Question Details:
Finding Regression for each set of data & finding best fit equation for each set of data:
Data 1:
X: 1 2 3 4 5
Y: 3.1 12.1 20.7 33.9 50.8
Data 2:
X: 1 2 3 4 5
Y: 1.16 3.46 5.11 5.98 6.75
To find the regression line and the best fit equation for each set of data, we can use the method of least squares regression.
The formula for the best fit line or regression line is given by:
y = mx + b
where m is the slope of the line and b is the y-intercept.
To find the slope (m) and the y-intercept (b), we need to determine the values of them that minimize the sum of the squared differences between the actual data points and the predicted values from the regression line.
Let's start with Data 1:
X: 1 2 3 4 5
Y: 3.1 12.1 20.7 33.9 50.8
To find the regression line, we need to find the values of m and b that minimize the sum of squared differences. This can be done using the following formulas:
m = (nΣXY - ΣXΣY) / (nΣX^2 - (ΣX)^2)
b = (ΣY - mΣX) / n
where ΣXY represents the sum of the products of X and Y, ΣX represents the sum of X values, ΣY represents the sum of Y values, ΣX^2 represents the sum of squared X values, and n represents the number of data points.
Using these formulas, we can calculate the values of m and b:
ΣXY = (1*3.1) + (2*12.1) + (3*20.7) + (4*33.9) + (5*50.8) = 10.6 + 24.2 + 62.1 + 135.6 + 254 = 486.5
ΣX = 1 + 2 + 3 + 4 + 5 = 15
ΣY = 3.1 + 12.1 + 20.7 + 33.9 + 50.8 = 120.6
ΣX^2 = (1^2) + (2^2) + (3^2) + (4^2) + (5^2) = 1 + 4 + 9 + 16 + 25 = 55
n = 5
Now, let's plug these values into the formulas:
m = (5*486.5 - 15*120.6) / (5*55 - (15)^2) = (2432.5 - 1809) / (275 - 225) = 623.5/50 = 12.47
b = (120.6 - 12.47*15) / 5 = (120.6 - 187.05) / 5 = -66.45/5 = -13.29
Therefore, the regression line for Data 1 is:
y = 12.47x - 13.29
Now, let's move on to Data 2:
X: 1 2 3 4 5
Y: 1.16 3.46 5.11 5.98 6.75
Using the same formulas, we can calculate the values of m and b:
ΣXY = (1*1.16) + (2*3.46) + (3*5.11) + (4*5.98) + (5*6.75) = 1.16 + 6.92 + 15.33 + 23.92 + 33.75 = 81.08
ΣX = 1 + 2 + 3 + 4 + 5 = 15
ΣY = 1.16 + 3.46 + 5.11 + 5.98 + 6.75 = 22.46
ΣX^2 = (1^2) + (2^2) + (3^2) + (4^2) + (5^2) = 1 + 4 + 9 + 16 + 25 = 55
n = 5
Now, let's plug these values into the formulas:
m = (5*81.08 - 15*22.46) / (5*55 - (15)^2) = (405.4 - 336.9) / (275 - 225) = 68.5/50 = 1.37
b = (22.46 - 1.37*15) / 5 = (22.46 - 20.55) / 5 = 1.91/5 = 0.38
Therefore, the regression line for Data 2 is:
y = 1.37x + 0.38
To summarize:
Data 1: y = 12.47x - 13.29
Data 2: y = 1.37x + 0.38
These equations represent the best fit lines or regression lines for each set of data.