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Flashcards · AP® Statistics · Unit 5

Unit 5: Regression Analysis flashcards

37 cards on topics 5.1 to 5.5. Pick a topic or study the whole unit.

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5.1 Graphical Representations Between Two Quantitative Variables

What are bivariate quantitative data?
Ordered pairs (x, y) of two quantitative variables measured on the same individuals, shown as a scatterplot with one dot per individual. Do not join the dots.
Which variable goes on each axis of a scatterplot?
The explanatory variable (used to explain or predict) goes on the x-axis; the response variable (being predicted) on the y-axis. To predict soup sales from temperature, sales go on the y-axis. "Explanatory" does not mean "causes".
What four features describe a scatterplot?
Form (linear or non-linear), direction (positive or negative), strength (strong, moderate, weak) and unusual features (unusual points, clusters, gaps), all in context.
How do you describe the direction of an association?
Positive: y tends to increase as x increases. Negative: y tends to decrease. Say "tends to", not "always".
Is a steep scatterplot pattern always strong?
No. Strength is how closely points follow the pattern; steepness is a different idea. Points very close to a gently falling line show a strong, negative, linear association.
What are the rules for scatterplot axes?
Label both axes with variable name and units and use equal steps. An axis need not start at zero: responses from 12 to 57 on an axis from 0 to 60 in steps of 10 is fine.
How do you write a model description of a scatterplot?
"There is a strong, negative, non-linear association between age and price of these cars: older cars tend to cost less, and there are no unusual points."

5.2 Correlation

What does the correlation coefficient r measure?
The direction and strength of a linear association between two quantitative variables. It is found with technology and interpreted in context, and it always lies between −1 and 1.
How do you compare the strength of two correlations?
By |r|, not the sign. r = −0.72 is a stronger linear association than r = 0.45. A rough working guide: |r| above 0.8 strong, 0.5 to 0.8 moderate, below 0.5 weak.
Does r = 0 mean there is no relationship?
No. It means no linear association. A perfect U-shape can have r = 0 and a strong non-linear association.
Does r near 1 mean the relationship is linear?
No. A curve can give r = 0.98, so always check the scatterplot (and residual plot) first.
How do units and swapping variables affect r?
r has no units, so converting heights from cm to inches leaves it unchanged (0.66 stays 0.66). Swapping x and y also leaves r unchanged.
Is r resistant to unusual points?
No. One unusual point can change r a lot.
Can a strong correlation show cause and effect?
Not from an observational study; name a possible lurking variable instead. Towns with more hospitals have more deaths (r = 0.88), but population size is a likely lurking variable. A well-designed experiment can show cause.
How do r and slope differ?
r measures how closely points follow a line, not how steep the line is.

5.3 Linear Regression Models

What is a linear regression model?
ŷ = a + bx, using the explanatory variable x to predict the response y when the scatterplot is linear. ŷ is a predicted value; observed values scatter around it.
What do a and b mean in ŷ = a + bx?
a is the y-intercept, ŷ when x = 0. b is the slope, the change in ŷ for each increase of 1 in x; b is the number multiplied by x.
How do you make a prediction from a regression line?
Substitute x, multiply by b, add a, and round at the end with units in context. ŷ = 4.5 + 2.2x at x = 10 gives 26.5.
What is the difference between interpolation and extrapolation?
Interpolation: x inside the interval from the smallest to the largest x in the data, end points included. Extrapolation: x outside it, less reliable the further out.
Why is extrapolation risky?
The linear pattern may not continue. ŷ = 120 − 3.5x fitted for x from 2 to 15 gives ŷ = −20 at x = 40, which may be impossible.
Why must a regression model be written with a hat?
ŷ shows predicted values. Writing y = a + bx says every point is on the line.
Can you predict x from a value of y with the same line?
No. The model predicts y from x; do not substitute a y-value for x.

5.4 Residuals

What is a residual?
The prediction error for one point: residual = y − ŷ = observed − predicted, measured vertically, in the units of the response. Observed 52 g, predicted 47.5 g: residual +4.5 g.
What do positive and negative residuals mean?
Positive: the point is above the line and the model underpredicts. Negative: below the line and the model overpredicts. A residual of 0 means the point is on the line.
How do you find the observed value from a residual?
y = ŷ + residual. ŷ = 30 minutes and residual −2.4 minutes give an observed 27.6 minutes.
How do you read a residual plot?
Residuals (vertical) against x or ŷ, with a line at 0. Random scatter: linear form, so a linear model is appropriate. Curvature (U or arch): non-linear form, so a linear model is not the best.
Which point has the worst prediction?
The one whose residual is furthest from 0, of either sign, not the most negative.
Does a high r mean a linear model is appropriate?
No. Curved data can have r close to −1 or 1, so always check the residual plot. Random scatter shows linear form, not that residuals are small.
How do you interpret a residual in context?
"The residual of −$1.67 means the actual fare for the 6.8 km ride was $1.67 less than the model predicted, so the model overpredicted it."

5.5 Least-Squares Regression

What is the least-squares regression line?
The one line ŷ = a + bx with the smallest sum of squared residuals, Σ(y − ŷ)². It always passes through (x̄, ȳ) and its residuals add to 0. Find a, b, r and r² with technology.
How do you interpret the slope of a regression line?
"For each additional [unit of x], the predicted [y] increases (or decreases) by [b units]." Always include "predicted".
How do you interpret the y-intercept, and when should you not?
"When x is 0, the predicted y is a." Say it has no sensible meaning when x = 0 is outside the data or the prediction is impossible, as with ŷ = −40 + 0.9x giving −40 g.
What does r² mean?
The coefficient of determination: about r² × 100% of the variation in y is explained by the linear relationship with x. It is not the percentage of points on the line or how often the line is right.
How do you find r from r²?
r = ±√r², with the same sign as the slope. r² = 0.49 with a negative slope gives r = −0.7.
How do you find the intercept from the means and slope?
The line passes through (x̄, ȳ), so a = ȳ − b·x̄. Slope 2, x̄ = 12, ȳ = 50: a = 26.
How do you read regression output from technology?
"Constant" is a, the row named after the x variable is b, and "R-sq" is r². Enter the explanatory variable as x: swapping x and y changes the line.
Are the slope and intercept of a regression line parameters?
No. a and b are statistics from one sample; a different sample would give different values. A regression line also cannot show cause and effect.

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