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Exam Questions Harvard Statistics

Harvard. Final Examination for topics in statistical theory. E.B. Wilson, 1938

 

Most course final examination questions at Harvard were officially printed, but for a variety of reasons some course final examinations questions were only duplicated using carbon paper or perhaps they were written on the black-board at the time of the examination. The Harvard archives collection of final examinations has boxes of the bound printed copies of final examinations and folders with the carbon or mimeographed copies of examinations for (some) of the other courses. We see from the enrollment data that there were only four graduate students enrolled in E. B. Wilson’s course on “Topics in Statistical Theory” so logistically it would have been no big deal for a secretary to type enough copies using carbon paper.  It appears to be the original copy of his examination questions for 1937-38 that I have transcribed for this post.

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Course Announcement

Economics 122b 2hf. (formerly 32b). Topics in Statistical Theory.
Half-course (second half-year). Tu., Th., 3 to 4.30. Professor E. B. Wilson.

 

Source:  Announcement of the Courses of Instruction Offered by the Faculty of Arts and Sciences during 1937-38 (First Edition). Official Register of Harvard University, Vol. 34, No. 5 (March 1, 1937), p. 149.

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Course Enrollment

[Economics] 122b 2hf. (formerly 32b). Professor E. B. Wilson.—Topics in Statistical Theory.

Total 4: 4 Graduates.

 

Source:  Harvard University. Report of the President of Harvard College for 1937-38, p. 86.

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Final Examination in Economics  122b2.
Wednesday June 8, 1938 at 2 P.M. in Amerson A

Students may individually use slide rules, logarithmic tables, books, notes, and their solutions of problems at their discretion.

  1. Why did Macaulay feel that he must include a twelve-months moving average as one process in his summation formula for smoothing monthly interest rates? Why did this make it advisable that he include another summation over an even number of elements?
  2. Prove that a running mean of a specified number of elements eliminates more of the random fluctuation from a time series than any other mean of the same number of elements.
  3. What does a 13-term running mean do to a sine curve with period of 40 (using the interval between terms as a unit)?
  4. Define a random series. Derive the relations which exist between the standard deviation of the random elements and the standard deviations of the 1st, 2nd, and 3rddifferences of those elements?
  5. What is the actuarial criterion of smoothness? What is the difficulty of using maximum smoothness as a criterion for smoothing a series?
  6. Assuming the expansion of \frac{x}{{{e}^{x}}-1} in a series with Bernoulli numbers as coefficients, derive formally the (asymptotic) expansion for \log n! or for \log \Gamma \left( n \right).
  7. What is the criterion of fidelity which is ordinarily imposed in graduating time series? Why is Spencer’s 21-term formula, which satisfies this criterion, used in place of that best 21-term eliminator or of that 21-term best smoother which satisfy this same criterion?
  8. Prove the ordinary formula for the standard deviation of a median.
  9. State R.A.Fisher’s method of finding the values of the constants (or parameters) of a frequency function of assumed type from the elements of a given sample. State also his rule for the standard deviations of the constants.
  10. Given any analytical frequency function with close contact at the ends, derive therefrom the expansion of another frequency function of the same mean and standard deviation good to fourth moments inclusive.
  11. Give a brief sketch of the symbolic method of treating advancing and retreating and central differences.
  12. Give an illustration of (a) a universe with median but no mean (b) another universe in which the median is a better criterion of center than the mean, (c) a universe in which the mean is a better criterion of center than the median, (d) a universe in which the average of the least and greatest elements of a sample is a better criterion of center than either the mean or the median. What do you mean by “a better criterion of center”?

 

Source:  Harvard University Archives. Final Examinations 1853-2001 (HUC 7000.28). Box 3, Folder “Final examinations, 1937-1938”.

Image Source:  Faculty portrait of of E. B. Wilson in Harvard Album 1939.