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Part I: Concept questions
These questions are all multiple choice or short answer. You don’t have to show any work. Work through them
quickly. Each answer is worth 2 points.
Concept 1. Which of the following represents a valid probability table?
Circle the best choice:
A. (i) B. (ii) C. (i) and (ii) D. Not enough information
Concept 2. True or false: Setting the prior probability of a hypothesis to 0 means that no amount of data
will make the posterior probability of that hypothesis the maximum over all hypotheses.
Circle one: True False
Concept 3. True or false: It is okay to have a prior that depends on more than one unknown
parameter.
Circle one: True False
Concept 4. Data is drawn from a normal distribution with unknown mean µ. We make the following
hypotheses: H0: µ = 1 and HA: µ > 1.
For (i)-(iii) circle the correct answers:
Problem

(i) Is H0 a simple or composite hypothesis? Simple Composite
(ii) Is HA a simple or composite hypothesis? Simple Composite
(iii) Is HA a one or two-sided? One-sided Two-sided
Concept 5. If the original data has n points then a bootstrap sample should have
A. Fewer points than the original because there is less information in the sample than in the
underlying distribution.
B. The same number of points as the original because we want the bootstrap statistic to mimic the
statistic on the original data.
C. Many more points than the original because we have the computing power to handle a lot of data.
Circle the best answer: A B C.
Concept 6. In 3 tosses of a coin which of following equals the event “exactly two heads”?
A = {T HH, HTH, HHT, HHH}
B = {T HH, HTH, HHT}
C = {HT H, T HH}
Circle the best answer: A B C B and C
Concept 7. These questions all refer to the following figure. For each one circle the best answer.

(i) The probability x represents A. P(A1) B. P(A1|B2) C. P(B2|A1) D. P(C1|B2 ∩ A1).
(ii) The probability y represents A. P(B2) B. P(A1|B2) C. P(B2|A1) D. P(C1|B2 ∩ A1).
(iii) The probability z represents A. P(C1) B. P(B2|C1) C. P(C1|B2) D. P(C1|B2 ∩ A1).
(iv) The circled node represents the event A. C1 B. B2 ∩ C1 C. A1 ∩ B2 ∩ C1 D. C1|B2 ∩ A1.
Concept 8. The graphs below give the pmf for 3 random variables.
Circle the answer that orders the graphs from smallest to biggest standard deviation.
ABC ACB BAC BCA CAB CBA
Concept 9. Suppose you have $100 and you need $1000 by tomorrow morning. Your only way to get the
money you need is to gamble. If you bet $k, you either win $k with probability p or lose $k with probability 1
− p. Here are two strategies:

Maximal strategy: Bet as much as you can, up to what you need, each time.
Minimal strategy: Make a small bet, say $10, each time.
Suppose p = 0.8.
Circle the better strategy: Maximal 2. Minimal
Concept 10. Consider the following joint pdf’s for the random variables X and Y . Circle the ones
where X and Y are independent and cross out the other ones.
Concept 11. Suppose X ∼ Bernoulli(θ) where θ is unknown. Which of the following is the correct
statement?
A. The random variable is discrete, the space of hypotheses is discrete.
B. The random variable is discrete, the space of hypotheses is continuous.
C. The random variable is continuous, the space of hypotheses is discrete.
D. The random variable is continuous, the space of hypotheses is continuous.
Circle the letter of the correct statement: A B C D
Concept 12. Let θ be the probability of heads for a bent coin. Suppose your prior f(θ) is Beta(6, 8). Also
suppose you flip the coin 7 times, getting 2 heads and 5 tails. What is the posterior pdf f(θ|x)? Circle the
best answer.
A. Beta(2,5) B. Beta(3,6) C. Beta(6,8) D. Beta(8,13) E. Not enough
information to say

Concept 13. Suppose the prior has been set. Let x1 and x2 be two sets of data. Circle true or false for each
of the following statements.
A. If x1 and x2 have the same likelihood function then they result in the same posterior.
B. If x1 and x2 result in the same posterior then they have the same likelihood function.
C. If x1 and x2 have proportional likelihood functions then they result in the same posterior.
Concept 14. Each day Jane arrives X hours late to class, with X ∼ uniform(0, θ). Jon models his initial belief
about θ by a prior pdf f(θ). After Jane arrives x hours late to the next class, Jon computes the likelihood
function f(x|θ) and the posterior pdf f(θ|x). Circle the probability computations a frequentist would consider
valid. Cross out the others.
True False
True False
True False
Concept 14. Each day Jane arrives X hours late to class, with X ∼ uniform(0, θ). Jon models his initial belief
about θ by a prior pdf f(θ). After Jane arrives x hours late to the next class, Jon computes the likelihood
function f(x|θ) and the posterior pdf f(θ|x).
Circle the probability computations a frequentist would consider valid. Cross out the others.
A. prior B. posterior C. likelihood
Concept 15. Suppose we run a two-sample t-test for equal means with significance level α = 0.05. If
the data implies we should reject the null hypothesis, then the odds that the two samples come from
distributions with the same mean are (circle the best answer)
A. 19/1 B. 1/19 C. 20/1 D. 1/20 E. unknown
Concept 16. Consider the following statements about a 95% confidence interval for a parameter θ.
A. P(θ0 is in the CI | θ = θ0) ≥ 0.95
B. P(θ0 is in the CI ) ≥ 0.95
C. An experiment produces the CI [−1, 1.5]: P(θ is in [−1, 1.5] | θ = 0) ≥ 0.95
Circle the letter of each correct statement and cross out the others:

