First, let's list the year and price
Year | Price (in $) |
---|---|
1996 | 105,000 |
2000 | 140,000 |
2003 | 200,000 |
2006 | 270,000 |
2007 | 290,000 |
Now, for each year, we will set to represent the number of years since 1996. So we adjust the years accordingly:
This will give us the following:
Year | Adjusted Year (x) | Price (y) |
---|---|---|
1996 | 0 | 105,000 |
2000 | 4 | 140,000 |
2003 | 7 | 200,000 |
2006 | 10 | 270,000 |
2007 | 11 | 290,000 |
Now let's calculate the necessary sums:
Now, using the formula for the slope:
Thus, the slope of the linear regression equation is approximately .
This means that for each year after 1996, the price per square meter increases by approximately $17,467.49.
Would you like more details or have any questions?
Here are 5 related questions to expand on this topic:
- How would you find the intercept of the linear regression equation?
- What does the slope represent in a real estate context?
- How would the regression change if prices in 2005 were also included?
- How can the residuals from this linear model be calculated?
- How would you calculate the correlation between year and price?
Tip: The slope in a regression line indicates how much the dependent variable (price) changes for a one-unit change in the independent variable (year).
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Math Problem Analysis
Mathematical Concepts
Linear Regression
Algebra
Statistics
Formulas
Slope formula for linear regression: slope = (n∑xy - (∑x)(∑y)) / (n∑x^2 - (∑x)^2)
Theorems
Linear regression formula
Suitable Grade Level
Grades 10-12