The \" squareroot of the sum of the squares\" uncertainty propagation equations

Question

The " squareroot of the sum of the squares" uncertainty propagation equations work for uncorrelated, random errors. When the uncertainties to be combined are independent of each other, the random errors will on average tend to partially offset each other, and the combined uncertainty will be less than the simple sum of uncertainties. a. In question 2c, you estimated the uncertainty in the volume of a spherical droplet from the uncertainty in its radius. Explain why you would not get the correct answer if you used the rule for propagation of uncertainty in multiplication: V = 4/3 pi(25.0 plusminus 1.5 mu m)(25.0 plusminus 1.5 mu m) (25.0 plusminus 1.5 mu m) b. For least-squares linear regression, explain why the best-fit slope and intercept are correlated with each other. (This is why, when estimating the uncertainty in an unknown concentration from a calibration curve, we have to use a more complicated equation that corrects for the "covariance" between these parameters.)

Explanation / Answer

(a)

This is because when quantities containing error are multiplied, the error cannot simply be added up as the propagation rule says. This is because when multiplication is done, it is the relative error which gets propagated, not the absolute error.

(b)

This is because the estimates of slope and intercept( termed as b0 and b1) are not independent. They are independent only when the regressor is centred on its sample mean. If the regressor is not centred, then the covariance will not be zero.

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