Describing Scatterplots & Correlation
When describing a scatterplot, what are the four characteristics you must mention?
Direction, Form, Strength, and Unusual Features (Outliers).
In the equation {y} = a + bx, what does the "hat" over the y signify?
It represents the predicted value of y.
What is the formula for calculating a residual?
Actual - Predicted (y-yhat)
What is the name of the statistic r^2?
The Coefficient of Determination.
What is the term for predicting a y-value using an x-value that is far outside the range of your data?
Extrapolation.
What is the range of possible values for the correlation coefficient (r)
Between $-1$ and $1$.
Define the "Least Squares" part of the LSRL.
It is the line that minimizes the sum of the squared residuals.
If a data point is located below the regression line, is its residual positive or negative?
Negative.
If r = 0.7, what is the value of r^2?
0.49 (or 49%).
Does a high correlation (r) prove that x causes y?
No. Correlation does not imply causation.
If r = -0.95, describe the strength and direction of the relationship.
Strong and negative.
If the LSRL is {Score} = 50 + 5 Hours interpret the slope of $5$ in context.
For every 1 additional hour studied, the predicted score increases by 5 points.
What should a "good" residual plot look like if a linear model is appropriate?
Randomly scattered points with no clear pattern or curve.
Interpret r^2 = 0.85 in context.
85% of the variation in the response variable (y) is explained by the linear relationship with the explanatory variable (x).
What do we call a point that, if removed, would significantly change the slope or y-intercept of the LSRL?
An influential point.
What happens to the correlation (r) if you switch the x and y variables?
t stays exactly the same.
What is the interpretation of the y-intercept (a) in a regression line?
The predicted value of y when x is zero.
If a residual is +10, did the model overpredict or underpredict the actual value?
Underpredict (the actual value was 10 units higher than predicted).
What does the standard deviation of the residuals (s) represent?
The typical distance that the actual y-values are from the predicted y-values.
A point that is an outlier in the x-direction (far to the left or right) is said to have high what?
High leverage.
Correlation only measures the strength of what specific type of relationship?
Linear relationships.
Every LSRL must pass through which specific coordinate point?
The point of averages bar{x}, bar{y}).
If a residual plot shows a clear "U-shaped" pattern, what does that indicate?
A linear model is not appropriate (the data is likely curved).
If r^2 is 0.64 and the slope of the regression line is negative, what is the value of r?
-0.8 (You must take the negative square root because the slope is negative).
What is the difference between an outlier and an influential point in regression?
An outlier has a large residual (far from the line); an influential point specifically changes the "tilt" (slope) of the line.