Simple Linear Regression and Inference on the Slope · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Lines that learn from data

Mathematics · Data & Statistics · ages 21-22
Name ______________________   Date ____________

  1. For one observation the actual y is 15 and the line predicts 14. Find the residual.

    Answer: ______________

  2. A fitted line is y-hat = 2 + 3x. Predict y when x is 4.

    Answer: ______________

  3. A fitted line is y hat equals 2 plus 3x. Predict y when x is 4.

    Answer: ______________

  4. What does the error term in the model collect?

    • The slope estimate
    • Everything the line cannot explain
    • The sample size

  5. A slope test gives p equals 0.3. Ana says this proves the slope is exactly zero.

    Circle one:   True   False

  6. Lena sees a clearly curved residual plot and says a straight line fits the data well. Is Lena right?

    Circle one:   True   False

  7. A slope test gives p = 0.3. Ana says this proves the slope is exactly zero. Is Ana right?

    Circle one:   True   False

  8. A residual plot fans out as x grows. What does that pattern signal?

    • The line fits perfectly
    • The slope must equal zero
    • The scatter around the line changes with x

  9. You predict at an x far beyond the observed range. What is the danger?

    • Interpolation always fails
    • Extrapolation may fail since the line can stop describing reality
    • The residual must be zero

  10. A slope interval is 0.4 to 1.1. A friend claims the slope could be zero. What is wrong?

    • Zero sits outside the interval, so the data rule it out
    • Intervals never bound slopes
    • Zero sits inside the interval
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Answer key

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

Lines that learn from data W1-mt_W2huNTKsMj-s1

  1. 1 · Residual is observed minus predicted: 15 minus 14 is 1.
  2. 14 · Plug in: 2 plus 3 times 4 is 14.
  3. 14 · Plug in: 2 plus 3 times 4 is 14.
  4. Everything the line cannot explain · The error term gathers all the influences outside the straight line part.
  5. False · A large p value only means the data do not rule zero out; it never proves equality.
  6. False · Lena is wrong. Curvature in the residuals means the straight line misses structure, so a curve or a new term is needed.
  7. False · Ana is wrong. A large p-value only means the data do not rule out zero; it never proves the slope equals zero.
  8. The scatter around the line changes with x · A widening fan means uneven spread, which breaks the steady spread condition.
  9. Extrapolation may fail since the line can stop describing reality · Far outside the data the true relationship can bend away from the fitted line.
  10. Zero sits outside the interval, so the data rule it out · The whole interval lies above zero, so zero is excluded at that confidence level.
Worksheet · LightMySky