A B C
Part II:
Problem 1.
(a) Let A and B be two events. Suppose that the probability that neither event occurs is 3/8. What is the
probability that at least one of the events occurs?
(b) Let C and D be two events. Suppose P(C) = 0.5, P(C∩D) = 0.2 and P((C∪D) c ) = 0.4. What is P(D)?
Problem 2.
An urn contains 3 red balls and 2 blue balls. A ball is drawn. If the ball is red, it is kept out of the urn
and a second ball is drawn from the urn. If the ball is blue, then it is put back in the urn and a red ball
is added to the urn. Then a second ball is drawn from the urn.
(a) What is the probability that both balls drawn are red?
(b) If the second drawn ball is red, what is the probability that the first drawn ball was blue?
Problem 3. (15) You roll a fair six sided die repeatedly until the sum of all numbers rolled is greater
than 6. Let X be the number of times you roll the die. Let F be the cumulative distribution function for
X. Compute F(1), F(2), and F(7).
Problem 4.
A test is graded on the scale 0 to 1, with 0.55 needed to pass. Student scores are modeled by the
following density:

(a) What is the probability that a random student passes the exam?
(b) What score is the 87.5 percentile of the distribution?
Problem 5.
Suppose X is a random variable with cdf
Problem 6. (15) Compute the mean and variance of a random variable whose distribution is uniform
on the interval [a, b].
It is not enough to simply state these values. You must give the details of the computation.

Problem 7. (20) Defaulting on a loan means failing to pay it back on time. The default rate among MIT
students on their student loans is 1%. As a project you develop a test to predict which students will default.
Your test is good but not perfect. It gives 4% false positives, i.e. prediciting a student will default who in fact
will not. If has a 0% false negative rate, i.e. prediciting a student won’t default who in fact will.
(a) Suppose a random student tests positive. What is the probability that he will truly default. (
b) Someone offers to bet me the student in part (a) won’t default. They want me to pay them $100 if the
student doesn’t default and they’ll pay me $400 if the student does default. Is this a good bet for me
to take?
Problem 8. (30) Data was taken on height and weight from the entire population of 700 mountain
gorillas living in the Democratic Republic of Congo:
Let X encode the weight, taking the values of a randomly chosen gorilla: 0, 1, 2 for light, average, and
heavy respectively.
Likewise, let Y encode the height, taking values 0 and 1 for short and tall respectively.
(a) Determine the joint pmf of X and Y and the marginal pmf’s of X and of Y .
(b) Are X and Y independent?
(c) Find the covariance of X and Y . For this part, you need a numerical (no variables) expression, but you
can leave it unevaluated.
(d) Find the correlation of X and Y . For this part, you need a numerical (no variables) expression, but you
can leave it unevaluated. Cov(X, Y )
(d) The definition of correlation is Cor(X, Y ) = . So we first need to compute σXσY the variances of X and
Y

Problem 9. (20) A political poll is taken to determine the fraction p of the population that would
support a referendum requiring all citizens to be fluent in the language of probability and statistics.
(a) Assume p = 0.5. Use the central limit theorem to estimate the probability that in a poll of 25
people, at least 14 people support the referendum. Your answer to this problem should be a
decimal.
(b) With p unknown and n the number of random people polled, let Xn be the fraction of the
polled people who support the referendum. What is the smallest sample size n in order to
have a 90% confidence that Xn is within 0.01 of the true value of p?
Your answer to this problem should be an integer.

Problem 10. (10 pts) Suppose a researcher collects x1, . . . , xn i.i.d. measurements of the background
radiation in Boston. Suppose also that these observations follow a Rayleigh distribution with parameter τ ,
with pdf given by
Find the maximum likelihood estimate for τ .
Problem 11. (15) Bivariate data (4, 10),(−1, 3),(0, 2) is assumed to arise from the model yi = b|xi −
3| + ei , where b is a constant and ei are independent random variables.
(a) What assumptions are needed on ei so that it makes sense to do a least squares fit of a
curve y = b|x − 3| to the data?
(b) Given the above data, determine the least squares estimate for b. For this problem we want
you to calculate all the way to a fraction b = r , where r and s s are integers.
Problem 12. (30) Data is collected on the time between arrivals of consecutive taxis at a downtown hotel.
We collect a data set of size 45 with sample mean x¯ = 5.0 and sample standard deviation s = 4.0.
(a) Assume the data follows a normal random variable.
(i) Find an 80% confidence interval for the mean µ of X.
(ii) Find an 80% χ 2 -confidence interval for the variance?
(b) Now make no assumptions about the distribution of of the data. By bootstrapping, we generate 500
values for the differences δ ∗ = x¯ ∗ − x¯. The smallest and largest 150 are written in non-decreasing
order on the next page. Use this data to find an 80% bootstrap confidence interval for µ.
(c) We suspect that the time between taxis is modeled by an exponential distribution, not a normal
distribution. In this case, are the approaches in the earlier parts justified?
(d) When might method (b) be preferable to method (a)?

