Download Exam 1 - Stetson University

DS280 – INTRODUCTION TO STATISTICS
SPRING SEMESTER 2003
“BIG QUIZ” #1
INSTRUCTIONS: Write your name at the top of this page. (It’s worth two points!) Writing
“Pledged” before your signature indicates your ongoing commitment to the Honor System.
There are 100 points worth of questions on this “quiz” – relative problem weights are given in
brackets. Answer the questions in the space provided. SHOW YOUR WORK on computational
problems. Enjoy!!
Question 1 [8 points; 2 each part]:
Select the best option in each of the following.
A) Alphonso Ferrabosco says the odds are 42-to-1 against the Cubs reaching the World Series
this year. This is an example of
probability
descriptive statistics
inferential statistics
B) Berengaria Naverre surveys 846 people, and concludes that 42% of Florida residents moved
to the state primarily because of the weather. This is an example of
probability
descriptive statistics
inferential statistics
C) Clorinda Cragdingle’s favorite “big” homework problem in DS280 was the “craps” problem,
where she worked out the probability of winning a “pass” bet at this dice game. This is
an example of what approach to probability?
classical
frequentist
subjective
D) Dietrich Buxtehude’s favorite basketball player “Brick” Hoopless has made 80 of 200 free
throw attempts so far this season – for a free throw percentage of 40%. This is an
example of what approach to probability?
classical
frequentist
subjective
Question 2 [8 points, 4 each part]:
A)
Mad Hatter University consists of three colleges: Arts & Leisure, Capitalism, and P.D.Q.
Bach Studies. These enroll 1200, 600, and 200 students, respectively, out of a total student body
of 2000. Which would be the best graph to use to show how the total enrollment is divided
among the three colleges?
pie chart
bar graph
line graph
histogram
B)
Euterpe Waldfogel, a student at Mad Hatter University, has obtained data from the
university archives on how tuition has changed during the 100-year history of the school. What
would be the best graph for her to use to show how tuition has grown over time?
pie chart
bar graph
line graph
histogram
Question 3 [2 points]:
Ferdinand Walpurgisnacht is working on the “Yahtzee” problem in Dr. Rasp’s evil
“challenging” homework assignment. He computes a probability of 4242.42. What should he
conclude?
Question 4 [2 points]:
Gracetta Squornshellous notes that Arizona has a lower death rate than Utah in every age
group, but Utah has a lower death rate overall. This is an example of … {choose one}
The Monty Hall Problem
The Chevalier de Mere’s Fallacy
Simpson’s Paradox
The Typing Monkey Problem
Question 5 [4 points]:
Horatio Wajberlinski is playing blackjack (a card game) at his favorite neighborhood
casino. He notes that the last two hands in a row have been “blackjack” (an Ace and a card
worth ten points). He reasons that there is a reduced chance of blackjack on this hand. Is his
reasoning correct? Explain.
Question 6 [8 points; 4 each part]:
Ismerelda Tempisfugit rolls a pair of standard, six-sided dice.
a) What is the probability that the two dice total 6?
b) Ismerelda will continue to roll the dice until she gets either a 6 or an 11. What is the
probability that she rolls a 6 before she rolls an 11?
Question 7 [8 points]:
What is the probability that, in a group of three people, at least two were born on the
same day of the week?
Question 8 [10 points]:
Jesperson P. Snood, demographer for the United Nations Special Commission on
National Statistics, obtains data on the death rate (per hundred thousand residents) for each of the
thirty provinces of Boravia, a third-world nation considered to be an underdeveloped nation even
by other underdeveloped nations. The data (a three-letter province abbreviation, and the death
rate) are given below. Sketch an appropriate graph to illustrate these data. Briefly describe what
the graph tells you about the data.
StL:
Mlt:
SQC:
Aar:
Blt:
Ncy:
846
882
904
918
935
942
FTD: 961
Mar: 978
WWF: 992
KoA: 1004
FtL: 1012
QQQ: 1042
IoU: 1058
RRt: 1072
RFd: 1089
Dak: 1105
How: 1128
Huh: 1142
Dfl:
Urv:
Pln:
Url:
Jre:
Mud:
1172
1189
1224
1240
1252
1254
Roc: 1263
Tue: 1289
Mon: 1305
Van: 1389
Chi: 1438
NyM: 1645
Question 9 [16 points; 8 each part]:
Players in “Pick Three” lotteries have a probability of 1/1000 of winning, on any given
day.
a) What is the probability of winning three times in a year (365 days)?
b) If 10,000 people play the “Pick Three” lottery daily, what is the probability that someone
wins three times?
Question 10 [16 points; 8 each part]:
Recall that we began the probability unit in class with the “problem of points.” Both you
and an opponent select a number, 1 through 6. You have four chances to roll your number on a
standard, six-sided die.
