Self-assessment for: Detecting gross errors

Mini course: Pre-adjustment screening for gross errors Detecting gross errors Self-assessment for: Detecting gross errors

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These self-assessment problems accompany the mini course Detecting gross errors.

Problem 1

Recall the closed leveling loop $A$ , $B$ , $C$ , $D$ , $E$ that we saw in the notes.

And recall that the variance of the measurements there was known to be 0.0025 $m^2$ , and that the errors in the measurements were equal and uncorrelated to each other.

And imagine that the measurements provided here were taken.

Using a confidence level of 95%, tell me whether you suspect a blunder in these measurements.

A good answer will:

Show your work
Sketch the situation
State the null hypothesis
Calculate the correct test statistic
Carry out the test
Draw a conclusion

leg	measurement (m)	variance (m)
$AB$	$1.4497$	$0.0025$
$BC$	$5.2046$	$0.0025$
$CD$	$-6.6709$	$0.0025$
$DE$	$-2.8610$	$0.0025$
$EA$	$3.1241$	$0.0025$

Problem 2

Now assume that after receiving the results from Problem 1 you send your crew chief back to review the log book. Imagine that he finds a recording error and returns these new measurements to you.

a) What now? Repeat the same process.

b) Does the outcome in a) mean there are no blunders in the data? Why or why not?

leg	measurement (m)	variance (m)
$AB$	$1.4497$	$0.0025$
$BC$	$5.2046$	$0.0025$
$CD$	$-6.6709$	$0.0025$
$DE$	$-2.9510$	$0.0025$
$EA$	$3.1241$	$0.0025$

Problem 3

You’re still nervous about the survey described above. So you send someone else out to measure leg $DE$ a bunch of times. The resulting measurement vector is as follows (all in metres):

$\begin{equation*} \mathbf{l}_{measured}= \begin{bmatrix} -3.0022 \\ -2.9094 \\ -2.9586 \\ -2.9949 \\ -2.7477 \\ -2.9968 \\ -3.0579 \\ -3.0602 \\ -2.9992 \end{bmatrix} \end{equation*}$

a) What is the sample mean?

b) What is the sample standard deviation?

c) What is the standard error of the sample mean?

d) Set up and carry out a test to detect gross errors. Use a confidence level of 95% and the known measurement variance given in Problem 1. A good answer will:

Show your work
State the null hypothesis
Indicate what distribution is to be used
Calculate and provide the test statistics
Carry out the test
Draw a conclusion

e) If you identified a possible blunder in d) then remove it and repeat the process.

f) What now is your best estimate of the height difference on leg $DE$ ?

g) What would be your estimate of its precision?

Problem 4

Would the answer in Problem 3 d) be any different if you didn’t know the variance of the measurements and had to use the sample variance? I want you to repeat the whole thing here to find out.

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