The 100 smallest and 100 largest values of δ∗ for problem 12.
1- 10 -0.534 -0.494 -0.491 -0.485 -0.422 -0.403 -0.382 -0.365 -0.347 -0.336
11- 20 -0.330 -0.328 -0.315 -0.304 -0.297 -0.293 -0.287 -0.279 -0.273 -0.273
21- 30 -0.271 -0.269 -0.262 -0.262 -0.260 -0.257 -0.256 -0.255 -0.249 -0.248
31- 40 -0.241 -0.240 -0.232 -0.226 -0.225 -0.223 -0.222 -0.220 -0.216 -0.216
41- 50 -0.213 -0.211 -0.211 -0.210 -0.209 -0.209 -0.208 -0.204 -0.202 -0.200
51- 60 -0.200 -0.200 -0.195 -0.193 -0.192 -0.192 -0.189 -0.188 -0.188 -0.183
61- 70 -0.182 -0.182 -0.181 -0.179 -0.179 -0.178 -0.176 -0.175 -0.174 -0.170
71- 80 -0.170 -0.166 -0.164 -0.163 -0.163 -0.162 -0.162 -0.160 -0.160 -0.159
81- 90 -0.159 -0.159 -0.158 -0.157 -0.156 -0.156 -0.155 -0.155 -0.154 -0.154
91-100 -0.153 -0.152 -0.151 -0.151 -0.150 -0.148 -0.148 -0.146 -0.145 -0.145
101-110 -0.144 -0.142 -0.142 -0.142 -0.138 -0.137 -0.135 -0.135 -0.134 -0.134
111-120 -0.133 -0.131 -0.129 -0.128 -0.124 -0.124 -0.124 -0.123 -0.123 -0.119
121-130 -0.118 -0.114 -0.114 -0.114 -0.112 -0.111 -0.109 -0.108 -0.108 -0.107
131-140 -0.105 -0.103 -0.103 -0.103 -0.102 -0.101 -0.099 -0.098 -0.098 -0.097
141-150 -0.097 -0.096 -0.095 -0.095 -0.095 -0.093 -0.093 -0.093 -0.093 -0.093
351-360 0.073 0.074 0.075 0.075 0.077 0.077 0.077 0.077 0.078 0.079
361-370 0.079 0.079 0.080 0.081 0.081 0.082 0.083 0.084 0.085 0.085
371-380 0.087 0.087 0.088 0.091 0.091 0.091 0.092 0.092 0.093 0.093
381-390 0.094 0.094 0.096 0.097 0.100 0.100 0.101 0.101 0.102 0.103
391-400 0.104 0.104 0.106 0.106 0.108 0.108 0.108 0.108 0.108 0.110
401-410 0.110 0.111 0.112 0.112 0.112 0.112 0.113 0.114 0.114 0.115
411-420 0.118 0.122 0.122 0.123 0.127 0.129 0.129 0.132 0.134 0.134
421-430 0.134 0.135 0.136 0.136 0.137 0.140 0.141 0.142 0.142 0.143
431-440 0.143 0.145 0.146 0.147 0.147 0.148 0.151 0.151 0.154 0.155
441-450 0.156 0.162 0.163 0.164 0.164 0.165 0.166 0.168 0.169 0.169
451-460 0.170 0.172 0.172 0.175 0.178 0.179 0.180 0.181 0.182 0.182
461-470 0.182 0.186 0.195 0.202 0.202 0.205 0.206 0.210 0.216 0.219
471-480 0.220 0.220 0.221 0.222 0.224 0.225 0.232 0.232 0.236 0.236
481-490 0.243 0.244 0.245 0.251 0.253 0.258 0.261 0.263 0.266 0.273
491-500 0.274 0.288 0.288 0.291 0.307 0.312 0.314 0.316 0.348 0.488
Keep going! There are more problems ahead.

Problem 14. You independently draw 100 data points from a normal distribution.
(a) Suppose you know the distribution is N(µ, 4) (4 = σ 2 ) and you want to test the null hypothesis H0 : µ =
3 against the alternative hypothesis HA : µ 6= 3. If you want a significance level of α = 0.05. What is
your rejection region? You must clearly state what test statistic you are using.
(b) Suppose the 100 data points have sample mean 5. What is the p-value for this data? Should you reject
H0?
(c) Determine the power of the test using the alternative HA : µ = 4.
Problem 13. Note. In this problem the geometric(p) distribution is defined as the total number of trials to
the first failure (the value includes the failure), where p is the probabilitiy of success.
(a) What sample statistic would you use to estimate p?
(b) Describe how you would use the parametric bootstrap to estimate a 95% confidence interval
for p. You can be brief, but you should give careful step-by-step instructions.
Problem 15. Suppose that you have molecular type with unknown atomic mass θ. You have an
atomic scale with normally-distributed error of mean 0 and variance 0.5.
(a) Suppose your prior on the atomic mass is N(80, 4). If the scale reads 85, what is your posterior
pdf for the atomic mass?
(b) With the same prior as in part (a), compute the smallest number of measurements needed so
that the posterior variance is less than 0.01.
Problem 16. Your friend grabs a die at random from a drawer containing two 6-sided dice, one 8-sided
die, and one 12-sided die. She rolls the die once and reports that the result is 7.
(a) Make a discrete Bayes table showing the prior, likelihood, and posterior for the type of die rolled
given the data.
(b) What are your posterior odds that the die has 12 sides?
(c) Given the data of the first roll, what is your probability that the next roll will be a 7?

