Before you read

Answer first — see what you already know.

A researcher models the relationship between student absences ( $x$ ) and final exam grade ( $y$ ) with the line of best fit $y = - 2.5 x + 100$ . What does the slope represent in this context?

A
Each absence raises the grade by 2.5 points
B
Each absence lowers the grade by 2.5 points
C
The student starts with a grade of 2.5
D
After 100 absences, the grade is 0

Check your answer

Answer:

Before you read

Answer first — see what you already know.

A scatterplot displays a strong negative correlation between hours of TV watched per week ( $x$ ) and GPA ( $y$ ). Which could be the equation of the line of best fit?

A
$y = 4 x + 10$
B
$y = - 0.8 x + 50$
C
$y = 0.1 x - 3$
D
$y = 25$

Check your answer

Answer:

Theory

Correlation in Scatterplots

A scatterplot shows pairs of data as points on a coordinate plane.

Positive correlation: As $x$ increases, $y$ tends to increase (upward trend).
Negative correlation: As $x$ increases, $y$ tends to decrease (downward trend).
No correlation: No clear pattern.

Positive correlation with line of best fit

Strength of correlation:
- Strong: Points are close to a line
- Weak: Points are loosely scattered around a line

Correlation $\neq$ Causation: Just because two variables are correlated doesn't mean one causes the other!

Theory

Line of Best Fit

The line of best fit (regression line) is the line that best represents the trend in a scatterplot. Its equation is $y = mx + b$ where:
- $m$ = slope (rate of change)
- $b$ = y-intercept (value when $x = 0$ )

Using the line of best fit:
- Interpolation: Predict $y$ for an $x$ value within the data range (reliable)
- Extrapolation: Predict $y$ for an $x$ value outside the data range (less reliable)

Residual $=$ Actual value $-$ Predicted value. A positive residual means the actual value is above the line.

Example 1

A line of best fit for study hours ( $x$ ) vs. test score ( $y$ ) is $y = 5x + 60$ . Predict the score for 6 hours of study. A student who studied 6 hours scored 85. What is the residual?

Predicted score: $y = 5(6) + 60 = 90$

Residual $= 85 - 90 = -5$

The student scored 5 points below the predicted value.

Predicted = 90, Residual = −5

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