Notes
Scatterplots and correlation
- The explanatory variable $x$ predicts the response variable $y$. Describe a scatterplot by direction, form, strength and unusual points.
- Correlation $r$ is between $-1$ and $1$ and measures only linear association. It has no units.
Regression
- Least-squares line: $\hat{y} = a + bx$ with $\displaystyle b = r\,\frac{s_y}{s_x}$ and $a = \bar{y} - b\bar{x}$. It passes through $(\bar{x}, \bar{y})$.
- Slope: predicted change in $y$ for each 1-unit increase in $x$.
- Residual $= y - \hat{y}$. A residual plot with a pattern means a linear model is not appropriate.
- $r^2$ is the fraction of variation in $y$ explained by the linear relationship with $x$.
Cautions
- Avoid extrapolation. Watch for outliers and influential points.
- Correlation is not causation. Lurking variables can explain an association.
Topics in this unit
Each topic has its own short notes and practice.
Practice
The response variable is the one being predicted.
The sign is negative and $0.85$ is close to 1.
$-10 + 2.5(20) = -10 + 50 = 40$.
$(-0.6)^2 = 0.36$, so $36\%$ of the variation in $y$ is explained.
Residual $= y - \hat{y} = 52 - 47 = 5$.
Older children have larger feet and also read better.
Free response
Write your answer first, then open the scoring guide.
Free-response question
A teacher studies the relationship between hours studied ($x$) and exam score ($y$) for a class. The summary: $\bar{x} = 4$ hours, $s_x = 2$ hours, $\bar{y} = 70$ points, $s_y = 8$ points and $r = 0.75$.
(a) Find the slope and intercept of the least-squares regression line and write the equation.
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$\displaystyle b = 0.75 \cdot \frac{8}{2} = 3$. $a = 70 - 3(4) = 58$. The line is $\hat{y} = 58 + 3x$.
Check off the points you earned:
(b) Predict the score for a student who studied 6 hours. That student actually scored 79. Find the residual.
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$\hat{y} = 58 + 3(6) = 76$. Residual $= 79 - 76 = 3$.
Check off the points you earned:
(c) Find $r^2$ and interpret it in context.
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$r^2 = 0.75^2 = 0.5625$, about $56\%$. About $56\%$ of the variation in exam scores is explained by the linear relationship with hours studied.
Check off the points you earned: