Linear Relationships
What does it mean if two variables are positively correlated?
When one variable increases, the other also increases.
In a linear model, what does the slope represent in words?
The rate of change of the dependent variable per unit change in the independent variable.
What is a residual?
The difference between the actual value and the predicted value from the linear model.
What does a correlation coefficient close to 1 indicate about the relationship between two variables?
A very strong positive linear relationship.
What is a real-world example where a linear model might not be appropriate?
Predicting human height based on age for all ages (because growth slows down). [any real-world example that is not linear is acceptable]
How do we determine if a linear model is appropriate for a given set of data?
By looking for a linear trend in a scatter plot and checking for outliers or other patterns.
If a linear model has a slope of –2, what does that mean in context?
As the independent variable increases by 1 unit, the dependent variable decreases by 2 units.
If the residual for a data point is negative, what does this mean?
The actual value is lower than the value predicted by the model.
What does a negative correlation coefficient (e.g., –0.85) indicate about the relationship between variables?
As one variable increases, the other decreases (negative relationship).
When might residuals be used to assess the fit of a linear model?
To determine if the model overestimates or underestimates the dependent variable at certain values.
How would you describe a weak linear relationship in terms of scatter plot appearance?
The points are scattered widely around the line, showing little to no pattern.
If a linear model's y-intercept is 0, what does this mean in the context of the problem?
When the independent variable is 0, the dependent variable is also 0.
What is the difference between the actual value and the predicted value?
The residual. It shows how far off the prediction is from the real data.
How does the correlation coefficient (r) affect the reliability of predictions made by a linear model?
A higher r (closer to 1 or –1) indicates a more reliable linear model for predictions.
Why is it important to assess both the correlation coefficient and residuals when evaluating a linear model?
To determine how well the model fits the data and whether predictions are reasonable.
What would a bimodal scatter plot suggest about the relationship between two variables?
The data might be better modeled by two separate groups rather than a single linear relationship.
What do we mean by the slope-intercept form of a linear equation?
The form y=mx+b, where m is the slope and b is the y-intercept.
How would you know if a linear model is not a good fit based on residuals?
If residuals are large and spread out inconsistently, it suggests the model does not fit the data well.
What is the range of possible values for the correlation coefficient (r)?
The values range from –1 to 1, where –1 is a perfect negative relationship and 1 is a perfect positive relationship.
Why might the y-intercept in a real-world model not always have a meaningful interpretation?
Because in many cases, the x = 0 value may not make sense in the context of the data.
What can we learn about the relationship between two variables if their scatter plot is shaped like a straight line?
It suggests a strong linear relationship, which can be modeled using a linear function.
If a model has a slope of –0.5 and a y-intercept of 10, what does the graph of this model look like?
It will slope downward from left to right, starting at 10 on the y-axis and decreasing by 0.5 for each unit increase in x.
Why would you calculate residuals for specific data points in an analysis?
To assess whether the model overestimates or underestimates the actual values, and to understand the model fit better.
How does the sign of the correlation coefficient indicate the direction of the relationship?
A positive sign means the variables increase together, while a negative sign means one increases as the other decreases.
In what situations would you consider using a different model (not linear), even if the data appears linear at first glance?
When the data is non-linear at larger or smaller ranges, or when there are clear outliers that suggest a different relationship.