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.

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:

\[
\text{slope} = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}
\]

Where:
- $x$ represents the years,
- $y$ represents the prices,
- $n$ 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:

\[
\text{slope} = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}
\]

Where:
- $x$ represents the years,
- $y$ represents the prices,
- $n$ 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 $x$ to represent the number of years since 1996. So we adjust the years accordingly:

\[
x = \text{year} - 1996
\]

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:

\[
\sum x = 0 + 4 + 7 + 10 + 11 = 32
\]
\[
\sum y = 105,000 + 140,000 + 200,000 + 270,000 + 290,000 = 1,005,000
\]
\[
\sum xy = (0 \times 105,000) + (4 \times 140,000) + (7 \times 200,000) + (10 \times 270,000) + (11 \times 290,000) = 0 + 560,000 + 1,400,000 + 2,700,000 + 3,190,000 = 7,850,000
\]
\[
\sum x^2 = 0^2 + 4^2 + 7^2 + 10^2 + 11^2 = 0 + 16 + 49 + 100 + 121 = 286
\]

Now, using the formula for the slope:

\[
\text{slope} = \frac{5(7,850,000) - (32)(1,005,000)}{5(286) - (32)^2}
\]
\[
\text{slope} = \frac{39,250,000 - 32,160,000}{1,430 - 1,024}
\]
\[
\text{slope} = \frac{7,090,000}{406}
\]
\[
\text{slope} \approx 17,467.49
\]

Thus, the slope of the linear regression equation is approximately $17,467.49$.

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:
1. How would you find the intercept of the linear regression equation?
2. What does the slope represent in a real estate context?
3. How would the regression change if prices in 2005 were also included?
4. How can the residuals from this linear model be calculated?
5. 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).

Learn how to compute the slope of a linear regression equation for Year in terms of price using real-world data from 1996 to 2007. This tutorial walks through each step of the calculation, including applying the linear regression slope formula to the given price and year data points. Suitable for students in Grades 10-12.

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