Standard normal table of left tail probabilities.
z Φ(z) z Φ(z) z Φ(z) z Φ(z)
-4.00 0.0000 -2.00 0.0228 0.00 0.5000 2.00 0.9772
-3.95 0.0000 -1.95 0.0256 0.05 0.5199 2.05 0.9798
-3.90 0.0000 -1.90 0.0287 0.10 0.5398 2.10 0.9821
-3.85 0.0001 -1.85 0.0322 0.15 0.5596 2.15 0.9842
-3.80 0.0001 -1.80 0.0359 0.20 0.5793 2.20 0.9861
-3.75 0.0001 -1.75 0.0401 0.25 0.5987 2.25 0.9878
-3.70 0.0001 -1.70 0.0446 0.30 0.6179 2.30 0.9893
-3.65 0.0001 -1.65 0.0495 0.35 0.6368 2.35 0.9906
-3.60 0.0002 -1.60 0.0548 0.40 0.6554 2.40 0.9918
-3.55 0.0002 -1.55 0.0606 0.45 0.6736 2.45 0.9929
-3.50 0.0002 -1.50 0.0668 0.50 0.6915 2.50 0.9938
-3.45 0.0003 -1.45 0.0735 0.55 0.7088 2.55 0.9946
-3.40 0.0003 -1.40 0.0808 0.60 0.7257 2.60 0.9953
-3.35 0.0004 -1.35 0.0885 0.65 0.7422 2.65 0.9960
-3.30 0.0005 -1.30 0.0968 0.70 0.7580 2.70 0.9965
-3.25 0.0006 -1.25 0.1056 0.75 0.7734 2.75 0.9970
-3.20 0.0007 -1.20 0.1151 0.80 0.7881 2.80 0.9974
-3.15 0.0008 -1.15 0.1251 0.85 0.8023 2.85 0.9978
-3.10 0.0010 -1.10 0.1357 0.90 0.8159 2.90 0.9981
-3.05 0.0011 -1.05 0.1469 0.95 0.8289 2.95 0.9984
-3.00 0.0013 -1.00 0.1587 1.00 0.8413 3.00 0.9987
-2.95 0.0016 -0.95 0.1711 1.05 0.8531 3.05 0.9989
-2.90 0.0019 -0.90 0.1841 1.10 0.8643 3.10 0.9990
-2.85 0.0022 -0.85 0.1977 1.15 0.8749 3.15 0.9992
-2.80 0.0026 -0.80 0.2119 1.20 0.8849 3.20 0.9993
-2.75 0.0030 -0.75 0.2266 1.25 0.8944 3.25 0.9994
-2.70 0.0035 -0.70 0.2420 1.30 0.9032 3.30 0.9995
-2.65 0.0040 -0.65 0.2578 1.35 0.9115 3.35 0.9996
-2.60 0.0047 -0.60 0.2743 1.40 0.9192 3.40 0.9997
-2.55 0.0054 -0.55 0.2912 1.45 0.9265 3.45 0.9997
-2.50 0.0062 -0.50 0.3085 1.50 0.9332 3.50 0.9998
-2.45 0.0071 -0.45 0.3264 1.55 0.9394 3.55 0.9998
-2.40 0.0082 -0.40 0.3446 1.60 0.9452 3.60 0.9998
-2.35 0.0094 -0.35 0.3632 1.65 0.9505 3.65 0.9999
-2.30 0.0107 -0.30 0.3821 1.70 0.9554 3.70 0.9999
-2.25 0.0122 -0.25 0.4013 1.75 0.9599 3.75 0.9999
-2.20 0.0139 -0.20 0.4207 1.80 0.9641 3.80 0.9999
-2.15 0.0158 -0.15 0.4404 1.85 0.9678 3.85 0.9999
-2.10 0.0179 -0.10 0.4602 1.90 0.9713 3.90 1.0000
-2.05 0.0202 -0.05 0.4801 1.95 0.9744 3.95 1.0000
Φ(z) = P (Z ≤ z) for N(0, 1).
(Use interpolation to estimate
z values to a 3rd decimal
place.)

