Regression Analysis

Five topics on the relationship between two quantitative variables, and on the interpretation that earns the points. Scatterplots described by form, direction, strength and unusual points, correlation and the four things r is routinely asked to do that it cannot, regression models whose slope is a predicted average change and not a promise about an individual, residuals and the plot that reveals a curve the scatterplot hid, and least squares: which distances are minimized, and what r-squared actually measures.

Exam weight 10-20%5 topics

Topics

5.1

Graphical Representations Between Two Quantitative Variables

Correlation

Linear Regression Models

Residuals

Least-Squares Regression

Lesson

Diagnostic

after diagnostic

Key forms For every problem in this unit

Complete description

form, direction, strength, unusual features, IN CONTEXT with variable names

Explanatory vs response

x predicts y. Swapping them changes the regression line, though not r

Correlation r

between −1 and 1. Strength is |r|, so r = −0.9 is STRONGER than r = 0.3

r is unitless

changing units, or swapping x and y, leaves r unchanged

r measures LINEAR strength only

a perfect parabola can have r near 0. Always look at the plot

Correlation is not causation

only a well-designed experiment with random ASSIGNMENT buys a causal claim

Influential points

a single far-out x can swing the slope and r. Report its effect, do not silently delete it

Fitted model

ŷ = a + bx, where ŷ is a PREDICTED value, not an observed one

Slope, interpreted

the predicted change in y, ON AVERAGE, for each one-unit increase in x, with units and context

Intercept, interpreted

the predicted y when x = 0. Often meaningless when x = 0 is outside the data

Slope from r

b = r · (s_y / s_x)

The line passes through

(x̄, ȳ), always

Residual

observed − predicted. POSITIVE means the model UNDER-predicted and the point sits above the line

Residual plot

random scatter supports a linear model; a curve or a fan says the model is wrong, whatever r says

Least squares

minimizes the sum of SQUARED VERTICAL residuals. Not perpendicular distances, not absolute errors

r-squared

the percent of the VARIATION in y accounted for by the model. Not the percent of points on the line

Extrapolation

predicting outside the observed x range is unsupported by the data, however good the fit looks

Unit 5 tools

Challenge bank

1 / 60

60 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

Question

Tap card to reveal explanation

Worked solution

Tap card to return to question

Cumulative assessment

Test the unit.

Twenty mixed items drawn from across all 5 topics, with guaranteed misconception-code coverage. Identifies which misconceptions still bite when you cannot see which topic the question came from.

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Check what stuck.

Units 1 through 5, drawn evenly so earlier units get the same share as this one. Twenty questions or a full 42-question section, your choice. Even coverage means this is a retention check rather than a score estimate.

20 or 42questions

55topics

65codes covered

Begin assessment →