GEOL0061: Statistics for Geoscientists
Answer all questions. You may use your notes and computer. Include any R code with your answer. Data sets
can be copied and pasted into your code. Solutions can be uploaded in any format accepted by WISEflow.
1. [8 marks] Compute the compositional average of the ternary dataset shown in Table 1.
Table 1: Data for question 1.
X Y Z
61.7 27.9 10.30
67.6 21.5 10.90
75.2 17.2 7.69
72.2 15.4 12.40
68.2 20.4 11.40
2. [6 marks] 20% of the rocks in a field area contain pyrite. 0.1% of the rocks in the same field area
contain gold. 90% of the gold-bearing rocks in the field area contain pyrite. What is the probability that a
pyrite-bearing rock from the field area contains gold
3. [8 marks] Figure 1 shows a nested set of decision trees with indication of the 10-fold cross-validation errors
(Xerr) and their standard errors (s[Xerr]). Which tree has optimal predictive power Explain your reasoning
in at most five sentences.
| (1)
Xerr =0.609
s[Xerr] =0.0518
| (2)
Xerr =0.587
s[Xerr] =0.0514
| (3)
Xerr =0.522
s[Xerr] =0.0501
| (4)
Xerr =0.522
s[Xerr] =0.0501
| (5)
Xerr =0.5
s[Xerr] =0.0495
| (6)
Xerr =0.514
s[Xerr] =0.0499
| (7)
Xerr =0.449
s[Xerr] =0.0481
| (8)
Xerr =0.471
s[Xerr] =0.0487
| (9)
Xerr =0.486
s[Xerr] =0.0492
Figure 1: Figure for question 3.
4. [7 marks] A geological consultant has predicted that there is a 40% chance of finding oil in a wildcat survey.
5 holes have been drilled and still no oil has been found. Was the geological consultant’s assessment wrong
Why
1
5. [8 marks] Consider the following equation:
z = 4 + 3 e
3x + 5 (xy)
2
where x = 3 ± 0.2 and y = 4 ± 0.2. Estimate z and propagate its uncertainty, assuming that the uncertainties
of x and y are uncorrelated.
6. [6 marks] On Figure 2, match each of the three probability density functions (a–c) with one of the six
cumulative distribution functions (1–6).
0.0 0.5 1.0
a)
0.0 0.5 1.0
b)
0.0 0.5 1.0
c)
0.0 0.5 1.0
1)
0.0 0.5 1.0
2)
0.0 0.5 1.0
3)
0.0 0.5 1.0
4)
0.0 0.5 1.0
5)
0.0 0.5 1.0
6)
Figure 2: Figure for question 6.
7. Consider the following set of bivariate data:
Table 2: Data for question 7.
1 2 3 4 5 6 7 8 9 10
x 25 18 25 68 22 26 28
25 27 28
y 28 24 27 77 28 22 28 21 24 26
a) [4 marks] What is the correlation coefficient Is the correlation statistically significant
b) [4 marks] Assess the scientific significance of this particular correlation (or lack thereof) in one or two
sentences.
8. In the UK, on average 2 people are killed by lightning per year.
a) [3 marks] What is the chance that more than 2 people will be killed by lightning next year
b) [3 marks] What is the highest number of annual deaths by lightning that we can expect with 95%
certainty
9. [8 marks] A geologic map contains 4 stratigraphic units. You have 6 coloured pencils. In how many ways
can you choose your colours
10. [8 marks] The principal component biplot of Figure 3 displays 10 samples analysed for 5 chemical elements.
Interpret this plot in no more than half a page of text.
2
0.2 0.0 0.2 0.4 0.6
PC1
1
2
3
4
5
6 7
8
9
10
5 0 5 10
Zn
Cu
Pb
Ti
Sr
Figure 3: Figure for question 10
11. [7 marks] Consider 200 numbers drawn from a normal distribution with mean of 3 and standard deviation
of 3. What is the probability that the sum of these numbers is greater than 654
12. [11 marks] Let Pv be the probability that a vaccinated person gets the Covid-19 virus, and let Pn be the
probability that a non-vaccinated person gets the Covid-19 virus. Then the effectiveness (E) of the vaccine is
defined as E = [Pn Pv]/Pn. Suppose that half of the participants in a clinical trial get the vaccine, and the
other half get a placebo. At the end of the trial, Nv=10 of the vaccinated people caught the virus and so did
Nn=30 people in the placebo group. Test whether the vaccine is more than 50% effective (i.e. test if E > 0.5)
at a 95% confidence level. Repeat the test for Nv=20 and Nn=60 and again for Nv=30 and Nn=90. Discuss
the implications in a few sentences.
13. [9 marks] A loaded six-sided die has a 1/7 chance of landing on 1 (i.e., p( ) = 1/7). This die is thrown
100 times to test the null hypothesis H0 : p( ) = 1/6 versus the alternative hypothesis Ha : p( ) < 1/6.
Using an α = 0.05 significance level, what is the probability of incurring a type-2 error
The total number of marks is 100
3
0.2 0.0 0.2 0.4 0.6
PC2
5 0
5 10