Alright! Let's go step by step. We want to understand how the house size relates to the number of residents. In other words, as the number of residents changes, how does the size of the house change? This relationship can be represented by a linear regression equation. The general form of a linear regression equation is:
y = m*x + b
Here:
- y is the dependent variable (in our case, the house size).
- x is the independent variable (in our case, the number of residents).
- m is the slope of the line (how much y changes for a unit change in x).
- b is the y-intercept (the value of y when x is 0).
We'll use the data you provided to calculate 'm' and 'b'. There are different ways to calculate these values, but I'll use a method that is relatively simple to understand:
m = (N * Σ(xy) - Σx * Σy) / (N * Σ(x^2) - (Σx)^2)
b = (Σy - m * Σx) / N
Where:
- N is the number of data points (in our case, 15).
- Σ stands for summation (sum of all values).
Now, let's calculate 'm' and 'b' using the data you provided:
Number of Residents(x) | House size (Sq. ft)(y) | xy | x^2
------------------------|------------------------|----|-----
3 | 1992 |5976|9
3 | 1754 |5262|9
3 | 1766 |5298|9
5 | 2060 |10300|25
6 | 2293 |13758|36
6 | 2139 |12834|36
3 | 1836 |5508|9
4 | 1924 |7696|16
6 | 2321 |13926|36
4 | 2060 |8240|16
3 | 1769 |5307|9
4 | 1955 |7820|16
5 | 2309 |11545|25
4 | 1857 |7428|16
4 | 1972 |7888|16
Σx = 66
Σy = 30999
Σxy = 120978
Σ(x^2) = 282
Plug these values into our formulas:
m = (15 * 120978 - 66 * 30999) / (15 * 282 - 66^2)
≈ 305.91
b = (30999 - 305.91 * 66) / 15
≈ 905.27
So our linear regression equation is:
House size = 305.91 * (Number of Residents) + 905.27
Now, let's predict the house size for a family of 5 residents:
House size = 305.91 * 5 + 905.27
≈ 2444.82 Sq. ft
This means that, according to our linear regression model, a family of 5 residents would need a house size of approximately 2445 square feet.