Linear Regression by Least Squares · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Drawing the line that misses least

Computing · Machine Learning · ages 19-20
Name ______________________   Date ____________

  1. What is a residual?

    • The vertical gap between a point and the line
    • The slope of the line
    • The number of data points

  2. The least squares line always passes through the point of the mean of x and the mean of y.

    Circle one:   True   False

  3. A fit gives s equals 52 plus 6.4 h for study hours and exam score. What does 6.4 mean?

    • The line crosses the axis at 6.4
    • Each extra hour links to about 6.4 more points
    • Every pupil gains exactly 6.4 points

  4. A least squares fit for study hours h and exam score s gives the line s = 52 + 6.4h. Which sentence correctly says what the slope means?

    • Each extra hour of study is linked to about 6.4 more points on the exam.
    • Everyone who studies one more hour gains exactly 6.4 points.
    • The line crosses the score axis at 6.4.
    • The score goes up 52 points for every hour studied.

  5. You fit a straight line to some data and then plot the residuals. The plot bends in a clear U shape instead of looking like random scatter. What does that pattern tell you?

    • A straight line is missing the curved shape in the data.
    • The fit is perfect and nothing needs checking.
    • The slope should be forced to zero.
    • Squaring the residuals was a mistake.

  6. Four friends recorded practice hours (x) and points scored (y): (1, 2), (2, 3), (3, 5), (4, 4). Using least squares, what is the slope of the best fit line? Give a decimal.

    Answer: ______________

  7. A two feature model predicts price from size x1 and age x2. What does its size coefficient mean?

    • The effect of size while age stays fixed
    • The price when the house vanishes
    • The total of size plus age

  8. The least squares regression line always passes through the point (mean of x, mean of y).

    Circle one:   True   False

  9. Residuals fan wider as x grows. What should you do?

    • Force the slope to zero
    • Trust the line fully
    • Rethink the model before trusting predictions

  10. A friend reads a slope of 12 dollars per hour as proof that one more hour always pays exactly 12. What is the error?

    • Units are wrong
    • Least squares gives typical change with scatter around it
    • Intercepts beat slopes
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Answer key

For grown-ups. Fold this page away before handing over the rest.

Drawing the line that misses least W1-mt_QRRkpIrwWs-s1

  1. The vertical gap between a point and the line · Residuals are the gaps you square and add.
  2. True · The intercept math puts the mean point on the line every time.
  3. Each extra hour links to about 6.4 more points · The slope reads per unit in data units, as a tendency.
  4. Each extra hour of study is linked to about 6.4 more points on the exam. · The slope 6.4 has units of points per hour, so it describes the typical change in predicted score for one more hour of study, not a guarantee for any one student.
  5. A straight line is missing the curved shape in the data. · Residuals should look like random noise when a straight line captures the shape. A U shape means a systematic curve is left over, so a straight line is the wrong shape for this data.
  6. 0.8 · The least squares slope is the sum of cross deviations divided by the sum of squared x deviations, and here it is 4 divided by 5, or 0.8.
  7. The effect of size while age stays fixed · Each coefficient holds the other features still.
  8. True · True. The intercept is computed as ȳ minus slope times x̄, so plugging x̄ into the line returns ȳ exactly, which puts the mean point on the line every time.
  9. Rethink the model before trusting predictions · A fan is a pattern, and patterns mean the line misses something.
  10. Least squares gives typical change with scatter around it · Points scatter, so the line trends without promising each case.
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