Free-Response - Penn Math

1. Below 1. are the ages of 15 students in a college class.
27, 50, 33, 25, 86, 25, 85, 31, 37, 44, 20, 36, 59, 34, 28
What is the median age? What is the the first quartile of the ages? What is the range for this data?
2. Below are the lengths (in minutes) of phone calls made on an 800 line to a business on one day. Find the fivenumber
summary for this data.
14, 6, 12, 19, 2, 35, 5, 4, 3, 7, 5, 8
3. Below is a histogram of car prices on a used car lot. An advertisement for the lot says the average price of a car
for sale is $12,000. Could the advertisement be true? Why or why not?
4. In the list of five measurements shown below, one is blurred. Supply a fifth value that would make the median 8.
8 5 10 3 #
5. Create a set of five positive numbers that have median 6 and mean 10.
6. Suppose the regression line for a scatterplot passes through the points (0, 10) and (5, 20). What type of
association exists between the variables?
7. What can you say about the slope of a regression line for variables that are negatively associated?
8. The least-squares regression line is the “best” line in that it minimizes a certain sum. What is this sum?
9. A spinner with regions numbered 1 to 4 is spun and a coin is tossed. Both the number spun and whether the coin
lands heads or tails is recorded. Write the sample space.
10. Suppose three fair coins are tossed and the number of heads that appear is recorded. What is the probability of
getting exactly two heads?
11. We must create a license plate code that consists of two letters followed by three digits. The letters cannot
repeat, but the digits may. How many such license plate codes can be formed? What is the probability that a
randomly chosen plate has all three digits the same?
12. A pizza can be made with any of the following toppings: cheese, pepperoni, sausage, mushrooms, anchovies,
green peppers, or olives. Doubling of any topping is not allowed. What is the probability that a randomly created
three-topping pizza will contain mushrooms?
13. A student is taking a five-question True/False test. If the student chooses answers at random, what is the
probability of getting all questions correct?
14. Suppose a trial consists of rolling two dice and reporting the smaller of the two numbers rolled? What are the
possible outcomes? Are they equally likely?

15. In order to choose which type of music to listen to in a student center music lounge, a Condorcet vote is held by
the 15 students present. Below are the preference schedules for the students. Is there a Condorcet winner and if so,
which music type?
Number of Students
6 5 4
First choice classical rock rock
Second choice jazz jazz classical
Third choice rock classical Jazz
16. Seventeen board members vote on four candidates, A, B, C, or D, for a new position on their board. Their
preference schedules are shown below.
Number of Members
7 6 4
First choice A D C
Second choice B A B
Third choice C B D
Fourth choice D C A
Which candidate will be selected if they use the Hare system?
17. Create a three-by-three matrix that represents a two-person zero-sum game in which each player has three
options and the game has a saddle point.
18. Create a three-by-three matrix that represents a two-person zero-sum game in which each player has three
options and the game has no saddle point.
19. In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to
column player II.
Does this game have a saddle point? What is each player's minimax or maximin strategy? Justify your
response.
20. In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to
column player II.
Does this game have a saddle point? What is each player's minimax or maximin strategy? Justify your
response.
The following problems are from the textbook. Answers can be found in the back of the book.
Page 308 : # 9, 11, 13, 17.
Page 502 : # 1, 3, 5 (ignore part (c) for 1, 3, 5), 9, 11 ((a) and (b)).

Answers to Prob. 1-20.
1. Median 34. First quartile 27. Range 66.
2. 2, 4.5, 6.5, 13, 35.
3. Yes, the advertisement could be true. The presence of a car or cars with a sale price of $200 (the outlier(s) on
the histogram) could lower the mean sale price to $12,000.
4. Answers may vary. Any value greater than or equal to 8 would be correct.
5. Answers may vary. One possibility is {2, 2, 6, 20, 20}.
6. A positive association
7. The slope is negative.
8. The sum of the squares of the vertical distances from each data point to the line.
9. {(1,H), (2,H), (3,H), (4,H), (1,T), (2,T), (3,T), (4 ,T)}
10. The probability of getting exactly two heads is 3/8 or 0.375.
11. 650,000. Probability 6500/650,000 = 1/100 = 0.01
12. 30/210 = 1/7 = 0.143
13. 1/32 = 0.03125
14. Possible outcomes are 1, 2, 3, 4, 5, 6. They are not equally likely.
15. Rock is the Condorcet winner.
16. D wins
17. For example,
18. For example,
19. No saddle point. Play the 2nd row and the 1st column.
20. It has a saddle point at 3. Play the 3rd row and the 1st column.