DS350 – QUANTITATIVE METHODS FOR BUSINESS DECISIONS
SPRING SEMESTER 2010
HOMEWORK #10
This data analysis assignment is worth 30 points toward your homework/quiz grade. It is
due at the beginning of class on Wednesday, April 28, the last day of class. Late papers are not
accepted. You may get assistance from others in analyzing the data. However, the final writeup
must be your own, in your own words – to demonstrate your understanding of the analysis and
your ability to communicate technical material.
PROBLEM 1:
“Benford’s Law” (first presented in 1938 in the Proceedings of the American
Philosophical Society, vol. 78, pp. 551-572) is a mathematical law describing the behavior of
“anomalous numbers” – that is, “those outlaw numbers that are without known relationship
rather than those than individually follow an orderly course.” The question for this data analysis
assignment is: Do stock market returns follow Benford’s Law?
Benford’s Law relates to the first non-zero digit in any decimal number. (Thus, for
example, the number -.000846 has 8 as its first non-zero digit; the number 42 has 4 as its first
non-zero digit.) According to Benford’s Law, the probability that a random decimal begins with
the digit d is given by the formula
 d  1
Pr( d )  log 

 d 
(for d = 1, 2, 3, …, 9).
So, for example, the probability that the first digit is a 4 is log(5/4) = .0969. (Use base-ten
logarithms in the formula.)
The data file DJIA.XLS is on my website. (Home page http://www2.stetson.edu/~jrasp/ .
Select the “Data” link.) It contains closing values of the Dow Jones Industrial Average for the
past ten years.
Begin the assignment by computing the daily return for each data point. (The end of this
handout gives information about that.) Then read off the first non-zero digit in that return. Upon
a few occasions the market was unchanged from one day to the next. Drop these days from the
analysis.
NOTE: Excel doesn’t directly give you the first non-zero digit in a number; you’re going
to need to trick it slightly. First, multiply the number by 10,000,000 or so – just to make sure the
number is greater than one, and hence the first nonzero digit is also the first digit. Second, take
the absolute value, to get rid of the negative sign. Then read off the leftmost first character in the
string. Thus, if the value you want to get the first nonzero digit of is in cell A1, the function
=LEFT(ABS(10000000*A1),1)
will give you what you want.
Once you’ve got your first digits: analyze them to determine whether stock market
returns obey Benford’s Law for “anomalous numbers.”
PROBLEM 2:
At the height of the Vietnam War, the U.S. government conducted a “lottery” to
determine the order in which men would be drafted into the military. Men were eligible for the
draft the year they turned eighteen years old. Draft order was determined by your birthday, with
the government “randomly” (they claimed) sequencing the days. Thus, for example, in 1970 the
“winner” of the lottery was September 14 – men born on that day in 1952 were the first to be
drafted in 1970. Second in line were those with April 24 birthdays, and so on through #366 in
line, June 8. The exact number of draftees needed was not known at the beginning of the year.
Men were advised that those whose birthday was among the first third chosen (#1 through #122)
almost certainly would be drafted. Those whose birthday was among the last third chosen (#245#366) almost certainly would not be drafted. Those with draft numbers near the middle of the
order were uncertain about their draft status.
The draft lottery was controversial from its inception. One criticism was that the lottery
results did not appear to be random. The table below indicates, for each month, how many days
in that month had lottery numbers in the beginning, middle, or end thirds of the draft order.
#1-#122
#123-#244
#245-#366
Draft Order in the 1970 Draft Lottery
Jan Feb Mar Apr May Jun Jul Aug Sep Oct
9
7
5
8
9
11
12
13
10
9
12
12
10
8
7
7
7
7
15
15
10
10
16
14
15
12
12
11
5
7
Nov
12
12
6
Dec
17
10
4
Outcry over the apparent non-randomness of the selection order was so severe that the
Pentagon revised the lottery process the following year. (REFERENCES: Science, vol. 171, pp.
255-61 and 306-308.) The table below gives a similar summary of results for the 1971 lottery.
#1-#122
#123-#244
#245-#365
Draft Order in the 1971 Draft Lottery
Jan Feb Mar Apr May Jun Jul Aug Sep Oct
12
9
10
8
9
9
12
12
7
12
14
8
10
14
15
11
7
4
10
8
5
11
11
8
7
10
12
15
13
11
Question 1: Does the 1970 draft lottery appear to be random?
Question 2: Does the 1971 draft lottery appear to be random?
Nov
13
7
10
Dec
9
14
8