Scatterplots and correlation
What does a scatterplot show?
The relationship between two quantitative variables.
The LSRL predicting quiz score from hours studied is ŷ = 65 + 5x.
What is the predicted score for a student who studies 3 hours?
ŷ = 65 + 5(3) = 80
What is a residual?
Actual y – Predicted y
What does the slope represent in context?
→ The predicted change in y for each 1-unit increase in x.
What does it mean if the residual is zero for a data point?
→ The predicted value equals the actual value — the model was exact for that point.
Describe what a positive association looks like.
As x increases, y tends to increase.
Write the general form of the LSRL equation
ŷ = a + bx
A predicted score is 82, actual is 78. What’s the residual?
-4
What does the intercept represent?
→ The predicted value of y when x = 0.
If all residuals were positive, what would that indicate about the model?
→ The model consistently underpredicted the actual values
What does correlation (r) measure?
The strength and direction of a linear relationship.
If slope b = 2.5 and intercept a = 10, what’s the predicted y when x = 4?
ŷ = 10 + 2.5(4) = 20
What does a positive residual mean?
The model underpredicted the actual value.
Why can the intercept sometimes be meaningless?
→ Because x = 0 may not be within the data’s context.
What is the mean of all residuals in a least-squares regression line?
→ 0
If r = –0.9, describe the association.
Strong negative linear relationship.
A regression line predicting weight (y) from height (x) is ŷ = –120 + 2.5x.
Predict the weight of someone who is 70 inches tall, and find the residual if their actual weight is 58.
Predicted: ŷ = –120 + 2.5(70) = 55 lbs
Residual = actual – predicted = 58 – 55 = +3
What pattern should residuals have if the model is appropriate?
No pattern — randomly scattered around 0.
If slope = 1.8, interpret it in context of hours studied vs. test score.
→ For each additional hour studied, the predicted test score increases by 1.8 points.
What does it mean if a residual plot shows a clear curved pattern?
The data have a nonlinear relationship, so a straight-line model (LSRL) isn’t a good fit.
True or False: Correlation is resistant to outliers.
False — correlation is not resistant to outliers
The regression line for predicting exam score from study time is ŷ = 40 + 8x.
A student who studied 5 hours actually scored 74.
Find the residual.
Did the model overpredict or underpredict?
Predicted = 40 + 8(5) = 80
Residual = 74 – 80 = –6, meaning the model overpredicted the score.
What type of residual pattern indicates nonlinearity?
A curved or systematic pattern.
A model predicts negative y-values for x = 0. What’s a likely issue?
→ The intercept is not realistic for this situation.
Why do we check a residual plot after making a regression line?
To see if a linear model is appropriate and ensure there’s no pattern in residuals — confirming the model fits the data well.