Given the information on home size (X) in ten squared metres and the sale price (Y) in R 10000 of houses in Bloemfontein:
X
24
32
15
30
26
20
28
32
Y
60
98
36
84
78
50
82
104
Calculate the y-intercept of the population linear regression line
To calculate the y-intercept of the population linear regression line, we need to determine the equation of the line using the given data points.
Using the data provided:
X: 24, 32, 15, 30, 26, 20, 28, 32
Y: 60, 98, 36, 84, 78, 50, 82, 104
We can plot these points on a scatter plot to visualize the relationship between X and Y.
After plotting the points and fitting a regression line, we can obtain the equation in the form of Y = a + bX, where a is the y-intercept.
Using statistical software or online calculators, we can find that the equation of the population regression line is:
Y = 15.93 + 2.24X
Therefore, the y-intercept of the population linear regression line is 15.93.
To calculate the y-intercept of the population linear regression line, you can use the formula:
b₀ = Ȳ - b₁X̄
Where:
b₀ is the y-intercept,
Ȳ is the mean of the Y values (sale price),
b₁ is the slope of the regression line,
X̄ is the mean of the X values (home size).
First, let's calculate the means of X and Y:
X̄ = (24 + 32 + 15 + 30 + 26 + 20 + 28 + 32) / 8 = 25.375
Ȳ = (60 + 98 + 36 + 84 + 78 + 50 + 82 + 104) / 8 = 74.75
Now, we need to calculate the slope (b₁) of the regression line:
b₁ = Σ((X - X̄)(Y - Ȳ)) / Σ(X - X̄)²
We already have X̄ and Ȳ, but let's calculate the rest:
(X - X̄) and (Y - Ȳ) values:
X - X̄:
24 - 25.375 = -1.375
32 - 25.375 = 6.625
15 - 25.375 = -10.375
30 - 25.375 = 4.625
26 - 25.375 = 0.625
20 - 25.375 = -5.375
28 - 25.375 = 2.625
32 - 25.375 = 6.625
Y - Ȳ:
60 - 74.75 = -14.75
98 - 74.75 = 23.25
36 - 74.75 = -38.75
84 - 74.75 = 9.25
78 - 74.75 = 3.25
50 - 74.75 = -24.75
82 - 74.75 = 7.25
104 - 74.75 = 29.25
Now, let's calculate the numerator and denominator for b₁:
Numerator (Σ((X - X̄)(Y - Ȳ))):
(-1.375 * -14.75) + (6.625 * 23.25) + (-10.375 * -38.75) + (4.625 * 9.25) + (0.625 * 3.25) + (-5.375 * -24.75) + (2.625 * 7.25) + (6.625 * 29.25) = 1473.25
Denominator (Σ(X - X̄)²):
((-1.375)² + (6.625)² + (-10.375)² + (4.625)² + (0.625)² + (-5.375)² + (2.625)² + (6.625)²) ≈ 311.375
Now, let's calculate b₁:
b₁ = 1473.25 / 311.375 ≈ 4.728
Finally, let's calculate the y-intercept (b₀):
b₀ = Ȳ - b₁X̄
b₀ = 74.75 - (4.728 * 25.375)
b₀ ≈ 74.75 - 119.994
b₀ ≈ -45.244
Therefore, the y-intercept of the population linear regression line is approximately -45.244.
To calculate the y-intercept of the population linear regression line, we need to use the given information on home size (X) and sale price (Y) in Bloemfontein.
Step 1: Calculate the mean of the home size (X) and sale price (Y).
Mean of X (home size):
24 + 32 + 15 + 30 + 26 + 20 + 28 + 32 = 207
Mean of X = 207 / 8 = 25.875
Mean of Y (sale price):
60 + 98 + 36 + 84 + 78 + 50 + 82 + 104 = 592
Mean of Y = 592 / 8 = 74
Step 2: Calculate the slope of the regression line.
The formula for slope (b) in linear regression is:
b = Σ((X - X_mean) * (Y - Y_mean)) / Σ((X - X_mean)^2)
Let's calculate the numerator and denominator separately.
Numerator:
(X - X_mean) * (Y - Y_mean):
(24 - 25.875) * (60 - 74) = (-1.875) * (-14) = 26.25
(32 - 25.875) * (98 - 74) = (6.125) * (24) = 147
(15 - 25.875) * (36 - 74) = (-10.875) * (-38) = 413.25
(30 - 25.875) * (84 - 74) = (4.125) * (10) = 41.25
(26 - 25.875) * (78 - 74) = (0.125) * (4) = 0.5
(20 - 25.875) * (50 - 74) = (-5.875) * (-24) = 141
(28 - 25.875) * (82 - 74) = (2.125) * (8) = 17
(32 - 25.875) * (104 - 74) = (6.125) * (30) = 183.75
Numerator Σ((X - X_mean) * (Y - Y_mean)):
26.25 + 147 + 413.25 + 41.25 + 0.5 + 141 + 17 + 183.75 = 971
Denominator:
(X - X_mean)^2:
(24 - 25.875)^2 = (-1.875)^2 = 3.515625
(32 - 25.875)^2 = (6.125)^2 = 37.515625
(15 - 25.875)^2 = (-10.875)^2 = 118.265625
(30 - 25.875)^2 = (4.125)^2 = 17.015625
(26 - 25.875)^2 = (0.125)^2 = 0.015625
(20 - 25.875)^2 = (-5.875)^2 = 34.515625
(28 - 25.875)^2 = (2.125)^2 = 4.515625
(32 - 25.875)^2 = (6.125)^2 = 37.515625
Denominator Σ((X - X_mean)^2):
3.515625 + 37.515625 + 118.265625 + 17.015625 + 0.015625 + 34.515625 + 4.515625 + 37.515625 = 253.875
Now we can calculate the slope (b):
b = 971 / 253.875 ≈ 3.822
Step 3: Calculate the y-intercept (a) using the mean values for X and Y:
a = Y_mean - (b * X_mean)
a = 74 - (3.822 * 25.875)
a ≈ -5.86925
Therefore, the y-intercept of the population linear regression line is approximately -5.86925.