18.05 Practice Exam 2b
No books or calculators. You may have one side of an 8×11 sheet of paper with any information you like on it.
7 problems, 7 pages
Simplifying expressions: You don’t need to simplify complicated expressions. For ex1 2 1 2
20!
ample, you can leave ⋅ + ⋅ exactly as is. Likewise for expressions like
.
4 3 3 5
18!2!
The 𝑧, 𝑡 and 𝜒2 tables are at the end of the exam if you need them.
Problem 1. Concept questions
(a) A certain august journal publishes psychological research. They will only publish results
that are statistically significant when tested at a significance level of 0.05.
Could all of their published results be true?
Yes
No
Could all of their published results be false?
Yes
No
(b) 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.
True
False
(c) A researcher collected data that fit the criteria for a two-sided 𝑍-test. He set the
significance level at 0.05. He ran 80 trials and got a 𝑧-value of 1.7. This gave a 𝑝-value of
0.0892, so he could not reject the null hypothesis. Convinced that his alternative hypothesis
was correct he ran 80 more trials. The combined data from the 160 trials now had a 𝑧-value
of 2.1. He wrote a paper carefully describing his experiments and submitted it to the journal
in part (a).
Will the journal publish his results?
Yes
No
(d) 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)?
Problem 2. The Pareto distribution with parameter 𝛼 has range [1, ∞) and pdf
𝑓(𝑥) =
𝛼
𝑥𝛼
Suppose the data
5, 2, 3
was drawn independently from such a distribution. Find the maximum likelihood estimate
(MLE) of 𝛼.
Problem 3.
Your friend grabs a die at random from a drawer containing two 4-sided dice, one 8-sided
die, and one 12-sided die. They roll the die once and report that the result is 5.
1
18.05 Practice Exam 2b
2
(a) Make a discrete Bayes table showing the prior, likelihood, and posterior for the type
of die rolled given the data.
(b) What is the prior predictive probability of rolling a 5?
(c) What are your posterior odds that the die has 12 sides?
(d) Given the data of the first roll, what is your probability that the next roll will be a
7?
Problem 4. Everyone knows that giraffes are tall, but how much do they weigh? Let’s
suppose that the weight of male giraffes is normally distributed with mean 1200 kg and
standard deviation 200 kg.
I volunteered at the zoo and was given the task of weighing their male giraffe Beau. Now
weighing a giraffe is not easy and the process produces random errors following a N(0, 1002 )
distribution. To compensate for the inaccuracy of the scale I weighed Beau three times and
got the following measurements:
1250 kg, 1300 kg, 1350 kg
.
What is the posterior expected value of Beau’s weight?
Problem 5. Data is drawn from a binomial(5, 𝜃) distribution, where 𝜃 is unknown. Here
is the table of probabilities 𝑝(𝑥 | 𝜃) for 3 values of 𝜃:
𝑥
𝜃 = 0.5
𝜃 = 0.6
𝜃 = 0.8
0
0.031
0.010
0.000
1
0.156
0.077
0.006
2
0.313
0.230
0.051
3
0.313
0.346
0.205
4
0.156
0.259
0.410
5
0.031
0.078
0.328
You want to run a significance test on the value of 𝜃. You have the following:
Null hypothesis: 𝜃 = 0.5.
Alternate hypotheses: 𝜃 > 0.5.
Significance level: 𝛼 = 0.1.
(a) Find the rejection region.
(b) Compute the power of the test for each of the two hypotheses 𝜃 = 0.6 and 𝜃 = 0.8.
(c) Suppose you run an experiment and the data gives 𝑥 = 4. Compute the 𝑝-value of this
data.
Problem 6. You have data drawn from a normal distribution with a known variance of
16. You set up the following NHST:
• 𝐻0 : data follows a 𝑁 (2, 42 )
• 𝐻𝐴 : data follows a 𝑁 (𝜇, 42 ) where 𝜇 ≠ 2.
• Test statistic: standardized sample mean 𝑧.
• Significance level set to 𝛼 = 0.05.
18.05 Practice Exam 2b
3
You then collected 𝑛 = 16 data points with sample mean 1.5.
(a) Find the rejection region. Draw a graph indicating the null distribution and the rejection
region.
(b) Find the 𝑧-value and add it to your picture in part (a).
(c) Find the 𝑝-value for this data and decide whether or not to reject 𝐻0 in favor of 𝐻𝐴 .
Problem 7. Someone claims to have found a long lost work by Jane Austen. She asks
you to decide whether or not the book was actually written by Austen.
You buy a copy of Sense and Sensibility and count the frequencies of certain common words
on some randomly selected pages. You do the same thing for the ‘long lost work’. You get
the following table of counts.
