DS280 – INTRODUCTION TO STATISTICS
FALL SEMESTER 2003
“Knowledge Festival” #1
Answer the following questions in the space provided. Show your work as appropriate. Relative
problem weights are given in brackets; these total 100 points. The word “Pledged” in front of
your signature on this “big quiz” indicates your ongoing commitment to the Stetson Honor
System.
Question 1 [8 points; 2 each part]:
a) You read on the back of your lottery ticket that the odds against winning the grand prize are
23-million-to-1. This is an example of which approach to probability?
classical
frequentist
subjective
b) Baseball player Joe Slabotnik has had thirty hits in one hundred chances so far this season.
Therefore has a batting average of 30/100 = .300, i.e., a probability of .300 of getting a
hit. This is an example of which approach to probability?
classical
frequentist
subjective
c) Based upon a survey of 1600 California residents, Gallup Poll notes that, if the election were
held today, over half of all voters would vote to recall Gov. Davis. This is an example of
probability
descriptive statistics
inferential statistics
d) Average SAT score among all students at the University of Southern North Dakota at Hoople
is 942. This is an example of
probability
descriptive statistics
inferential statistics
Question 2 [6 points; 3 each part]:
Indicate the best type of graph to use in each of the following:
a) You want to show how the total undergraduate enrollment at Stetson is divided among Arts &
Sciences, Business, and Music majors.
pie chart
bar chart
line graph
histogram
b) You want to show the trend in the number of cases of SARS reported worldwide.
pie chart
bar chart
line graph
histogram
Question 3 [2 points]:
The nobleman whose notorious gambling problem provided the impetus for formal study
of probability was …
Pascal
Simpson
Question 4 [4 points]:
4242
Compute C 2 . Show your work.
de Mere
Fermat
Question 5 [12 points]:
Starting salaries for twenty-four recent graduates of the University of Southern North
Dakota at Hoople are given below. Sketch an appropriate graph to illustrate these data. Interpret
your graph: what does it tell you about the distribution of salaries?
Name
Anastasia
Berengaria
Clorinda
Desdemona
Euterpe
Francine
Gracetta
Hermione
Salary
42,125
40,100
38,750
37,500
36,125
35,500
34,800
33,750
Name
Ismerelda
Jemima
Keturah
Leonora
Murgatroyd
Neldwin
Otho
Percival
Salary
32,900
32,500
31,400
29,600
29,100
28,375
27,100
26,000
Name
Quentin
Radmar
Siegfired
Torvald
Ulysses
Valmont
Wogrim
Xerxes
Salary
25,500
25,000
24,500
23,000
21,500
20,000
17,000
14,500
Question 6 [20 points; 10 each part]:
The typing monkey is back at work … one step at a time. His super-charged, bananapowered word processor has 27 keys – 26 letters and a space key. He sits down to type the word
“Palatka.”
a) What is the probability that he gets it right on his first attempt?
b) What is the probability that he gets it right at least once, in one billion attempts?
Question 7 [4 points]:
Balph Snerdwell got an answer of 42 for Question 6b (above). What conclusion should
he draw? Explain.
Question 8 [4 points]:
Clyde Arthur Fazenbaker reads in the newspaper that Rollins College has a higher
average SAT score than Stetson University. Careful comparison of the data, however, shows
that among each major (liberal arts, business, music), the average for Stetson is higher than that
for Rollins. This is an example of …
_____ Chevalier de Mere’s fallacy
_____ Simpson’s paradox
_____ Monty Hall problem
_____ false positive rate
_____ a computational mistake – it cannot happen
Question 9 [10 points]:
Dr. Rasp notes that, at any given point in time, thirty percent of the students in his
statistics class are asleep. What’s the probability that, if he calls on ten random students to work
a lecture review problem on the front board, three of them will be asleep?
Question 10 [8 points]:
Recall that the World Series is a best-of-seven tournament. In other words, the two teams
play until one has won four games. Alphonso and Clyde are betting on the World Series. Each
has wagered 500 pennies – his life savings. Alphonso is betting on the National League team;
Clyde is betting on the American League team. They believe the two teams to be evenly
matched. However, the National League team wins the first three games of the Series. Before
the fourth game starts, Clyde “suddenly remembers an important out-of-town commitment” and
wants to call off the bet. How should the total stake of 1000 pennies be fairly divided at this
point? Explain.
Question 11 [4 points]:
While at the supermarket checkout counter picking up copies of her favorite newspapers,
Euterpe spots a booklet on winning strategies for the lottery. Euterpe is always on the lookout
for strategies to improve the performance of her portfolio (which mostly consists of investments
in ping-pong ball futures, with occasional forays into call options on canine velocities). She
picks up a copy of the booklet. The booklet advocates betting on “hot” numbers – ones that have
appeared recently. “Nearly two-thirds of all winning lottery numbers have appeared in the
previous two months,” notes the booklet. Is this strategy of betting on “hot” numbers in the
lottery a valid one? Explain.
Question 12 [14 points, divided as indicated]:
Students in the Roland McGeorge Investment Program at Mad Hatter University aren’t
perfect – but they do have a good track record in identifying good stocks for investing. Over the
past five years, eighty percent of the stocks that they have recommended to “buy” have increased
in value during the following year. They’re even better on the downside: ninety percent of the
stocks that they recommend to “sell” have declined in value during the year. At present, your
portfolio consists of twenty stocks with Roland McGeorge “buy” recommendations and ten with
Roland McGeorge “sell” recommendations.
a) [10] How many of your thirty stocks can you expect to increase in value in the coming year?
Of these, what percentage will be ones with “buy” recommendations?
b) [4] Are Roland McGeorge recommendation and actual stock performance independent or
dependent events? Explain.
Question 13 [4 points]:
Strafe Airlines has adopted a program of giving all its pilots breathalyzer tests
immediately before a flight. If the breathalyzer says the pilot is sober (i.e., negative test result),
s/he flies the plane. If the breathalyzer says the pilot is drunk (positive test result), a replacement
pilot is found. In this context, what are the consequences of a false positive? Of a false
negative? Which is more costly to the airline?
DS280 – FALL 2003 – “Big Quiz” #1 - SOLUTIONS
1a) classical
2a) pie chart
4)
C
1c) inferential
3) de Mere
1d) descriptive
4242  4241
4242!

