Download First ballot and final verdict

Math 170
Midterm 2
March 29, 2005
Prof. Gerstenhaber
This take-home exam is due at the end of class on April 5, 2005. Each student must submit an
individual paper Answer each question on a separate page and show all work; no credit will be
granted without it. Maximum score is 350 points. Put your name and Penn ID number at the top
of each page. Attach the cover sheet and staple all pages together securely, in the order of the
number of the question being answered.
(Some of the questions are adapted from the text, indicated by “FAPP”, some from the
recommended collateral reading, Statistics by Freedman, Pisani and Purves, indicated by “FPP”,
and some from Statistics for Lawyers 2nd Edition by Finkelstein and Levin, indicated by “FL”.)
1. (15 pts; 5 each part) A palindrome is a word, number or sentence which reads
the same in either direction, e.g, “deified”, “12321”, or Adam’s introduction to
Eve, “Madam, I’m Adam”. How many palindromic numbers are there consisting
of four digits (counting, e.g., “0000”)? Of five digits? Of six digits?
2. (30 pts; 10+20) An urn contains a large quantity of black and white balls in
equal numbers. You are blindfolded and draw out a sample of 5 balls.
a. What is the probability that exactly 2 are black?
b. Suppose now that the urn actually contained only 16 balls, 8 black and 8
white. Now what is the probability of drawing exactly 2 black balls in a
sample of 5?
3. (15 pts; 10+5) The following list of test scores has an average of 50 and a
standard deviation (SD) of 10. (Note: This is the entire population.)
39 41 47 58 65 37 37 49 56 59 62 36 48
52 64 29 44 47 49 52 53 54 72 50 50
a. Use the normal approximation to estimate the number of scores within 1.25
SDs of the mean.
b. b. How many scores were really within 1.25 SDs of the mean? (FPP)
4. (25 pts) You are looking at a computer printout of 100 test scores which have
been converted to standard units (z-scores). The first 10 entries are
1
-6.2 3.5 1.2 -0.13 4.3 -5.1 -7.2 -11.3 1.8 6.3
Does the printout look reasonable, or is something wrong with the computer?
Explain briefly. (FPP)
5. (15 pts; 5+10) An SRS of 500 motorcycle registrations finds that 68 of the
motorcycles are Harley-Davidsons. Give a 95% confidence interval for the
proportion of all motorcycles that are Harleys by the quick estimate of Chapter 5
of FAPP and by the more accurate estimate of Chapter 8.
6. (35 pts; 15+20) Find the correlation coefficient for the data in the following
table.
x
4
5
7
8
8
10
y
7
0
9
9
13
6
On the basis of this, if you observed an x value of 6, what would be your best
prediction of the corresponding y value?
7. (45 pts; 5+10+10+10+10) A die is rolled 100 times. The numbers of spots
appeared with the following frequencies:
spots
1
2
3
4
5
6
frequency 21
15
13
17
19
15
FPP says that the average of the numbers rolled is approximately 3.43 with
standard deviation approximately 1.76. Can you verify this? Should the results be
analyzed using a z-test or a chi-square test? Explain. What do you conclude about
the fairness of the die? Suppose now that the number of rolls is 200 and each of
the frequencies is also exactly doubled. How do your answers change? Suppose
that the number of rolls is increased to 1000 and that each frequency is multiplied
by exactly ten. How do your answers change?
8. (20 pts) Petit juries (those that actually hear cases) need not have 12 members.
Suppose that minority M constitutes 5% of the pool from which the juries are
drawn at random. What is the smallest jury size that insures that in the long run
25% of the juries will have at least one member drawn from M? (Explain)
2
9. (75 pts; 25+25+25) In Louisiana the unanimous 12-person jury has been
replaced by the following system: a unanimous 12-person jury is required for
conviction of the most serious felonies, a 9-3 jury for less serious ones, and a
unanimous 5 person jury for conviction of the least serious ones. Assume that in
Louisiana juries are selected at random from a population that is 20% minority.
We also have the following data concerning jury balloting from a sample of 225
cases in Chicago where a unanimous verdict of the 12-person jury was required. A
total of 1,828 out of 2,700 voted for conviction on the first ballot.
Final verdict
Not guilty
Hung
Guilty
No. of cases
First ballot and final verdict
Number of guilty votes on the first ballot
0
1-5
6
7-11
100%
91%
50%
5%
0
7
0
9
0
2
50
86
26
41
10
105
12
0%
0
100
43
From H. Kalven and H. Zeisel, The American Jury
488 Table 139 (1966)
a. As the attorney for a minority-group defendant charged with a felony in the
middle group, use a binomial model to argue that the probability that a
minority juror will be required to concur in the verdict is substantially
reduced by the shift from unanimous to 9-3 verdicts.
b. Use a binomial model to argue that a 9-3 conviction vote on a first ballot is
easier to obtain than a unanimous 5 for conviction of lesser offenses.
c. As a prosecutor, use the Kalven and Zeisel data to attack the binomial
model. (from FL)
10. (45 pts; 20+15+10) Here is the matrix of a two-person zero-sum game; the
entries are the payoffs to the first player.
3
2
1
-1
-2
-1
2
Here the first player has two pure strategies and the second player has three.
Suppose that the first player adopts a mixed strategy, using his first strategy with
probability p and his second with probability 1-p. What value of p is best for the
first player? Against this mixed strategy, what mixed strategy should the second
player adopt? What is the value of the game? (Solve each of the 2 x 2 games
obtained by eliminating one of the second player’s strategies. These will give
different mixed strategies for the first player. One of these will be the correct one.)
11. (30 pts; 20+10) Using the RSA algorithm with n = 9797 and r = 7, encode
one by one the last 4 digits of your Penn ID; if one of the 1ast 4 digits in your ID
is 0, replace it by 10. (Do not encode the four digits as a single number; the
encoding should consist of four numbers. ) Now take the same four numbers
(again using 10 for 0) and write them in binary notation. Put as many 0s in front as
necessary to make up a string of four binary digits. Encode these into strings of
seven 0s and 1s by the method of Chapter 10, Section 1.
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