Use the equation of the regression line to predict y when x = 11.
x - 2 4 3 5 1
y - 9 12 11 14 6
To predict the value of y when x = 11 using the equation of the regression line, we first need to perform linear regression analysis on the given data points. The regression line is represented by the equation y = mx + b, where m is the slope of the line and b is the y-intercept.
1. Calculate the mean of the x-values (x̄) and the mean of the y-values (ȳ).
x̄ = (2 + 4 + 3 + 5 + 1) / 5 = 3
ȳ = (9 + 12 + 11 + 14 + 6) / 5 = 10.4
2. Calculate the deviations from the means for each data point.
For x: (2-3), (4-3), (3-3), (5-3), (1-3) = -1, 1, 0, 2, -2
For y: (9-10.4), (12-10.4), (11-10.4), (14-10.4), (6-10.4) = -1.4, 1.6, 0.6, 3.6, -4.4
3. Calculate the product of the deviations from the means for each data point.
(-1) * (-1.4), (1) * (1.6), (0) * (0.6), (2) * (3.6), (-2) * (-4.4) = 1.4, 1.6, 0, 7.2, 8.8
4. Calculate the sum of the squared deviations of x from x̄.
(-1)^2 + (1)^2 + (0)^2 + (2)^2 + (-2)^2 = 10
5. Calculate the sum of the products of the deviations of x and y.
1.4 + 1.6 + 0 + 7.2 + 8.8 = 19
6. Calculate the slope (m) of the regression line.
m = sum of products of deviations / sum of squared deviations of x
m = 19 / 10 = 1.9
7. Calculate the y-intercept (b) of the regression line.
b = ȳ - m * x̄
b = 10.4 - (1.9 * 3) = 10.4 - 5.7 = 4.7
8. Substitute the value of x = 11 into the regression line equation y = mx + b.
y = 1.9 * 11 + 4.7
y = 20.9 + 4.7 = 25.6
Therefore, when x = 11, the predicted value of y using the equation of the regression line is 25.6.