= 8,995,161. (I gave you one that your calculator won’t do,
4240! 2!
2 1
to see if you actually [gasp!] understand the calculation.)
4242
2
1b) frequentist
2b) line graph

5) The key is to do a histogram of the data (not a bar chart). Many different scales and choices
for cutpoints on the horizontal axis are possible; any reasonable histogram is OK. The
graph below is done by Microsoft Excel, which has the world’s worst histogram
procedure. You should make some sort of interpretive comment about the data.
6
Frequency
5
4
3
2
1
0
14000
22000
30000
38000
46000
Salaries
6a) Pr(right on one attempt) = Pr(1st letter right AND 2nd letter right AND … AND 7th right)
7
1 1
1
 1 

 ... 
=
=   = 9.5599 * 10-11
27 27
27  27 
6b) Pr(right at least once) = 1 – Pr(always wrong) = 1 – Pr(wrong AND wrong AND …)
=1 – (1 – Answer to A)1 billion = 1 – (.999999999904401)1 billion = 1 - .909 = .091
7) If Balph got a probability of 42, he did something wrong. Probabilities must be between 0
and 1.
8) Simpson’s paradox
9) Binomial.
C
10
3
 (.3) 3  (.7) 7 = .269
10) For Clyde to win, the AL team must win four in a row – which happens with probability
(.5*.5*.5*.5) = .0625. Hence, Clyde gets 6.25% of the total wager, or .0625*1000 = 62.5
pennies, while Alphonso gets the remaining 1000-62.5 = 937.5 pennies. (We’ll let them
decide how to deal with the half-cent.)
11) No. Ping-pong balls have no memory; lottery drawings are independent.
12a)
|
Up
|
“Buy” | .8*20=16 |
“Sell” |
1
|
|
17
|
Down
4
.9*10=9
13
|
|
|
|
20
10
30
Overall, 17 of the 30 stocks are expected
to increase in value during the year.
16/17 = 94% of these are stocks with
“buy” recommendations
12b) These are dependent events. Knowing whether the stock has a “buy” or “sell”
recommendation changes its probability for going up or down.
13) A false positive grounds a sober pilot. A false negative lets a drunk one fly. False negatives
are worse.