Word
Sense and Sensibility
Long lost work
a
150
90
an
30
20
this
30
10
that
90
80
Using this data, set up and evaluate a significance test of the claim that the long lost book
is by Jane Austen. Use a significance level of 0.1.
18.05 Practice Exam 2b
4
Standard normal table of left tail probabilities.
𝑧
-4.00
-3.95
-3.90
-3.85
-3.80
-3.75
-3.70
-3.65
-3.60
-3.55
-3.50
-3.45
-3.40
-3.35
-3.30
-3.25
-3.20
-3.15
-3.10
-3.05
-3.00
-2.95
-2.90
-2.85
-2.80
-2.75
-2.70
-2.65
-2.60
-2.55
-2.50
-2.45
-2.40
-2.35
-2.30
-2.25
-2.20
-2.15
-2.10
-2.05
Φ(𝑧)
0.0000
0.0000
0.0000
0.0001
0.0001
0.0001
0.0001
0.0001
0.0002
0.0002
0.0002
0.0003
0.0003
0.0004
0.0005
0.0006
0.0007
0.0008
0.0010
0.0011
0.0013
0.0016
0.0019
0.0022
0.0026
0.0030
0.0035
0.0040
0.0047
0.0054
0.0062
0.0071
0.0082
0.0094
0.0107
0.0122
0.0139
0.0158
0.0179
0.0202
𝑧
-2.00
-1.95
-1.90
-1.85
-1.80
-1.75
-1.70
-1.65
-1.60
-1.55
-1.50
-1.45
-1.40
-1.35
-1.30
-1.25
-1.20
-1.15
-1.10
-1.05
-1.00
-0.95
-0.90
-0.85
-0.80
-0.75
-0.70
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
Φ(𝑧)
0.0228
0.0256
0.0287
0.0322
0.0359
0.0401
0.0446
0.0495
0.0548
0.0606
0.0668
0.0735
0.0808
0.0885
0.0968
0.1056
0.1151
0.1251
0.1357
0.1469
0.1587
0.1711
0.1841
0.1977
0.2119
0.2266
0.2420
0.2578
0.2743
0.2912
0.3085
0.3264
0.3446
0.3632
0.3821
0.4013
0.4207
0.4404
0.4602
0.4801
𝑧
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
1.20
1.25
1.30
1.35
1.40
1.45
1.50
1.55
1.60
1.65
1.70
1.75
1.80
1.85
1.90
1.95
Φ(𝑧)
0.5000
0.5199
0.5398
0.5596
0.5793
0.5987
0.6179
0.6368
0.6554
0.6736
0.6915
0.7088
0.7257
0.7422
0.7580
0.7734
0.7881
0.8023
0.8159
0.8289
0.8413
0.8531
0.8643
0.8749
0.8849
0.8944
0.9032
0.9115
0.9192
0.9265
0.9332
0.9394
0.9452
0.9505
0.9554
0.9599
0.9641
0.9678
0.9713
0.9744
𝑧
2.00
2.05
2.10
2.15
2.20
2.25
2.30
2.35
2.40
2.45
2.50
2.55
2.60
2.65
2.70
2.75
2.80
2.85
2.90
2.95
3.00
3.05
3.10
3.15
3.20
3.25
3.30
3.35
3.40
3.45
3.50
3.55
3.60
3.65
3.70
3.75
3.80
3.85
3.90
3.95
Φ(𝑧)
0.9772
0.9798
0.9821
0.9842
0.9861
0.9878
0.9893
0.9906
0.9918
0.9929
0.9938
0.9946
0.9953
0.9960
0.9965
0.9970
0.9974
0.9978
0.9981
0.9984
0.9987
0.9989
0.9990
0.9992
0.9993
0.9994
0.9995
0.9996
0.9997
0.9997
0.9998
0.9998
0.9998
0.9999
0.9999
0.9999
0.9999
0.9999
1.0000
1.0000
Φ(𝑧) = 𝑃 (𝑍 ≤ 𝑧) for N(0, 1).
(Use interpolation to estimate
𝑧 values to a 3rd decimal
place.)
18.05 Practice Exam 2b
5
Table of Student 𝑡 critical values (right-tail)
The table shows 𝑡𝑑𝑓, 𝑝 = the 1 − 𝑝 quantile of 𝑡(𝑑𝑓).
We only give values for 𝑝 ≤ 0.5. Use symmetry to find the values for 𝑝 > 0.5, e.g.
