Notes
Scatterplots
Describe direction (positive or negative), form (linear or curved), strength, and unusual features, in context. The explanatory variable goes on the $x$-axis and the response on the $y$-axis.
Correlation
The correlation $r$ measures the direction and strength of a linear relationship. $-1 \le r \le 1$.
- $r$ has no units and doesn't change if you switch $x$ and $y$ or change units.
- $r$ is not resistant: one outlier can change it a lot.
- Correlation doesn't imply causation. A lurking variable may drive both.
Least-squares regression
$\hat y = a + bx$, where $\hat y$ is the predicted response.
- Slope $\displaystyle b = r\frac{s_y}{s_x}$: "for each additional 1 [unit of $x$], the predicted [$y$] increases by $b$ [units]."
- $y$-intercept $a$: the predicted $y$ when $x = 0$ (only meaningful if $x = 0$ makes sense).
- The line always passes through $(\bar x, \bar y)$ and minimizes the sum of squared residuals.
- Residual $= y - \hat y$ (actual minus predicted). Positive means the model underpredicted.
- Residual plot: no pattern means a linear model is appropriate; a curve means it isn't.
- $r^2$: the percent of the variation in $y$ explained by the linear model with $x$.
- Standard deviation of residuals $s$: the typical prediction error.
- Extrapolation (predicting outside the range of $x$ values in the data) is unreliable.
Topics in this unit
Each topic has its own short notes and practice.
Practice
$|r|$ near 1 means strong; the sign gives the direction.
$12 + 3.5(4) = 26$.
Residual $= y - \hat y = 23 - 26$.
That's the meaning of the coefficient of determination.
A pattern in the residuals means the form isn't linear.
Correlation doesn't show causation; summer heat drives both.
80 is far outside the observed $x$ values.
$\displaystyle b = r\frac{s_y}{s_x} = 0.6 \cdot 5 = 3$.
Free response
Write your answer first, then open the scoring guide.
Free-response question
For 20 used cars, a regression of price (in thousands of dollars) on age (in years) gives $\widehat{\text{price}} = 28.4 - 1.9(\text{age})$, with $r^2 = 0.72$ and $s = 2.6$.
(a) Interpret the slope in context.
Show the answer and scoring guide
For each additional year of age, the predicted price of a used car decreases by about 1.9 thousand dollars.
Check off the points you earned:
(b) Predict the price of a 6-year-old car and find the residual for one that sold for 18.1 thousand dollars.
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$28.4 - 1.9(6) = 17.0$ thousand dollars. Residual $= 18.1 - 17.0 = 1.1$ thousand; the model underpredicted by about 1,100 dollars.
Check off the points you earned:
(c) Interpret $s = 2.6$.
Show the answer and scoring guide
The actual prices typically differ from the prices predicted by the model by about 2.6 thousand dollars.
Check off the points you earned: