a small mountain near tectonic plate line andes has been growing consistently in height each year. the first measured height was recorded i 1997.
1997; 997.7300
2000;997.7606
2005;997.7810
determine the regression equation for the data
To determine the regression equation for the given data, we will use a linear regression model. This will help us find the equation that best fits the relationship between the years and the corresponding heights.
Step 1: Calculate the number of years since 1997. We will use this as our independent variable (x).
1997 - 1997 = 0
2000 - 1997 = 3
2005 - 1997 = 8
Years (x): [0, 3, 8]
Step 2: Use the given heights as the dependent variable (y).
Heights (y): [997.7300, 997.7606, 997.7810]
Step 3: Calculate the means of x and y.
Mean of x (x̄) = (0 + 3 + 8) / 3 = 11 / 3 ≈ 3.67
Mean of y (ȳ) = (997.7300 + 997.7606 + 997.7810) / 3 ≈ 997.7572
Step 4: Calculate the deviations from the means for both x and y.
deviations from mean of x (Δx): [-3.67, -0.67, 4.33]
deviations from mean of y (Δy): [-0.0272, 0.0034, 0.0238]
Step 5: Calculate the sum of the products of the deviations.
Σ(Δx * Δy) = (-3.67 * -0.0272) + (-0.67 * 0.0034) + (4.33 * 0.0238) ≈ 0.0889
Step 6: Calculate the sum of the squared deviations for x.
Σ(Δx^2) = (-3.67)^2 + (-0.67)^2 + (4.33)^2 ≈ 26.4049
Step 7: Calculate the slope of the regression line.
Slope (b) = Σ(Δx * Δy) / Σ(Δx^2) ≈ 0.0889 / 26.4049 ≈ 0.00337
Step 8: Calculate the y-intercept of the regression line.
y-intercept (a) = ȳ - b * x̄ ≈ 997.7572 - 0.00337 * 3.67 ≈ 997.7452
Step 9: Write the equation for the regression line.
Regression equation: y = a + bx
y = 997.7452 + 0.00337x
Therefore, the regression equation for the given data is y ≈ 997.7452 + 0.00337x.