For the following dataset, enter the data into a calculator and find the least square regression line, correlation coefficient, and coefficient of determination. step by step please
X1 Y1
10 8.04
8 6.95
13 7.58
9 8.81
11 8.33
14 9.96
6 7.24
4 4.26
12 10.84
7 4.82
5 5.68
If you do this using a calculator, you will have to use the calculator's instructions for finding a regression equation, correlation coefficient (r) and coefficient of determination (r^2).
To find the least squares regression line, correlation coefficient, and coefficient of determination, follow these steps:
Step 1: Calculate the mean (average) of both X and Y values.
X_mean = (10 + 8 + 13 + 9 + 11 + 14 + 6 + 4 + 12 + 7 + 5) / 11 = 9.00
Y_mean = (8.04 + 6.95 + 7.58 + 8.81 + 8.33 + 9.96 + 7.24 + 4.26 + 10.84 + 4.82 + 5.68) / 11 = 7.50
Step 2: Calculate the deviations from the mean for both X and Y values.
Deviation_X = (10 - 9) + (8 - 9) + (13 - 9) + (9 - 9) + (11 - 9) + (14 - 9) + (6 - 9) + (4 - 9) + (12 - 9) + (7 - 9) + (5 - 9) = 4
Deviation_Y = (8.04 - 7.50) + (6.95 - 7.50) + (7.58 - 7.50) + (8.81 - 7.50) + (8.33 - 7.50) + (9.96 - 7.50) + (7.24 - 7.50) + (4.26 - 7.50) + (10.84 - 7.50) + (4.82 - 7.50) + (5.68 - 7.50) = 1.76
Step 3: Calculate the sum of the squared deviations for both X and Y values.
Squared_Deviation_X = (10 - 9)^2 + (8 - 9)^2 + (13 - 9)^2 + (9 - 9)^2 + (11 - 9)^2 + (14 - 9)^2 + (6 - 9)^2 + (4 - 9)^2 + (12 - 9)^2 + (7 - 9)^2 + (5 - 9)^2 = 66
Squared_Deviation_Y = (8.04 - 7.50)^2 + (6.95 - 7.50)^2 + (7.58 - 7.50)^2 + (8.81 - 7.50)^2 + (8.33 - 7.50)^2 + (9.96 - 7.50)^2 + (7.24 - 7.50)^2 + (4.26 - 7.50)^2 + (10.84 - 7.50)^2 + (4.82 - 7.50)^2 + (5.68 - 7.50)^2 = 26.4964
Step 4: Calculate the product of deviations for each X and Y values.
Deviation_Product = (10 - 9) * (8.04 - 7.50) + (8 - 9) * (6.95 - 7.50) + (13 - 9) * (7.58 - 7.50) + (9 - 9) * (8.81 - 7.50) + (11 - 9) * (8.33 - 7.50) + (14 - 9) * (9.96 - 7.50) + (6 - 9) * (7.24 - 7.50) + (4 - 9) * (4.26 - 7.50) + (12 - 9) * (10.84 - 7.50) + (7 - 9) * (4.82 - 7.50) + (5 - 9) * (5.68 - 7.50) = 28.32
Step 5: Calculate the slope (b) of the least squares regression line.
b = Deviation_Product / Squared_Deviation_X = 28.32 / 66 = 0.4291
Step 6: Calculate the y-intercept (a) of the least squares regression line.
a = Y_mean - b * X_mean = 7.50 - 0.4291 * 9.00 = 7.50 - 3.8619 = 3.6381
Step 7: The equation of the least squares regression line is:
Y = a + b * X = 3.6381 + 0.4291 * X
Step 8: Calculate the correlation coefficient (r).
r = Deviation_Product / (Squared_Deviation_X * Squared_Deviation_Y)^0.5 = 28.32 / (66 * 26.4964)^0.5 = 0.8164
Step 9: Calculate the coefficient of determination (r^2).
r^2 = r^2 = 0.8164^2 = 0.6664
Therefore, the least squares regression line equation is Y = 3.6381 + 0.4291 * X, the correlation coefficient is r = 0.8164, and the coefficient of determination is r^2 = 0.6664.
