Math 170



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

-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)

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.

First ballot and final verdict

Number of guilty votes on the first ballot

|Final verdict |0 |1-5 |6 |7-11 |12 |

|Not guilty |100% |91% |50% |5% |0% |

|Hung |0 |7 |0 |9 |0 |

|Guilty |0 |2 |50 |86 |100 |

|No. of cases |26 |41 |10 |105 |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.

|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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