Prof. David Draper
Department of Statistics
University of California, Santa Cruz
STAT 131: Quiz 4 90 points total
Name:
Here is Your background information, translatable into B, for this problem.
▶ (Fact 1) As a broad generalization (which you can verify empirically), statisticians
tend to have shy personalities more often than economists do — let’s quantify this
observation by assuming (based on previous psychological studies) that 87% of statis-
ticians are shy but the corresponding percentage among economists is only 9%.
▶ (Fact 2) Conferences on the topic of econometrics are almost exclusively attended
by economists and statisticians, with the majority of participants being economists
— let’s approximately quantify this fact by assuming (based on data from previous
conferences) that 96% of the attendees are economists (and the rest statisticians,
except for a tiny proportion of people from other professions, which can be ignored).
Suppose that you (a physicist, say) go to an econometrics conference — you strike up a
conversation with the first person you (haphazardly) meet, and find that this person is shy.
The point of this problem is to show that the (conditional) probability p that you’re talking
to a statistician, given this data and the above background information, is only about 29%,
which most people find surprisingly low, and to understand why this is the right answer.
Let St = (person is statistician), E = (person is economist), and Sh = (person is shy).
(a) Identify (in the form of a proposition B1, one of the elements of B) the most important
assumption needed in this problem to permit its solution to be probabilistic; explain
briefly. 5 points
(b) Using the St, E and Sh notation, express the three numbers (87%, 9%, 96%) above,
and the probability we’re solving for, in conditional probability terms, remembering
to condition appropriately on B. 5 points
1
Table 1: 2 by 2 table cross-tabulating truth (statistician, economist) against data (shy, not
shy) for the people at the conference, assuming a total number of attendees of 10,000.
Truth
Statistician Economist Total
Data
Shy — — —
Not Shy — — —
Total — — 10,000
(c) Briefly explain why calculating the desired probability is a good job for Bayes’s The-
orem. 5 points
The goal in the rest of the problem is for you to use all three of the methods developed in
class — the 2 by 2 table cross-tabulating truth against data; Bayes’s Theorem in odds ratio
form; and calculating the denominator using the Law of Total Probability, by partitioning
over the unknown truth — to compute P (St |Sh,B), the posterior probability that the
haphazard person is a statistician given that this person is shy (and given the contextual
information in B).
(d) Use the three numerical facts (87%, 9%, 96%) given at the beginning of the quiz to
fill in all 8 of the entries marked ‘—’ in Table 1, taking the total number of attendees
at the conference to be 10,0001 (Hint: All of these numbers are integers), thereby
showing that P (St |Sh,B) = 348
1212
= 29
101
.
= 28.7%; show your work. 20 points
(e) Briefly explain why the following expression is a correct use of Bayes’s Theorem on
the odds ratio scale in this problem. 5 points
[
P (St |Sh,B)
P (E |Sh,B)
]
=
[
P (St | B)
P (E | B)
]
·
[
P (Sh |St,B)
P (Sh |E,B)
]
(1) = (2) · (3)
1The biggest annual economics meetings worldwide typically have between 6,000 and 13,000 attendees.
2
(f) Here are three terms that are relevant to the quantities in part (e) above:
– (Prior odds ratio in favor of St over E, given B)
– (Bayes factor in favor of St over E, given B)
– (Posterior odds ratio in favor of St over E, given B)
Match these three terms with the numbers (1), (2), (3) in the second line of the equa-
tion in part (e). 5 points
(g) Compute the three ratios in part (e), briefly explaining your reasoning, thereby
demonstrating that the posterior odds ratio o in favor of St over E (given B) is
o = 29
72
.
= 0.403. 15 points
(h) Use the expression p = o
1+o
to show that the desired probability in this problem —
the conditional probability that you’re talking to a statistician (given Sh and B) —
is p = 29
101
.
= 0.287. 5 points
(i) Briefly explain why the following expression is a correct use of Bayes’s Theorem on
the probability scale in this problem. 5 points
P (St |Sh, B) = P (St | B)P (Sh |St, B)
P (Sh | B) . (1)
(j) Notice as usual that you know both of the numerator probabilities in equation (1) but
you don’t (yet) know the denominator P (Sh | B). Use the Law of Total Probability,
partitioning over the unknown truth, to show that
P (Sh | B) = 303
2500
= 0.1212 , (2)
3
and use this to show that
P (St |Sh, B) = (0.04)(0.87)
0.1212
=
29
101
.
= 0.287 . (3)
15 points
(k) Someone says, “That 28.7% probability can’t be right: 87% of statisticians are shy,
versus 9% for economists, so your probability p of talking to a statistician has to be
over 50%.” Briefly explain why this line of reasoning is wrong, and why p should
indeed be less than 50%. 5 points