To find the least square regression line, correlation coefficient, and coefficient of determination for the given dataset, you can follow these step-by-step instructions:
Step 1: Calculate the mean of X and Y
To calculate the mean of X, add up all the X1 values and divide by the total number of data points:
Mean of X = (10 + 8 + 13 + 9 + 11 + 14 + 6 + 4 + 12 + 7 + 5) / 11 = 9.0
To calculate the mean of Y, add up all the Y1 values and divide by the total number of data points:
Mean of Y = (8.04 + 6.95 + 7.58 + 8.81 + 8.33 + 9.96 + 7.24 + 4.26 + 10.84 + 4.82 + 5.68) / 11 = 7.5
Step 2: Calculate the deviation of each X value from the mean of X (X - X̄)
Subtract the mean of X (9.0) from each X1 value:
X1 deviations: 1, -1, 4, 0, 2, 5, -3, -5, 3, -2, -4
Step 3: Calculate the deviation of each Y value from the mean of Y (Y - Ȳ)
Subtract the mean of Y (7.5) from each Y1 value:
Y1 deviations: 0.54, -0.55, 0.08, 1.31, 0.83, 2.46, -0.26, -3.24, 3.34, -2.68, -1.82
Step 4: Calculate the products of X deviations and Y deviations
Multiply each X deviation with its corresponding Y deviation:
Product of deviations: 0.54, 0.55, 0.32, 0, 1.66, 12.3, 0.78, 16.2, 10.02, 5.36, 7.28
Step 5: Calculate the squared deviations of X (X - X̄)²
Square each X1 deviation:
Squared X deviations: 1, 1, 16, 0, 4, 25, 9, 25, 9, 4, 16
Step 6: Calculate the sum of the squared deviations of X, squared deviations of Y, and the product of X and Y deviations
Add up the squared X deviations, squared Y deviations, and the product of X and Y deviations:
Sum of squared X deviations = 1 + 1 + 16 + 0 + 4 + 25 + 9 + 25 + 9 + 4 + 16 = 110
Sum of squared Y deviations = 0.54² + 0.55² + 0.32² + 1.31² + 0.83² + 2.46² + 0.26² + 3.24² + 3.34² + 2.68² + 1.82² = 35.33
Sum of product of deviations = 0.54 + 0.55 + 0.32 + 0 + 1.66 + 12.3 + 0.78 + 16.2 + 10.02 + 5.36 + 7.28 = 54.41
Step 7: Calculate the slope (b) of the regression line
The slope (b) of the regression line can be calculated using the formula:
b = sum of product of deviations / sum of squared X deviations
b = 54.41 / 110 = 0.4946
Step 8: Calculate the y-intercept (a) of the regression line
The y-intercept (a) of the regression line can be calculated using the formula:
a = mean of Y - (b * mean of X)
a = 7.5 - (0.4946 * 9) = 7.5 - 4.4514 = 3.0486
Therefore, the equation of the regression line is:
Y = 3.0486 + 0.4946X
Step 9: Calculate the correlation coefficient (r)
The correlation coefficient (r) can be calculated using the formula:
r = (sum of product of deviations) / sqrt((sum of squared X deviations) * (sum of squared Y deviations))
r = 54.41 / sqrt(110 * 35.33) = 54.41 / sqrt(3881.3) = 54.41 / 62.2791 = 0.8727
Step 10: Calculate the coefficient of determination (r²)
The coefficient of determination (r²) can be calculated by squaring the correlation coefficient:
r² = 0.8727² = 0.7614
Therefore, the least square regression line equation is Y = 3.0486 + 0.4946X, the correlation coefficient is 0.8727, and the coefficient of determination is 0.7614.