Answer :
The equation of the regression line is approximately:
y = -5.347x + 24.405
To find the equation of the regression line, we need to perform linear regression on the given data points.
The equation of a regression line is typically represented as:
y = mx + b
where y is the dependent variable, x is the independent variable, m is the slope, and b is the y-intercept.
In this case, the independent variable is the family income (x) and the dependent variable is the ulcer rate (y).
We can calculate the slope (m) and y-intercept (b) using the following formulas:
m = (Σ(xy) - (Σx)(Σy) / n(Σx²) - (Σx)²)
b = (Σy - m(Σx)) / n
Let's calculate the slope (m) and y-intercept (b) using the given data points:
Income (x) Ulcer rate (y)
4000 12.9
6000 12.8
8000 12.4
12000 11.9
16000 12.3
20000 11.1
30000 10.5
45000 8.5
60000 7.2
Σx = 4000 + 6000 + 8000 + 12000 + 16000 + 20000 + 30000 + 45000 + 60000 = 224000
Σy = 12.9 + 12.8 + 12.4 + 11.9 + 12.3 + 11.1 + 10.5 + 8.5 + 7.2 = 100.6
Σxy = (4000 * 12.9) + (6000 * 12.8) + (8000 * 12.4) + (12000 * 11.9) + (16000 * 12.3) + (20000 * 11.1) + (30000 * 10.5) + (45000 * 8.5) + (60000 * 7.2) = 2,685,000
Σx² = (4000²) + (6000²) + (8000²) + (12000²) + (16000²) + (20000²) + (30000²) + (45000²) + (60000²) = 8,144,000,000
n = 9 (number of data points)
Now, let's substitute these values into the formulas to find m and b:
m = (Σ(xy) - (Σx)(Σy) / n(Σx²) - (Σx)²)
= (2,685,000 - (224000 * 100.6) / (9 * 8,144,000,000) - (224000)²
≈ -5.347
b = (Σy - m(Σx)) / n
= (100.6 - (-5.347) * 224000) / 9
≈ 24.405
Therefore, the equation of the regression line is approximately:
y = -5.347x + 24.405
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