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

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

  1. A slope estimate is 0.8 and its standard error is 0.2. Find t for testing a population slope of zero.

    Answer: ______________

  2. For a simple regression fit, Sxx = 20 and Sxy = 30. Find the least-squares slope estimate.

    Answer: ______________

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

    Answer: ______________

  4. In the population model y = a + bx + e, what does e represent?

    • The mean response a + bx
    • The random departure y - (a + bx)
    • The slope estimated from the sample

  5. A valid two-sided slope test gives p = 0.3. At the 5% significance level, this establishes that the population slope is exactly zero.

    Circle one:   True   False

  6. The normal-error assumption for exact regression t inference requires the pooled response values to follow a normal distribution, regardless of x.

    Circle one:   True   False

  7. A pattern-free residual plot against x can leave dependence between successive errors undetected.

    Circle one:   True   False

  8. A residual plot widens into a fan as x increases. Which assumption does this call into question?

    • Errors from one observation are independent of those from another
    • Errors follow a normal distribution at each x
    • Errors have constant variance across all values of x

  9. Observed x values range from 2 to 8. You use the fitted line to predict at x = 20. What is the concern?

    • Interpolation must fail inside the data range
    • The relationship may change beyond the observed range
    • The predicted response must equal an observed response

  10. A valid two-sided 95% t interval for the slope is 0.4 to 1.1. What follows for the matching two-sided test of a zero slope at 5%?

    • Reject a zero slope in the matching two-sided 5% test
    • Accept a zero slope in the matching two-sided 5% test
    • Treat the slope estimate as the exact population slope
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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. 4 · For a zero-slope null, t = b1/SE(b1) = 0.8/0.2 = 4.
  2. 1.5 · The slope estimate is Sxy/Sxx = 30/20 = 1.5.
  3. 14 · Plug in: 2 plus 3 times 4 is 14.
  4. The random departure y - (a + bx) · The model's mean response at x is a + bx. The error e is the random departure of y from that mean.
  5. False · Since 0.3 > 0.05, do not reject a zero slope. This does not establish that the population slope is zero.
  6. False · Normality concerns errors at each x, or responses around their conditional mean. The pooled response distribution need not be normal.
  7. True · A plot against x alone cannot establish independence. Dependence related to observation order may remain undetected.
  8. Errors have constant variance across all values of x · The widening fan suggests greater scatter at larger x, so constant error variance is doubtful.
  9. The relationship may change beyond the observed range · Since 20 is outside 2 to 8, this is extrapolation. The observed data do not check the relationship there.
  10. Reject a zero slope in the matching two-sided 5% test · Zero is outside the interval, so the matching two-sided test rejects at 5%. This is evidence against zero, not proof.
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