Regression and Correlation
Linear regression fits a line $y=a+bx$ to paired data for estimation and prediction.
$$b=\frac{\sum (x-\bar{x})(y-\bar{y})}{\sum (x-\bar{x})^2}, \qquad a=\bar{y}-b\bar{x}$$
Linear regression fits a line $y=a+bx$ to paired data for estimation and prediction.
Fit a line to points $(1,2)$, $(2,5)$, and $(3,7)$. Estimate $y$ when $x=4$.
Using least squares, $\bar{x}=2$ and $\bar{y}=14/3$.
At $x=4$, $y=9.67$.
A fitted line for project cost is y = 120 + 8x, where y is in thousand pesos and x is floor area in 100 m2. Predict y when x = 15.
Answer: The predicted cost is P240,000.
For the regression line y = 35 + 2.5x, state the change in y for each one-unit increase in x.
The slope gives the predicted change in the response variable.
Answer: y increases by 2.5 units for each additional unit of x.
A data set gives r = -0.86 between compaction delay and achieved density. Interpret the sign and strength.
The negative sign means the variables move in opposite directions, and the magnitude 0.86 is close to 1.
Answer: There is a strong negative linear relationship.
A regression model predicts 74.5, but the observed value is 72.0. Find the residual using observed minus predicted.
Answer: The residual is -2.5.