Linear regression is a statistical method for modeling the relationship between a dependent variable and one or more independent variables by fitting a straight line that minimizes the sum of squared distances from data points to the line (least squares method).
Given scattered points, draw the 'best' line through them. 'Best' means the line that's closest to all points on average. This line lets you predict Y from X.
Showing a random 20 of 50 problems.
Example 1
medium
A regression of ice-cream sales on temperature has slope +2 and r=0.9. Does temperature cause sales?
Example 2
medium
You shift all x values by +10 without changing y. Does the regression slope change?
Example 3
easy
In y^=bx+a, what does a represent?
Example 4
hard
You scale all y values by 2 but leave x unchanged. How do slope and intercept of the new regression compare to the old?
Example 5
easy
In y^=4x+11, by how much does predicted y change when x increases by 1?Slope of ŷ = 4x + 11: predicted y rises by 4 for each unit increase in x
Example 6
hard
A regression equation is y^=50−2.3x. Interpret the slope. If x=8, find y^.
Example 7
easy
In y^=5x+8, what is the intercept and what does it mean?ŷ = 5x + 8; the intercept a = 8 is the predicted y at x = 0
Example 8
medium
Regression line y^=2.5x+4. The number of hours studied is x and the test score is y^. How much does each extra hour raise the predicted score?Slope of ŷ = 2.5x + 4: each additional hour raises predicted score by 2.5 points
Example 9
easy
In a regression of test score on hours studied, the slope is 5. Interpret it in plain English.
Example 10
easy
Regression line y^=−2x+30. Predict y at x=4.Regression line ŷ = −2x + 30; find the predicted value at x = 4
Example 11
hard
For data with ∑x=20, ∑y=60, ∑xy=250, ∑x2=90, n=5, find the regression line.
Example 12
challenge
Regression y^=4x+2 from data with x from 1 to 6. A student predicts y^ at x=6 as 26 and at x=20 as 82. Which prediction is trustworthy and why?
Example 13
medium
Why does a regression equation describe association, not causation?
Example 14
easy
Linear regression fits a line by minimizing what?
Example 15
medium
Regression line y^=7x+50. Solve for x when y^=134.Regression line ŷ = 7x + 50; solve for x when ŷ = 134
Example 16
easy
Regression line y^=2.5x+4. Find y^ at x=0.ŷ = 2.5x + 4; the intercept a = 4 is the predicted y at x = 0
Example 17
medium
A regression slope is b=3 with sx=4 and sy=20. Find the correlation r.
Example 18
medium
Data: (xˉ,yˉ)=(5,30), sx=2, sy=8, r=0.75. Write the regression equation.Regression line ŷ = 3x + 15 passes through the centroid (x̄=5, ȳ=30)
Example 19
hard
Regression residuals are plotted vs. y^ and show a clear curved pattern. What does this suggest about the linear model?
Example 20
medium
A regression model of plant height (cm) on water (mL) gives y^=0.1x+4. Interpret slope and intercept in context.Plant height vs water: ŷ = 0.1x + 4; slope = 0.1 cm/mL, intercept = 4 cm