t-table of left tail probabilities. (The tables show P (T < t) for T ∼ t(df ).)
tdf
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3.8
4.0
1
0.5000
0.5628
0.6211
0.6720
0.7148
0.7500
0.7789
0.8026
0.8222
0.8386
0.8524
0.8642
0.8743
0.8831
0.8908
0.8976
0.9036
0.9089
0.9138
0.9181
0.9220
2
0.5000
0.5700
0.6361
0.6953
0.7462
0.7887
0.8235
0.8518
0.8746
0.8932
0.9082
0.9206
0.9308
0.9392
0.9463
0.9523
0.9573
0.9617
0.9654
0.9686
0.9714
3
0.5000
0.5729
0.6420
0.7046
0.7589
0.8045
0.8419
0.8720
0.8960
0.9152
0.9303
0.9424
0.9521
0.9598
0.9661
0.9712
0.9753
0.9788
0.9816
0.9840
0.9860
4
0.5000
0.5744
0.6452
0.7096
0.7657
0.8130
0.8518
0.8829
0.9076
0.9269
0.9419
0.9537
0.9628
0.9700
0.9756
0.9800
0.9835
0.9864
0.9886
0.9904
0.9919
5
0.5000
0.5753
0.6472
0.7127
0.7700
0.8184
0.8581
0.8898
0.9148
0.9341
0.9490
0.9605
0.9692
0.9759
0.9810
0.9850
0.9880
0.9904
0.9922
0.9937
0.9948
6
0.5000
0.5760
0.6485
0.7148
0.7729
0.8220
0.8623
0.8945
0.9196
0.9390
0.9538
0.9649
0.9734
0.9797
0.9844
0.9880
0.9907
0.9928
0.9943
0.9955
0.9964
7
0.5000
0.5764
0.6495
0.7163
0.7750
0.8247
0.8654
0.8979
0.9232
0.9426
0.9572
0.9681
0.9763
0.9823
0.9867
0.9900
0.9925
0.9943
0.9956
0.9966
0.9974
8
0.5000
0.5768
0.6502
0.7174
0.7766
0.8267
0.8678
0.9005
0.9259
0.9452
0.9597
0.9705
0.9784
0.9842
0.9884
0.9915
0.9937
0.9953
0.9965
0.9974
0.9980
9
0.5000
0.5770
0.6508
0.7183
0.7778
0.8283
0.8696
0.9025
0.9280
0.9473
0.9617
0.9723
0.9801
0.9856
0.9896
0.9925
0.9946
0.9961
0.9971
0.9979
0.9984
tdf 10 11 12 13 14 15 16 17 18 19
0.0 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000
0.2 0.5773 0.5774 0.5776 0.5777 0.5778 0.5779 0.5780 0.5781 0.5781 0.5782
0.4 0.6512 0.6516 0.6519 0.6522 0.6524 0.6526 0.6528 0.6529 0.6531 0.6532
0.6 0.7191 0.7197 0.7202 0.7206 0.7210 0.7213 0.7215 0.7218 0.7220 0.7222
0.8 0.7788 0.7797 0.7804 0.7810 0.7815 0.7819 0.7823 0.7826 0.7829 0.7832
1.0 0.8296 0.8306 0.8315 0.8322 0.8329 0.8334 0.8339 0.8343 0.8347 0.8351
1.2 0.8711 0.8723 0.8734 0.8742 0.8750 0.8756 0.8762 0.8767 0.8772 0.8776
1.4 0.9041 0.9055 0.9066 0.9075 0.9084 0.9091 0.9097 0.9103 0.9107 0.9112
1.6 0.9297 0.9310 0.9322 0.9332 0.9340 0.9348 0.9354 0.9360 0.9365 0.9370
1.8 0.9490 0.9503 0.9515 0.9525 0.9533 0.9540 0.9546 0.9552 0.9557 0.9561
2.0 0.9633 0.9646 0.9657 0.9666 0.9674 0.9680 0.9686 0.9691 0.9696 0.9700
2.2 0.9738 0.9750 0.9759 0.9768 0.9774 0.9781 0.9786 0.9790 0.9794 0.9798
2.4 0.9813 0.9824 0.9832 0.9840 0.9846 0.9851 0.9855 0.9859 0.9863 0.9866
2.6 0.9868 0.9877 0.9884 0.9890 0.9895 0.9900 0.9903 0.9907 0.9910 0.9912
2.8 0.9906 0.9914 0.9920 0.9925 0.9929 0.9933 0.9936 0.9938 0.9941 0.9943
3.0 0.9933 0.9940 0.9945 0.9949 0.9952 0.9955 0.9958 0.9960 0.9962 0.9963

