Fitting a Model to Data: Least Squares, Chi-Square and Goodness of Fit · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.
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Fitting lines and judging the fit

Science · Scientific Inquiry · ages 22-23
Name ______________________   Date ____________

  1. A line predicts distance y in metres from time x in seconds: y-hat = a + b x. What units must slope b have?

    • Seconds per metre
    • Metres times seconds
    • Metres per second
    • Metres per second squared

  2. For one data point, the observed value is y and the model predicts y-hat. What is the residual?

    • y minus y-hat
    • y-hat minus x
    • The largest value in the data set

  3. For a linear fit under the lesson's normal-error and known-uncertainty assumptions, reduced chi-square near 1 is consistent with the stated error scale, but does not prove the model correct.

    Circle one:   True   False

  4. A scientist fits a model using ordinary, unweighted least squares. How does the method choose its free parameters?

    • It forces the model through every observation, whatever the residuals
    • It chooses the parameters by appearance without calculating residuals
    • It minimises the sum of squared residuals

  5. With n > 2 observations, at least two distinct exact x values and SSE > 0, an unweighted line can estimate coefficient standard errors without reported point uncertainties, assuming a correct linear mean and independent errors of equal positive variance.

    Circle one:   True   False

  6. A weighted line meets the lesson's exact-x and known-variance error assumptions. Its slope is b = 1.5 m/s and Cxx = 16 s²/m². Using SE(b) = 1/sqrt(Cxx), what standard error in m/s should accompany the slope?

    Answer: ______________

  7. A full-rank linear model fits 3 coefficients to 20 measurements. Its chi-square is 34. What is its reduced chi-square?

    • 2, using 34 divided by (20 - 3)
    • 1.7, dividing by all measurements without subtracting fitted coefficients
    • 11.33, dividing only by the number of fitted coefficients
    • 17, reporting the residual degrees of freedom as the statistic

  8. A weighted line fit has x-bar = 2, y-bar = 9 and slope b = 3. What is its intercept a?

    • 15, from adding the slope contribution to y-bar
    • 3, from calculating 9 - 3 × 2
    • 6, from treating the slope contribution as the intercept
    • 4.5, from dividing y-bar by x-bar

  9. A full-rank linear fit has 100 residual degrees of freedom and reduced chi-square 0.05. Which possibility is reasonable to investigate?

    • The small value proves that the fitted equation describes every possible measurement exactly
    • The small value proves that every observation has zero random error and no systematic effects
    • Uncertainties may be too large, or independence may fail

  10. Three readings with equal uncertainties are 5.0, 6.0 and 10.0. Fit the constant model y-hat = a by least squares. Type the best value of a.

    Answer: ______________

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Answer key

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

Fitting lines and judging the fit W1-mt_NN0WYlp0md-s1

  1. Metres per second · The product b x must have distance units. Dividing metres by seconds gives the slope units.
  2. y minus y-hat · The residual is the observed value minus the predicted value.
  3. True · Reduced chi-square has mean 1 under those assumptions. A value near that mean is a consistency check, not proof.
  4. It minimises the sum of squared residuals · Ordinary least squares chooses parameters that make the sum of squared residuals as small as possible.
  5. True · Distinct x values identify both coefficients. With n > 2, estimate s² = SSE/(n - 2), then scale the coefficient standard-error formulas by s. Positive SSE gives a positive estimated scale.
  6. 0.25 · SE(b) = 1/sqrt(16) = 0.25 m/s. Report slope 1.5 m/s with standard error 0.25 m/s.
  7. 2, using 34 divided by (20 - 3) · There are 20 - 3 = 17 residual degrees of freedom. Reduced chi-square is 34/17 = 2.
  8. 3, from calculating 9 - 3 × 2 · The intercept is a = y-bar - b x-bar = 9 - 3 × 2 = 3.
  9. Uncertainties may be too large, or independence may fail · With 100 residual degrees of freedom, this very small value warrants checking the calculation and error assumptions. It does not prove exact measurements or model truth.
  10. 7 · With equal weights, the best constant is the mean: (5 + 6 + 10)/3 = 7.
Worksheet · LightMySky