𝑡5, 0.975 = −𝑡5, 0.025
In R notation 𝑡𝑑𝑓, 𝑝 = qt(1-p, df).
df\p
1
2
3
4
5
6
7
8
9
10
16
17
18
19
20
21
22
23
24
25
30
31
32
33
34
35
40
41
42
43
44
45
46
47
48
49
0.005
63.66
9.92
5.84
4.60
4.03
3.71
3.50
3.36
3.25
3.17
2.92
2.90
2.88
2.86
2.85
2.83
2.82
2.81
2.80
2.79
2.75
2.74
2.74
2.73
2.73
2.72
2.70
2.70
2.70
2.70
2.69
2.69
2.69
2.68
2.68
2.68
0.010
31.82
6.96
4.54
3.75
3.36
3.14
3.00
2.90
2.82
2.76
2.58
2.57
2.55
2.54
2.53
2.52
2.51
2.50
2.49
2.49
2.46
2.45
2.45
2.44
2.44
2.44
2.42
2.42
2.42
2.42
2.41
2.41
2.41
2.41
2.41
2.40
0.015
21.20
5.64
3.90
3.30
3.00
2.83
2.71
2.63
2.57
2.53
2.38
2.37
2.36
2.35
2.34
2.33
2.32
2.31
2.31
2.30
2.28
2.27
2.27
2.27
2.27
2.26
2.25
2.25
2.25
2.24
2.24
2.24
2.24
2.24
2.24
2.24
0.020
15.89
4.85
3.48
3.00
2.76
2.61
2.52
2.45
2.40
2.36
2.24
2.22
2.21
2.20
2.20
2.19
2.18
2.18
2.17
2.17
2.15
2.14
2.14
2.14
2.14
2.13
2.12
2.12
2.12
2.12
2.12
2.12
2.11
2.11
2.11
2.11
0.025
12.71
4.30
3.18
2.78
2.57
2.45
2.36
2.31
2.26
2.23
2.12
2.11
2.10
2.09
2.09
2.08
2.07
2.07
2.06
2.06
2.04
2.04
2.04
2.03
2.03
2.03
2.02
2.02
2.02
2.02
2.02
2.01
2.01
2.01
2.01
2.01
0.030
10.58
3.90
2.95
2.60
2.42
2.31
2.24
2.19
2.15
2.12
2.02
2.02
2.01
2.00
1.99
1.99
1.98
1.98
1.97
1.97
1.95
1.95
1.95
1.95
1.95
1.94
1.94
1.93
1.93
1.93
1.93
1.93
1.93
1.93
1.93
1.93
0.040
7.92
3.32
2.61
2.33
2.19
2.10
2.05
2.00
1.97
1.95
1.87
1.86
1.86
1.85
1.84
1.84
1.84
1.83
1.83
1.82
1.81
1.81
1.81
1.81
1.80
1.80
1.80
1.80
1.79
1.79
1.79
1.79
1.79
1.79
1.79
1.79
0.050
6.31
2.92
2.35
2.13
2.02
1.94
1.89
1.86
1.83
1.81
1.75
1.74
1.73
1.73
1.72
1.72
1.72
1.71
1.71
1.71
1.70
1.70
1.69
1.69
1.69
1.69
1.68
1.68
1.68
1.68
1.68
1.68
1.68
1.68
1.68
1.68
0.100
3.08
1.89
1.64
1.53
1.48
1.44
1.41
1.40
1.38
1.37
1.34
1.33
1.33
1.33
1.33
1.32
1.32
1.32
1.32
1.32
1.31
1.31
1.31
1.31
1.31
1.31
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
1.30
0.200
1.38
1.06
0.98
0.94
0.92
0.91
0.90
0.89
0.88
0.88
0.86
0.86
0.86
0.86
0.86
0.86
0.86
0.86
0.86
0.86
0.85
0.85
0.85
0.85
0.85
0.85
0.85
0.85
0.85
0.85
0.85
0.85
0.85
0.85
0.85
0.85
0.300
0.73
0.62
0.58
0.57
0.56
0.55
0.55
0.55
0.54
0.54
0.54
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.53
0.400
0.32
0.29
0.28
0.27
0.27
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.26
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.500
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
18.05 Practice Exam 2b
6
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).