tdf
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
20
0.5000
0.5782
0.6533
0.7224
0.7834
0.8354
0.8779
0.9116
0.9374
0.9565
0.9704
0.9801
0.9869
0.9914
0.9945
0.9965
21
0.5000
0.5783
0.6534
0.7225
0.7837
0.8357
0.8782
0.9119
0.9377
0.9569
0.9707
0.9804
0.9871
0.9916
0.9946
0.9966
22
0.5000
0.5783
0.6535
0.7227
0.7839
0.8359
0.8785
0.9123
0.9381
0.9572
0.9710
0.9807
0.9874
0.9918
0.9948
0.9967
23
0.5000
0.5784
0.6536
0.7228
0.7841
0.8361
0.8788
0.9126
0.9384
0.9575
0.9713
0.9809
0.9876
0.9920
0.9949
0.9968
24
0.5000
0.5784
0.6537
0.7229
0.7842
0.8364
0.8791
0.9128
0.9387
0.9578
0.9715
0.9812
0.9877
0.9921
0.9950
0.9969
25
0.5000
0.5785
0.6537
0.7230
0.7844
0.8366
0.8793
0.9131
0.9389
0.9580
0.9718
0.9814
0.9879
0.9923
0.9951
0.9970
26
0.5000
0.5785
0.6538
0.7231
0.7845
0.8367
0.8795
0.9133
0.9392
0.9583
0.9720
0.9816
0.9881
0.9924
0.9952
0.9971
27
0.5000
0.5785
0.6538
0.7232
0.7847
0.8369
0.8797
0.9136
0.9394
0.9585
0.9722
0.9817
0.9882
0.9925
0.9953
0.9971
28
0.5000
0.5785
0.6539
0.7233
0.7848
0.8371
0.8799
0.9138
0.9396
0.9587
0.9724
0.9819
0.9884
0.9926
0.9954
0.9972
29
0.5000
0.5786
0.6540
0.7234
0.7849
0.8372
0.8801
0.9139
0.9398
0.9589
0.9725
0.9820
0.9885
0.9927
0.9955
0.9973
tdf 30 31 32 33 34 35 36 37 38 39
0.0 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000
0.2 0.5786 0.5786 0.5786 0.5786 0.5787 0.5787 0.5787 0.5787 0.5787 0.5787
0.4 0.6540 0.6541 0.6541 0.6541 0.6542 0.6542 0.6542 0.6543 0.6543 0.6543
0.6 0.7235 0.7236 0.7236 0.7237 0.7238 0.7238 0.7239 0.7239 0.7240 0.7240
0.8 0.7850 0.7851 0.7852 0.7853 0.7854 0.7854 0.7855 0.7856 0.7857 0.7857
1.0 0.8373 0.8375 0.8376 0.8377 0.8378 0.8379 0.8380 0.8381 0.8382 0.8383
1.2 0.8802 0.8804 0.8805 0.8807 0.8808 0.8809 0.8810 0.8811 0.8812 0.8813
1.4 0.9141 0.9143 0.9144 0.9146 0.9147 0.9148 0.9150 0.9151 0.9152 0.9153
1.6 0.9400 0.9401 0.9403 0.9404 0.9406 0.9407 0.9408 0.9409 0.9411 0.9412
1.8 0.9590 0.9592 0.9594 0.9595 0.9596 0.9598 0.9599 0.9600 0.9601 0.9602
2.0 0.9727 0.9728 0.9730 0.9731 0.9732 0.9733 0.9735 0.9736 0.9737 0.9738
2.2 0.9822 0.9823 0.9824 0.9825 0.9826 0.9827 0.9828 0.9829 0.9830 0.9831
2.4 0.9886 0.9887 0.9888 0.9889 0.9890 0.9891 0.9892 0.9892 0.9893 0.9894
2.6 0.9928 0.9929 0.9930 0.9931 0.9932 0.9932 0.9933 0.9933 0.9934 0.9935
2.8 0.9956 0.9956 0.9957 0.9958 0.9958 0.9959 0.9959 0.9960 0.9960 0.9960
3.0 0.9973 0.9974 0.9974 0.9974 0.9975 0.9975 0.9976 0.9976 0.9976 0.9977

t df 40 41 42 43 44 45 46 47 48 49
0.0 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000 0.5000
0.2 0.5788 0.5788 0.5788 0.5788 0.5788 0.5788 0.5788 0.5788 0.5788 0.5788
0.4 0.6544 0.6544 0.6544 0.6544 0.6545 0.6545 0.6545 0.6545 0.6545 0.6546
0.6 0.7241 0.7241 0.7241 0.7242 0.7242 0.7242 0.7243 0.7243 0.7243 0.7244
0.8 0.7858 0.7858 0.7859 0.7859 0.7860 0.7860 0.7861 0.7861 0.7862 0.7862
1.0 0.8383 0.8384 0.8385 0.8385 0.8386 0.8387 0.8387 0.8388 0.8388 0.8389
1.2 0.8814 0.8815 0.8816 0.8816 0.8817 0.8818 0.8819 0.8819 0.8820 0.8820
1.4 0.9154 0.9155 0.9156 0.9157 0.9157 0.9158 0.9159 0.9160 0.9160 0.9161
1.6 0.9413 0.9414 0.9415 0.9415 0.9416 0.9417 0.9418 0.9419 0.9419 0.9420
1.8 0.9603 0.9604 0.9605 0.9606 0.9606 0.9607 0.9608 0.9609 0.9609 0.9610
2.0 0.9738 0.9739 0.9740 0.9741 0.9742 0.9742 0.9743 0.9744 0.9744 0.9745
2.2 0.9832 0.9833 0.9833 0.9834 0.9834 0.9835 0.9836 0.9836 0.9837 0.9837
2.4 0.9894 0.9895 0.9895 0.9896 0.9897 0.9897 0.9898 0.9898 0.9898 0.9899
2.6 0.9935 0.9935 0.9936 0.9936 0.9937 0.9937 0.9938 0.9938 0.9938 0.9939
2.8 0.9961 0.9961 0.9962 0.9962 0.9962 0.9962 0.9963 0.9963 0.9963 0.9964
3.0 0.9977 0.9977 0.9977 0.9978 0.9978 0.9978 0.9978 0.9978 0.9979 0.9979

