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Analyze scatterplots, determine lines of best fit, interpret slope and intercepts in context, and make predictions using linear and nonlinear models.
Learn step-by-step with practice exercises built right in.
A scatterplot displays the relationship between two quantitative variables as points on a coordinate plane.
| Pattern | Description | Example |
|---|---|---|
| Positive |
A scatterplot shows the relationship between hours of sleep () and alertness score (). The points trend upward from left to right and cluster closely around a line. How would you describe this association?
Direction: Points trend upward → positive association
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| As increases, increases |
| Height vs. weight |
| Negative | As increases, decreases | Temperature vs. hot chocolate sales |
| None | No clear pattern | Shoe size vs. GPA |
The line of best fit is the straight line that best approximates the data.
"For each additional [unit of ], [the variable] is predicted to [increase/decrease] by [slope] [units of ]."
Example: If models the relationship between hours studied () and test score (): "For each additional hour of study, the test score is predicted to increase by 2.5 points."
"When [ variable] is 0, the predicted [ variable] is []."
Note: The -intercept may not always make practical sense (e.g., "0 hours of study" may be unrealistic).
A good model has residuals that are randomly scattered around zero. A pattern in residuals suggests the model is not a good fit.
| value | Meaning |
|---|---|
| Perfect positive linear relationship | |
| Perfect negative linear relationship | |
| No linear relationship | |
| $ | r |
| $ | r |
Remember:
"Which best describes the relationship?" → positive/negative, strong/weak, linear/nonlinear
"In context, what does the slope represent?" → rate of change per unit
Given a point and the line equation, calculate residual = actual − predicted.
Use the line equation to predict for a given value.
The point farthest from the line of best fit (largest residual).
Strength: Points cluster closely → strong association
Form: Follows a line → linear association
Answer: Strong, positive, linear association.
In context: As hours of sleep increase, alertness scores tend to increase.
A scatterplot shows the relationship between hours of sleep () and alertness score (). The points trend upward from left to right and cluster closely around a line. How would you describe this association?
Direction: Points trend upward → positive association
Strength: Points cluster closely → strong association
Form: Follows a line → linear association
Answer: Strong, positive, linear association.
In context: As hours of sleep increase, alertness scores tend to increase.
The line of best fit for a scatterplot relating years of experience () to salary in thousands () is . Interpret the slope in context.
The slope is 3.2.
Interpretation: For each additional year of experience, the predicted salary increases by $3,200 (3.2 thousand dollars).
Template: "For each 1-unit increase in [x-variable], the [y-variable] is predicted to [increase/decrease] by [slope] [units]."
Note: The -intercept of 32 means a person with 0 years of experience has a predicted salary of $32,000.
The line of best fit for a scatterplot relating years of experience () to salary in thousands () is . Interpret the slope in context.
The slope is 3.2.
Interpretation: For each additional year of experience, the predicted salary increases by $3,200 (3.2 thousand dollars).
Template: "For each 1-unit increase in [x-variable], the [y-variable] is predicted to [increase/decrease] by [slope] [units]."
Note: The -intercept of 32 means a person with 0 years of experience has a predicted salary of $32,000.
The line of best fit is . A data point has coordinates . What is the residual for this point?
Step 1: Find the predicted value at :
The line of best fit is . A data point has coordinates . What is the residual for this point?
Step 1: Find the predicted value at :
A researcher collects data and finds a correlation coefficient of between ice cream sales and drowning incidents. Can the researcher conclude that eating ice cream causes drowning?
Answer: No!
Explanation: A strong correlation () shows that ice cream sales and drowning incidents are associated — they tend to increase together. However, correlation does NOT prove causation.
What's really happening: There is a confounding variable — hot weather. When it's hot:
The heat is the common cause. Ice cream doesn't cause drowning.
SAT Rule: Only a randomized controlled experiment can establish causation. Observational studies can only show association.
A researcher collects data and finds a correlation coefficient of between ice cream sales and drowning incidents. Can the researcher conclude that eating ice cream causes drowning?
Answer: No!
Explanation: A strong correlation () shows that ice cream sales and drowning incidents are associated — they tend to increase together. However, correlation does NOT prove causation.
What's really happening: There is a confounding variable — hot weather. When it's hot:
The heat is the common cause. Ice cream doesn't cause drowning.
SAT Rule: Only a randomized controlled experiment can establish causation. Observational studies can only show association.
A line of best fit is for data where ranges from to . Which of the following predictions is most reliable?
A) Predicting when B) Predicting when C) Predicting when D) Predicting when
Key concept: Interpolation vs. Extrapolation
The data ranges from to .
A) : This is WITHIN the data range → → most reliable ✓ Beyond the range → extrapolation → less reliable ✗ Far beyond the range → extrapolation → unreliable ✗ Below the range → extrapolation → unreliable ✗
A line of best fit is for data where ranges from to . Which of the following predictions is most reliable?
A) Predicting when B) Predicting when C) Predicting when D) Predicting when
Key concept: Interpolation vs. Extrapolation
The data ranges from to .
A) : This is WITHIN the data range → → most reliable ✓ Beyond the range → extrapolation → less reliable ✗ Far beyond the range → extrapolation → unreliable ✗ Below the range → extrapolation → unreliable ✗
Step 2: Calculate the residual:
Answer: The residual is .
Interpretation: The actual value (88) is 3 units ABOVE the predicted value (85), so this point lies above the line of best fit.
Step 2: Calculate the residual:
Answer: The residual is .
Interpretation: The actual value (88) is 3 units ABOVE the predicted value (85), so this point lies above the line of best fit.
Answer: A
SAT Tip: Predictions within the data range (interpolation) are more trustworthy than predictions outside it (extrapolation). The farther outside the range, the less reliable the prediction.
Answer: A
SAT Tip: Predictions within the data range (interpolation) are more trustworthy than predictions outside it (extrapolation). The farther outside the range, the less reliable the prediction.