The value of r² for the following data to three decimal places is 0.994.
To calculate r², we first need to find the following values:
Mean of x: (1 + 2 + 3 + 4 + 5) / 5 = 3
Mean of y: (2 + 5 + 7 + 10 + 20) / 5 = 8
Sum of squared deviations from the mean for x:
((1 - 3)² + (2 - 3)² + (3 - 3)² + (4 - 3)² + (5 - 3)²) = 4
Sum of squared deviations from the mean for y:
((2 - 8)² + (5 - 8)² + (7 - 8)² + (10 - 8)² + (20 - 8)²) = 32
The total sum of squares:
((1 - 3)² + (2 - 3)² + (3 - 3)² + (4 - 3)² + (5 - 3)²) + ((2 - 8)² + (5 - 8)² + (7 - 8)² + (10 - 8)² + (20 - 8)²) = 36
r² is then calculated using the following formula:
r² = 1 - (Sum of squared deviations from the mean for x) / (Total sum of squares)
r² = 1 - 4 / 36 = 0.994
Therefore, the value of r² for the following data to three decimal places is 0.994.
r² is a measure of how well a regression line fits the data. It is calculated by dividing the sum of squared deviations from the mean for y by the total sum of squares. A value of r² close to 1 indicates that the regression line fits the data very well. A value of r² close to 0 indicates that the regression line does not fit the data very well.
In this case, the value of r² is 0.994, which is very close to 1. This indicates that the regression line fits the data very well.