Table of χ2 critical values (right-tail)
The table shows cdf, p = the 1 − p quantile of χ2(df ). In R
notation cdf, p = qchisq(1-p, df).
dfp 0.010 0.025 0.050 0.100 0.200 0.300 0.500 0.700 0.800 0.900 0.950 0.975 0.990
1 6.63 5.02 3.84 2.71 1.64 1.07 0.45 0.15 0.06 0.02 0.00 0.00 0.00
2 9.21 7.38 5.99 4.61 3.22 2.41 1.39 0.71 0.45 0.21 0.10 0.05 0.02
3 11.34 9.35 7.81 6.25 4.64 3.66 2.37 1.42 1.01 0.58 0.35 0.22 0.11
4 13.28 11.14 9.49 7.78 5.99 4.88 3.36 2.19 1.65 1.06 0.71 0.48 0.30
5 15.09 12.83 11.07 9.24 7.29 6.06 4.35 3.00 2.34 1.61 1.15 0.83 0.55
6 16.81 14.45 12.59 10.64 8.56 7.23 5.35 3.83 3.07 2.20 1.64 1.24 0.87
7 18.48 16.01 14.07 12.02 9.80 8.38 6.35 4.67 3.82 2.83 2.17 1.69 1.24
8 20.09 17.53 15.51 13.36 11.03 9.52 7.34 5.53 4.59 3.49 2.73 2.18 1.65
9 21.67 19.02 16.92 14.68 12.24 10.66 8.34 6.39 5.38 4.17 3.33 2.70 2.09
10 23.21 20.48 18.31 15.99 13.44 11.78 9.34 7.27 6.18 4.87 3.94 3.25 2.56
16 32.00 28.85 26.30 23.54 20.47 18.42 15.34 12.62 11.15 9.31 7.96 6.91 5.81
17 33.41 30.19 27.59 24.77 21.61 19.51 16.34 13.53 12.00 10.09 8.67 7.56 6.41
18 34.81 31.53 28.87 25.99 22.76 20.60 17.34 14.44 12.86 10.86 9.39 8.23 7.01
19 36.19 32.85 30.14 27.20 23.90 21.69 18.34 15.35 13.72 11.65 10.12 8.91 7.63
20 37.57 34.17 31.41 28.41 25.04 22.77 19.34 16.27 14.58 12.44 10.85 9.59 8.26
21 38.93 35.48 32.67 29.62 26.17 23.86 20.34 17.18 15.44 13.24 11.59 10.28 8.90
22 40.29 36.78 33.92 30.81 27.30 24.94 21.34 18.10 16.31 14.04 12.34 10.98 9.54
23 41.64 38.08 35.17 32.01 28.43 26.02 22.34 19.02 17.19 14.85 13.09 11.69 10.20
24 42.98 39.36 36.42 33.20 29.55 27.10 23.34 19.94 18.06 15.66 13.85 12.40 10.86
25 44.31 40.65 37.65 34.38 30.68 28.17 24.34 20.87 18.94 16.47 14.61 13.12 11.52
30 50.89 46.98 43.77 40.26 36.25 33.53 29.34 25.51 23.36 20.60 18.49 16.79 14.95
31 52.19 48.23 44.99 41.42 37.36 34.60 30.34 26.44 24.26 21.43 19.28 17.54 15.66
32 53.49 49.48 46.19 42.58 38.47 35.66 31.34 27.37 25.15 22.27 20.07 18.29 16.36
33 54.78 50.73 47.40 43.75 39.57 36.73 32.34 28.31 26.04 23.11 20.87 19.05 17.07
34 56.06 51.97 48.60 44.90 40.68 37.80 33.34 29.24 26.94 23.95 21.66 19.81 17.79
35 57.34 53.20 49.80 46.06 41.78 38.86 34.34 30.18 27.84 24.80 22.47 20.57 18.51
40 63.69 59.34 55.76 51.81 47.27 44.16 39.34 34.87 32.34 29.05 26.51 24.43 22.16
41 64.95 60.56 56.94 52.95 48.36 45.22 40.34 35.81 33.25 29.91 27.33 25.21 22.91
42 66.21 61.78 58.12 54.09 49.46 46.28 41.34 36.75 34.16 30.77 28.14 26.00 23.65
43 67.46 62.99 59.30 55.23 50.55 47.34 42.34 37.70 35.07 31.63 28.96 26.79 24.40
44 68.71 64.20 60.48 56.37 51.64 48.40 43.34 38.64 35.97 32.49 29.79 27.57 25.15
45 69.96 65.41 61.66 57.51 52.73 49.45 44.34 39.58 36.88 33.35 30.61 28.37 25.90
46 71.20 66.62 62.83 58.64 53.82 50.51 45.34 40.53 37.80 34.22 31.44 29.16 26.66
47 72.44 67.82 64.00 59.77 54.91 51.56 46.34 41.47 38.71 35.08 32.27 29.96 27.42
48 73.68 69.02 65.17 60.91 55.99 52.62 47.34 42.42 39.62 35.95 33.10 30.75 28.18
49 74.92 70.22 66.34 62.04 57.08 53.67 48.33 43.37 40.53 36.82 33.93 31.55 28.94

Part I:
Solution
These questions are all multiple choice or short answer. You don’t have to show any work. Work through
them quickly. Each answer is worth 2 points.
Concept 1. answer: C. (i) and (ii)
Concept 2. answer: True
Concept 3. answer: True
Concept 4. answer: (i) Simple (ii) Composite (iii) One-sided
Concept 5. answer: B. Concept 6. answer: 2. B
Concept 7. (i) answer: A. P(A1).
(ii) answer: C. P(B2|A1).
(iii) answer: D. P(C1|B2 ∩ A1).
(iv) answer: C. A1 ∩ B2 ∩ C1.
Concept 8. answer: BAC
Concept 9. answer: p = 0.8 use minimal strategy. If you use the minimal strategy the law of large numbers
says your average winnings per bet will almost certainly be the expected winnings of one bet.