df\p
1
2
3
4
5
6
7
8
9
10
16
17
18
19
20
21
22
23
24
25
30
31
32
33
34
35
40
41
42
43
44
45
46
47
48
49
0.010
6.63
9.21
11.34
13.28
15.09
16.81
18.48
20.09
21.67
23.21
32.00
33.41
34.81
36.19
37.57
38.93
40.29
41.64
42.98
44.31
50.89
52.19
53.49
54.78
56.06
57.34
63.69
64.95
66.21
67.46
68.71
69.96
71.20
72.44
73.68
74.92
0.025
5.02
7.38
9.35
11.14
12.83
14.45
16.01
17.53
19.02
20.48
28.85
30.19
31.53
32.85
34.17
35.48
36.78
38.08
39.36
40.65
46.98
48.23
49.48
50.73
51.97
53.20
59.34
60.56
61.78
62.99
64.20
65.41
66.62
67.82
69.02
70.22
0.050
3.84
5.99
7.81
9.49
11.07
12.59
14.07
15.51
16.92
18.31
26.30
27.59
28.87
30.14
31.41
32.67
33.92
35.17
36.42
37.65
43.77
44.99
46.19
47.40
48.60
49.80
55.76
56.94
58.12
59.30
60.48
61.66
62.83
64.00
65.17
66.34
0.100
2.71
4.61
6.25
7.78
9.24
10.64
12.02
13.36
14.68
15.99
23.54
24.77
25.99
27.20
28.41
29.62
30.81
32.01
33.20
34.38
40.26
41.42
42.58
43.75
44.90
46.06
51.81
52.95
54.09
55.23
56.37
57.51
58.64
59.77
60.91
62.04
0.200
1.64
3.22
4.64
5.99
7.29
8.56
9.80
11.03
12.24
13.44
20.47
21.61
22.76
23.90
25.04
26.17
27.30
28.43
29.55
30.68
36.25
37.36
38.47
39.57
40.68
41.78
47.27
48.36
49.46
50.55
51.64
52.73
53.82
54.91
55.99
57.08
0.300
1.07
2.41
3.66
4.88
6.06
7.23
8.38
9.52
10.66
11.78
18.42
19.51
20.60
21.69
22.77
23.86
24.94
26.02
27.10
28.17
33.53
34.60
35.66
36.73
37.80
38.86
44.16
45.22
46.28
47.34
48.40
49.45
50.51
51.56
52.62
53.67
0.500
0.45
1.39
2.37
3.36
4.35
5.35
6.35
7.34
8.34
9.34
15.34
16.34
17.34
18.34
19.34
20.34
21.34
22.34
23.34
24.34
29.34
30.34
31.34
32.34
33.34
34.34
39.34
40.34
41.34
42.34
43.34
44.34
45.34
46.34
47.34
48.33
0.700
0.15
0.71
1.42
2.19
3.00
3.83
4.67
5.53
6.39
7.27
12.62
13.53
14.44
15.35
16.27
17.18
18.10
19.02
19.94
20.87
25.51
26.44
27.37
28.31
29.24
30.18
34.87
35.81
36.75
37.70
38.64
39.58
40.53
41.47
42.42
43.37
0.800
0.06
0.45
1.01
1.65
2.34
3.07
3.82
4.59
5.38
6.18
11.15
12.00
12.86
13.72
14.58
15.44
16.31
17.19
18.06
18.94
23.36
24.26
25.15
26.04
26.94
27.84
32.34
33.25
34.16
35.07
35.97
36.88
37.80
38.71
39.62
40.53
0.900
0.02
0.21
0.58
1.06
1.61
2.20
2.83
3.49
4.17
4.87
9.31
10.09
10.86
11.65
12.44
13.24
14.04
14.85
15.66
16.47
20.60
21.43
22.27
23.11
23.95
24.80
29.05
29.91
30.77
31.63
32.49
33.35
34.22
35.08
35.95
36.82
0.950
0.00
0.10
0.35
0.71
1.15
1.64
2.17
2.73
3.33
3.94
7.96
8.67
9.39
10.12
10.85
11.59
12.34
13.09
13.85
14.61
18.49
19.28
20.07
20.87
21.66
22.47
26.51
27.33
28.14
28.96
29.79
30.61
31.44
32.27
33.10
33.93
0.975
0.00
0.05
0.22
0.48
0.83
1.24
1.69
2.18
2.70
3.25
6.91
7.56
8.23
8.91
9.59
10.28
10.98
11.69
12.40
13.12
16.79
17.54
18.29
19.05
19.81
20.57
24.43
25.21
26.00
26.79
27.57
28.37
29.16
29.96
30.75
31.55
0.990
0.00
0.02
0.11
0.30
0.55
0.87
1.24
1.65
2.09
2.56
5.81
6.41
7.01
7.63
8.26
8.90
9.54
10.20
10.86
11.52
14.95
15.66
16.36
17.07
17.79
18.51
22.16
22.91
23.65
24.40
25.15
25.90
26.66
27.42
28.18
28.94
MIT OpenCourseWare
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18.05 Introduction to Probability and Statistics
Spring 2022
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