Unit 5: Regression Analysis

15 terms. Say what each one means, then open the card to check.

• Explanatory and response variables

  The explanatory variable (x) is the one used to predict or explain; the response variable (y) is the outcome being predicted.

  Example. Predicting a runner's race time from weekly training miles makes miles the explanatory variable and race time the response.

• Scatterplot

  A graph of two quantitative variables measured on the same individuals, with one dot per individual. Describe it by direction, form, strength and unusual features.

  Example. Each dot shows one student's hours of sleep (x) and reaction time (y) from the same night.

• Direction, form and strength

  Direction is positive or negative; form is linear or curved; strength is how closely the points follow that form.

  Example. Points that rise in a tight straight band show a strong, positive, linear relationship.

• Correlation r

  A number from -1 to 1 that measures the direction and strength of a linear relationship. It has no units and is not changed by swapping x and y.

  Example. r = -0.92 for car age and resale price means a strong negative linear relationship: older cars tend to sell for less.

• Correlation is not causation

  A strong correlation in observational data does not show that x causes y. A lurking variable may drive both.

  Example. Ice cream sales and sunburn cases rise together, but hot sunny weather drives both.

• Influential point

  A point that, if removed, would change the slope, intercept or correlation a lot. Points with x-values far from the rest are often influential.

  Example. One point far to the right and well below the pattern drags the line down and drops r from about 0.997 to 0.13.

• Least-squares regression line

  The line $\hat{y} = a + bx$ that makes the sum of the squared residuals as small as possible. It always passes through $(\bar{x}, \bar{y})$.

  Example. For practice hours and free-throw percentage, the line $\hat{y} = 50.33 + 2.83x$ has the smallest sum of squared residuals.

• Slope b

  The predicted change in y for each 1-unit increase in x. It equals $r \frac{s_y}{s_x}$ and has the sign of r.

  Example. A slope of 14.5 in predicted rentals versus temperature means about 14.5 more predicted rentals for each extra degree.

• Intercept a

  The predicted value of y when x = 0. It is meaningful only when x = 0 makes sense and is near the observed x-values.

  Example. An intercept of -92.4 drinks at 0 degrees is not meaningful when the data run only from 62 to 96 degrees.

• Extrapolation

  Using a regression line to predict for x-values outside the range of the data. The pattern may not continue there, so such predictions are unreliable.

  Example. A growth line fit to children aged 3 to 10 predicts a 30-year-old would be 260 cm tall.

• Regression output

  A software table for a regression. The Constant row gives the intercept, the row named for x gives the slope, and S and R-sq are listed too.

  Example. Output with Constant 41.8 and Age -2.6 gives the line: predicted range = 41.8 - 2.6(age).

• Residual

  Observed y minus predicted y, $y - \hat{y}$. A positive residual means the point is above the line, so the line underpredicted it.

  Example. If the line predicts 36.6 and the observed value is 30, the residual is -6.6: the line overpredicted by 6.6.

• Residual plot

  A plot of residuals against x (or against predicted y). Random scatter around 0 supports a linear model; a curve or a fan shape signals a problem.

  Example. Residuals that are high, then low, then high again in a U shape show that a curved model would fit better.

• Coefficient of determination r-squared

  The proportion of the variation in y that is accounted for by the linear relationship with x. It is the square of r.

  Example. r-squared = 0.78 means about 78% of the variation in drinks sold is accounted for by the linear relationship with temperature.

• Standard deviation of the residuals s

  The typical size of a prediction error: about how far actual y-values fall from the line, in the units of y.

  Example. s = 31.6 drinks means the line's predictions are typically off by about 31.6 drinks.