The expected value when p = 0.8 is 6. Since this is positive you’d like to make a lot of bets and let the
law of large numbers (practically) guarantee you will win an average of $6 per bet. So you use the
minimal strategy.
Concept 11. answer: B. A Bernoulli random variable takes values 0 or 1. So X is discrete. The
parameter θ can be anywhere in the continuous range [0,1]. Therefore the space of hypotheses is
continuous.
Concept 12. answer: D. By the form of the posterior pdf we know it is beta(8, 13).
Concept 13. A. True, B. False C. True
Concept 14. answer: A. Not valid B. not valid C. valid
Both the prior and posterior measure a belief in the distribution of hypotheses about the value of θ. The
frequentist does not consider them valid.
The likelihood f(x|theta) is perfectly acceptable to the frequentist. It represents the probability of data from
a repeatable experiment, i.e. measuring how late Jane is each day. Conditioning on θ is fine. This just
fixes a model parameter θ. It doesn’t require computing probabilities for θ.

Concept 15. answer: E. unknown. Frequentist methods only give probabilities for data under an assumed
hypothesis. They do not give probabilities or odds for hypotheses. So we don’t know the odds for
distribution means
Concept 16. A. Correct, This is the definition of a confidence interval.
B. Incorrect. Frequentist methods do not give probabilities for hypotheses.
C. Correct. Given θ = 0 the probability θ is in [-1, 1.5] is 100%.
Part II:

Problem 3. F(1): Since you never get more than 6 on one roll we have
Problem 4. (20) (a) Let X = score of a random student.
(b) Geometric method: We need the shaded area in the figure
to be 0.125 1 Shaded area = area of triangle = (1 − x)(4 −
4x)=0.125. 2 Solving for x we get

Analytic mehtod: We want a such that F(a)=7/8. Since f(x) is defined in two pieces we have to
compute F(a) in two pieces.
(Which we knew geometrically already.)
For a ≥ 1/2 we then have
Solving for a such that F(a)=7/8 we get
Since 5 4 is not in the range of X we have (The same answer as with the geometric
method.)

Problem 5. (a) f(x) = F/ (x)=2 − 2x on [0, 1]. Therefore

Problem 7. (a) We organize the problem in a tree. Here:
A positive expected winnings means it’s a good bet.

(d) The definition of correlation is Cor(X, Y ) So we first need to compute
the variances of X and Y .

Problem 9. (20) (a) Let X ∼ binomial(25, 0.5) = the number supporting the referendum. We
know that

where the last probability was looked up in the Z-table.
(b) The rule of thumb CI is
Problem 10.

We find the MLE for τ by taking a derivative of the log likelihood with respect to τ and setting equal to 0.
Problem 11. (a) We assume the random error terms ei are independent, have mean 0 and all have
the same variance (homoscedastic).
The least squares fit is found by setting the derivative (with respect to b) to 0,

Therefore ˆ the least squares estimate of b is
Problem 12.
(a) Since σ is unknown we use the Studentized mean
which follows a t distribution with 44 degrees of freedom.

where c0.9 and c0.1 are the right critical values from the chi-square distribution with 44 degrees of
freedom.
(b) The 80% bootstrap CI is [x − δ0 ∗ .1, x − δ0 ∗ .9], where δ0 ∗ .1 and δ0 ∗ .9 are empirical right tail
critical points for δ∗
(c) The approach in (b) is fine since it makes no assumptions about the underlying distrix − μ bution.
The approach in (a) is more problematic since √ does not follow a Student-t s/ n distribution. However
for an exponential distribution and n = 45 the approximation is not too bad.
(d) Method (b) is preferable if the underlying distribution is highly asymmetric.
Problem 13. (15) (a) Since μ = 1/p we should use the approximation
(b) Step 1. Approximate p by pˆ = 1/x. ∗ ∗ Step 2. Generate a bootstrap sample x1,...,x from geom(ˆp). n ∗
∗ Step 3. Compute p = 1/x and δ∗ = p∗ − pˆ. Repeat steps 2 and 3 many times (say 104 times. ∗ Step 4.
List all the δ and find the critical values. Let δ∗ = 0.025 critical value = 0.975 quantile. 0.025 Let δ∗ = 0.975
critical value = 0.025 quantile. 0.975 ∗ Step 5. The bootstrap confidence interval is [ˆp − δ p − δ∗ 0.025, ˆ
0.975].

Problem 14. (30) (a) We will use the standardized mean based on H0 as a test statistic:
At α = 0.05 we reject H0 if
(Or we could have used x as a test statistic and got the corresponding rejection region.)

We standardize using the given mean μ = 4
The probabilities were looked up in the z-table. We used Φ(−6.9) ≈ 0.
(We could have used much less calculation to find that the non-rejection range is x between −7σx
and −3σx from the mean μ = 4.)
Problem 15. (a) This is a normal/normal conjugate prior/likilihood update.

(c) We extend the table in order to compute the posterior predictive probability

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