a) What is the probability that you get your number at least once, in your four rolls?
b) Recall that a point is scored if one player gets his/her number and the other does not. What is
the probability that a point is scored in any given round?
Question 11 [16 points; divided as indicated]:
Even though Dr. Rasp encourages students to get a lot of sleep the night before a “big
quiz,” he finds that historically about 40% of the class ignores his advice and neglect sleep.
Some after-the-party data analysis last semester revealed that 75% of the students who got plenty
of sleep did well on the “quiz” (B or higher), while only 40% of those who neglected sleep did.
The rest performed suboptimally.
a) [6] Overall, what percentage of the class did well on the “quiz” (B or higher)?
b) [6] If someone did suboptimally on the “quiz,” what’s the probability that he/she neglected
sleep?
b) [4] Are the events “doing well on the quiz” and “getting enough sleep” independent or
dependent? Explain.
DS280 – SPRING SEMESTER 2003 – “BIG QUIZ” #1
Second die
1a) Alphonso is probability. (“Odds” are one way of expressing a probability.)
1b) Berengaria is inferential statistics. (A conclusion about all residents is made based upon
data from only 846 of them.)
1c) Clorinda has a classical probability. (Games of chance – each side of the die equally likely.)
1d) Dietrich has a frequentist probability. (Long-run behavior.)
2a) Euterpe uses a pie chart to show how the “whole” is divided into “parts.”
2b) She uses a line graph to show trends in tuition.
3) Ferdinand should conclude he made a mistake – probabilities cannot be bigger than 1!!!!!
4) Gracetta has an example of Simpson’s paradox – including a third variable (state) reverses
the probabilities here.
5) Remember that cards are dependent events. Playing cards changes the odds on subsequent
hands. By playing Aces, you reduce the chance of subsequent Aces. (In class we noted
that there could be no winning “system” at roulette, craps, or the lottery – but there could
be winning systems at blackjack precisely because the events were dependent.)
6a) This example was done in Lecture #2. Set up a table of possible die rolls:
1
2
3
4
5
6
1
First die:
2 3 4 5
6
2
3
4
5
6
7
3
4
5
6
7
8
4
5
6
7
8
9
5
6
7
8
9
10
6
7
8
9
10
11
7
8
9
10
11
12
Note that there are five ways to get a total of 6, so the probability is 5/36.
6b) This was used in the “craps” problem on “Big” Homework #2. Note that there are 5 ways to
get a six, and 2 ways to get an eleven. We ignore everything else. So the probability is
5/(5+2) = 5/7
7) This is a “birthday” problem. You were asked to do a problem very similar to this on the
daily review from Lecture #3.
Pr(at least one match) = 1 – Pr(all different)
= 1 – Pr(any AND diff’t AND diff’t)
7 6 5
 1      = 1 - .612 = .388
7 7 7
Frequency
8) Do a histogram, NOT A BAR GRAPH!!!!! (Bar graphs are not effective ways of this much
data – we cannot realistically “by eye” compare that many data points.) Any reasonable
grouping on the horizontal axis is fine. Your graph might look something like this:
8
6
4
2
0
800
1000
1200
1400
Death Rate
1600
More
Note something about what the graph is indicating: Most provinces have death rates
between 900 and 1300. Data are slightly skewed. There’s one outlier on the high side.
9a) Binomial. Probability of three wins is:
365
3
362
C 3  .001  .999 = .005596
Many of you expressed concern that you got error messages when you tried to do the
combination on your calculator. Most calculators can’t compute numbers larger than 63!
without overflowing the memory. You can still calculate this number, though. (In fact,
one of the points of the problem was to see if you understood the computation rather than
simply feeding numbers to a calculator.) Remember that:
C
365
3

365  364  363  362  361...  2 1 365  364  363
365!


 8,038,030
3!  362!
(3  2 1)  (362  361 ...  2 1)
3  2 1
9b) Pr(at least one three-time winner) = 1 – Pr(no three-time winner)
= 1 – Pr(non3winner AND non3winner AND …)
10, 000
 1  1  .005596
1–0=1
(It’s almost certain that someone will win the daily lottery three times in a year – although
any individual’s chance of doing so is small.)
10a) We did this one in class.
5 5 5 5
Pr(at least one #) = 1 – Pr(no #) = 1       = 1 - .482 = .518
6 6 6 6
10b) Pr(point is scored) = Pr([you get # & opponent doesn’t] OR [you don’t & opponent does])
= [.518 * .482] + [.482 * .518] – 0
{no overlap}
= .499
11) Set up a table:
SleepUn-sleep
Total
Did well:
Did suboptimally:
Total:
.75*60 = 45
15
60
.40*40 = 16
24
40
|
|
|
|
61
39
100
11a) Overall total doing well on the “quiz”: 61/100 = .61
11b) Pr(neglect sleep IF did suboptimally) = 24/39 = .615
11c) These are DEPENDENT events. Sleep affects people’s chances of doing well on the
“quiz.”