To calculate the residual for the data point (28, 19) using the least-squares regression line, we need to follow the steps below:
1. Use the regression equation to predict the value of y (which we denote as ŷ) for the given x-value, which in this case is 28.
2. Calculate ŷ using the equation of the line: ŷ = -25.5 + 1.5x.
3. Substitute x = 28 into the equation to compute ŷ.
4. Once we obtain the predicted value, calculate the residual by subtracting the predicted value of y (ŷ) from the actual value of y (which is given as 19).
Let's perform the calculations:
The predicted value of y (ŷ) is:
ŷ = -25.5 + 1.5(28)
ŷ = -25.5 + (1.5 * 28)
ŷ = -25.5 + 42
ŷ = 16.5
Now we compute the residual:
residual = actual y - predicted y
residual = 19 - 16.5
residual = 2.5
Therefore, the residual for the data point (28, 19) is 2.5.
