Unit 4: Data, Statistics, and Probability

Topic 7

Interpreting Trend Lines and Regression (Conceptual)

A trend line (or line of best fit) shows the general direction of data in a scatterplot. It helps predict values and understand the relationship between two variables.

Positive correlation: as one variable increases, the other tends to increase.
Negative correlation: as one variable increases, the other tends to decrease.
No correlation: points show no clear pattern.

A line of best fit is often written as:

\hat{y} = mx + b

where

m

is the slope (rate of change) and

b

is the

y

-intercept (predicted

y

when

x=0

For example, a trend line

\hat{y} = 5x + 60

fit through hours studied vs. test score looks like this:

Positive Correlation

Negative Correlation

No Correlation

Core Skills

Identify direction (positive, negative, none) of correlation.
Interpret slope and intercept in real-world context.
Use the line to make predictions or estimates.
Recognize that correlation does not imply causation.

Example 1: Positive Correlation

A scatterplot shows an upward trend between hours studied and test score. As study hours increase, scores increase. The trend line shows a positive slope.

Example 2: Negative Correlation

A scatterplot shows a downward trend between absences and GPA. As absences increase, GPA decreases. Negative slope, negative correlation.

Example 3: No Correlation

A scatterplot shows a random scatter with no clear pattern. No consistent relationship between the two variables.

Example 4: Interpreting the Equation of a Trend Line

For the trend line

\hat{y} = 5x + 60

$m = 5$ : each additional study hour increases score by about 5 points.
$b = 60$ : if 0 hours studied, predicted score is 60.

Example 5: Making Predictions

x = 4

hours, predicted score

= 5(4)+60=80

Example 6: Correlation Coefficient $r$ (Conceptual)

r ≈ 1 (strong positive)

r ≈ -1 (strong negative)

r ≈ 0 (no relationship)

The correlation coefficient

r

measures strength and direction of a linear relationship:

-1 \le r \le 1

$r \approx 1$ : strong positive correlation
$r \approx -1$ : strong negative correlation
$r \approx 0$ : little or no linear relationship

Example 7: Causation Reminder

Ice cream sales
Drowning incidents

Even with strong correlation, one variable may not cause the other. Example: Ice cream sales and drownings both rise in summer—correlated due to a third factor (temperature).

Key Takeaways

Trend lines describe relationships and allow prediction.
The slope shows rate of change; the intercept gives baseline prediction.
$r$ measures strength of linear correlation.
Correlation $\neq$ causation; always interpret context carefully.