Chicago. Ph.D. qualifying exam in statistics. 1932

In his memo of February 1985 (Columbia University, A. G. Hart papers: Box 60, Folder “Sec I Notes on teaching materials, Learning”) Albert G. Hart wrote “I ducked the qualifying exam in statistics (in which for that date I was very well trained) because I disapproved of the focus of previous exams upon minor technicalities—hence I exploited the loophole which made ‘financial organization’ a separate field even though in principle the ‘theory’ exam included monetary economics.” The previous three postings give the examination questions for theory, economic history and financial organization (i.e. money and banking) for the qualifying exams Hart did take. I presume the exam of this posting is one he examined and then decided to duck statistics.

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[Handwritten note: University of Chicago (H Schultz)]

STATISTICS
Written Examination for the Ph.D.
Spring Quarter, 1932

Time – 3 1/2 hours

Answer seven questions: one question in Part I and two questions in each of the other parts.

PART I. Time Series

1. Discuss the possibility of applying the theory of probability or of sampling to the study of the statistical characteristics of time series.
2. Explain the factors that have to be taken into consideration in determining the best trend of a time series. What analyses can be made of a time series from which the trend and seasonal variation have been removed.
3. Discuss the advantages and limitations of the elimination of seasonals (a) by subtracting, (b) by dividing.

PART II. Index Numbers

1. Discuss the problem of assigning a precise and unambiguous meaning to a change in the price level (or to a change in some specified section of the price level, e.g., the wholesale price level of metals), touching on the contributions of Edgeworth, Fisher, Divisia, Keynes, and Bortkevitch.
2. If you were attempting to construct a 15 commodity wholesale price index which would precede the general B.L.S. wholesale price index by at least two months as consistently as possible (a) how would you select your commodities, (b) how would you wait them in the index?
3. Explain fully:

(a) Does Fisher’s ideal Index measure precisely and unambiguously the change in price level from one period to another of the commodities included in the index?
(b) What significance would you attach to the Factor Reversal test in the selection of the formula for price index?
(c) What significance would you attach to the Time Reversal test in the selection of a formula for a price index?

PART III. Correlation

1. Let

x₁ = annual per capita cigarette consumption

x₂ = deflated average annual wholesale price of cigarettes

x₃ = deflated annual expenditure on advertising

x₄ = time in years

R_1.234 = .998 for the period 1922-1929 inclusive

r₁₄= .95

(a)  What meaning would you attach to R_1.234?
(b) How reliable would you consider forecasts of x₁  for subsequent years based on the regression of x₁ on x₂ , x₃ , and x₄ ?
(c) Adjust R_1.234  for loss of degrees of freedom. Explain this adjustment.
(d) Calculate R_{1´.2´3´4´} in which the 1´, 2´, and 3´refer to the deviations from linear trends of the variables 1, 2 and 3.

2.  Prove and explain the following relations:     (The B’s are Greek Betas.)

(a)  R²_1.23 = B_12.3 r₁₂  + B_13.2 r₁₃

(b)  R²_1.23  = B²_12.3 + B²_13.2 + 2B_12.3 B_13.2  r₂₃

What meaning can be given to the Br’s in this connection when the equation of regression is of the type

x₁ = a + bx₂ + ct + dt² where t stands for time?

3.  Critically appraise the attempts that have been made to apply the method of multiple correlation to one of the following:

(a) Statistical studies of demand
(b) Statistical studies of supply
(c) Any field selected by yourself.

PART IV. Probability and Sampling

1. Indicate the best procedures and tables to use in determining the reliability of the following constants, when the number of observations from which they have been derived is small (i.e., less than 50):

(a)  the mean
(b)  the standard deviation
(c)  the simple coefficient of correlation
(d)  the multiple coefficient of correlation
(e)  the coefficients of progression in a multiple correlation equation
(f)  the agreement of a hypothesis with observation
(g)  the presence or absence of dependence

2. In a straw vote 200,000 ballots are sent out. 100,000 are returned and of the 60,000 or marked in favor of the proposition submitted.

(a) What can you say about the reliability of this vote?
(b) If the original mailing had been increased to 800,001 increase in reliability would have been secured in the returns?
(c) List the types of errors to which straw votes are subject.

3.   189 cases were treated with tetanus serum and 80 of them were cured. 199 cases were not treated with tetanus serum and only 42 of them were cured. What is the probability that the serum has had no effect, the difference in recoveries being due to fluctuations in sampling? (Outline your solution.)

4. A factory produces a certain screw which is collected at the machine inboxes of 1200 each. Long experience has shown that the proportion of boxes which contain various percentages of bad screws is as follows:
Per Cent of Bad Screws in Box

Per Cent of Bad Screws in Box	Proportion of Boxes Observed to Contain this Percentage of Bad Screws
0	0.780
1	0.170
2	0.034
3	0.009
4	0.005
5	0.002
6	0.000

 

The manufacturing standard is to consider any box which contains 2% or less of bad screws is satisfactory. The normal inspection consists in the examination of 50 screws out of each box. In particular box showed six bad screws under normal inspection. What is the probability that the manufacturing standard has not been maintained in the production of this box (i.e., that the box contains more than 2% defective screens)?

N. B. – Outline your solution giving formulas, indicating required tables, etc., But do not carry out the actual computations.

Source: Columbia University Libraries, Manuscript Collections. Albert Gailord Hart Collection. Box 60; Folder “Exams: Chi[cago] Qualifying”.

Image Source: Detail from the Social Science Research Building. University of Chicago Photographic Archive, apf2-07448, Special Collections Research Center, University of Chicago Library.