Self-assessment for: Local measures of precision, and measures in one dimension

These self-assessment problems accompany our in-class discussions on Local measures of precision, and measures in one dimension.

Problem 1

In class we discussed precision vs. accuracy and, in doing so, we looked at four instances: a, b, c, and d. Then I asked you whether b or c was better. And whether b or d was better.

In both cases we answered that “it depends”. Explain why this is. Provide supporting examples for each case.

Problem 2

In your earlier courses you will have looked at a handful of so-called “standard” measures of precision.

a) Provide examples of such measures

b) Explain the meaning of the term “standard”

Problem 3

For random variable that is normally distributed:

a) What confidence level corresponds to 1\sigma?

b) What confidence level corresponds to 2\sigma?

c) What confidence level corresponds to 3\sigma?

d) How many \sigma‘s correspond to a 95% confidence level?

e) How many \sigma‘s correspond to a 50% confidence level?

Problem 4

Sketch each of the above cases and include appropriately shaded areas and labeling of the key values and probabilities

Problem 5

What is the “multiplier”, \alpha, in each of the above cases? Describe in words the meaning of this multiplier and why we need it.

Problem 6

Imagine you do a leveling survey and a subsequent adjustment to estimate the heights of three points, \mathbf{x}=\begin{bmatrix}h_1 & h_2 & h_3\end{bmatrix}^T.

And imagine that your least squares estimate yields the following variance-covariance matrix of the adjusted unknowns in units of mm^2:

    \begin{equation*} \mathbf{C}_{\hat{\mathbf{x}}} = \begin{bmatrix} 0.49 & 0.30 & 0.18 \\ 0.30 & 0.64 & 0.30 \\ 0.18 & 0.30 & 0.36 \end{bmatrix} \end{equation*}

a) What is the Linear Error Probable (LEP) for each point?

b) What is the standard deviation for each?

c) Assuming a normal distribution, which of the estimated heights would be acceptable if the 3-\sigma error limit in height was 2.00 mm?

Problem 7

Imagine that your least squares estimate of the coordinates of 15 points yields a variance-covariance matrix of the adjusted unknowns. A subset of that matrix is given below for just the heights of points 4 and 9 in units of mm^2:

    \begin{equation*} \mathbf{C}_{\hat{\mathbf{x}}} = \begin{bmatrix} 3 & 3 \\ 3 & 5 \end{bmatrix} \end{equation*}

a) Provide a measure of uncertainty in height for point 4 at the 95.4% confidence level

b) Provide a measure of uncertainty in height for point 9 at the 95.4% confidence level

c) What is the meaning of the 95.4% confidence level?

d) Provide a measure of the relative uncertainty for the heights of points 4 and 9 at the 95.4% confidence level

e) Is a measure of uncertainty or relative uncertainty more meaningful? Explain.


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