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:

slope=nxy(x)(y)nx2(x)2\text{slope} = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}

Where:

  • xx represents the years,
  • yy represents the prices,
  • nn 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:

slope=nxy(x)(y)nx2(x)2\text{slope} = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}

Where:

  • xx represents the years,
  • yy represents the prices,
  • nn is the number of data points.

First, let's list the year and price

YearPrice (in $)
1996105,000
2000140,000
2003200,000
2006270,000
2007290,000

Now, for each year, we will set xx to represent the number of years since 1996. So we adjust the years accordingly:

x=year1996x = \text{year} - 1996

This will give us the following:

YearAdjusted Year (x)Price (y)
19960105,000
20004140,000
20037200,000
200610270,000
200711290,000

Now let's calculate the necessary sums:

x=0+4+7+10+11=32\sum x = 0 + 4 + 7 + 10 + 11 = 32 y=105,000+140,000+200,000+270,000+290,000=1,005,000\sum y = 105,000 + 140,000 + 200,000 + 270,000 + 290,000 = 1,005,000 xy=(0×105,000)+(4×140,000)+(7×200,000)+(10×270,000)+(11×290,000)=0+560,000+1,400,000+2,700,000+3,190,000=7,850,000\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 x2=02+42+72+102+112=0+16+49+100+121=286\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:

slope=5(7,850,000)(32)(1,005,000)5(286)(32)2\text{slope} = \frac{5(7,850,000) - (32)(1,005,000)}{5(286) - (32)^2} slope=39,250,00032,160,0001,4301,024\text{slope} = \frac{39,250,000 - 32,160,000}{1,430 - 1,024} slope=7,090,000406\text{slope} = \frac{7,090,000}{406} slope17,467.49\text{slope} \approx 17,467.49

Thus, the slope of the linear regression equation is approximately 17,467.4917,467.49.

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).

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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