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

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

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

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

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

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: ______________

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

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

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

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: ______________