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Problem
1
2
3
4
5
6
7
8
9
10
Marks
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
Topic
Bayesian
Poisson
Distributions
Test for μ; σ known
Test for μ; σ unknown
Nonparametric
Regression
Joint Density
Source of data
Probability
25 total marks
Chapter
5
4
4
5
7
14
11
3
10
2
Our prior distribution, for the probability of tails with a coin, is:
We toss the coin twice, and we get tails both times.
Under the resulting posterior distribution, complete the table:
p
0.28
0.53
0.78
f(p)
f(p) = 1; ie uniform on (0,1)
Hurricane occurrences, per year, are Poisson-distributed with λ = 1.9
Complete the table:
Y
H
1
1
1
10
100
1
2
3
19
163
P(at least H hurricanes in Y years)
(you are allowed 1% tolerance)
(you are allowed 1% tolerance)
A random variable, X, is known to have a mean of 6
Complete the table:
If X follows
Uniform(0,n)
Poisson
Geometric
Exponential
Chi-square
Binomial(p = .1)
then Variance(X) is
H0:
population mean =
The underlying population is normal with variance =
Test H0 using a 5% chance of Type 1 error (two tails)
732
312
sample mean
5% critical values:
low
high
decision
(“reject” or “cannot reject”)
Sample (n = 40); Problems 4-5-6 all use this same sample
702
734
735
722
732
732
733
740
715
699
726
710
725
730
735
721
738
733
720
728
731
735
724
724
734
736
732
709
734
732
714
695
720
735
741
736
777
660
712
709
H0:
population mean =
732
The underlying population is normal with unknown variance
Test H0 using a 5% chance of Type 1 error (two tails)
sample mean
5% critical values:
low
high
decision
(“reject” or “cannot reject”)
Sample (n = 40); Problems 4-5-6 all use this same sample
702
734
735
722
732
732
733
740
715
699
726
710
725
730
735
721
738
733
720
728
731
735
724
724
734
736
732
709
734
732
714
695
720
735
741
736
777
660
712
709
H0: population median =
732
We do not know anything about the underlying population
Test H0 using a 5% chance of Type 1 error (two tails)
test statistic
5% critical values:
low
high
decision
(“reject” or “cannot reject”)
Sample (n = 40); Problems 4-5-6 all use this same sample
702
734
735
722
732
732
733
740
715
699
726
710
725
730
735
721
738
733
720
728
731
735
724
724
734
736
732
709
734
732
714
695
720
735
741
736
777
660
712
709
Using the x and y data below, run a linear regression to find
intercept
slope
s2
std dev of slope estimator
p-value for testing H0: slope = 0 using 2 tails
x
y
4234
3811
3430
3087
2778
2500
2250
2025
1823
1641
1477
1329
1196
1076
968
871
784
706
635
572
515
256.9
211.6
238.1
211.8
194.1
124.5
187.3
110.5
233.1
150.3
124.7
41.2
182.1
118.1
31.9
114.3
144.9
59.7
126.9
43.9
136.3
A speedboat has two engines, A and B.
The joint density function of the lifetimes of the two engines, measured in hours, is:
f(A, B) = .125 (A + B)

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