Self-assessment for: Detecting gross errors

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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