Page 219 - Introduction to Mineral Exploration

202 M.K.G. WHATELEY & B. SCOTT

95% chance that the population mean is in this TABLE 10.2 Two-sided and one-sided confidence
calculated interval and five chances in a hun- intervals at 95% probability.
dred that it is outside. The relationship is:
Two-sided One-sided
. 196σ . 1 96σ
>
−
+
>
X Y Y X 10.55% Zn 10.55% Zn
n n S 3.60% 3.60%
n 121 121
This method is correct but useless because we n 11 11
do not know σ and the expression becomes: (S/ n) 0.33 0.33

1.96 NA
t 2.5%
S S
+
>
−
>
X t 25 Y X t 25 t 5% NA 1.65
.%
.%
n n t(S/ n) 0.65 0.54
Upper limit 11.20% NA
where σ is replaced by its best estimate S, and Lower limit 9.90% 10.01%
the normal distribution is replaced by a dis-
tribution which has a wider spread than the 1. The total number of samples needed to halve the
standard error of the mean (S/ n), and thus the range of the

normal case. In this t distribution (Davis 2003) confidence interval, is 484 [= (2 × 11) ].
2
the shape of its distribution curve varies ac- 2. With a one-sided confidence interval all the risk is
cording to the number of samples (n) used to es- placed into one side of the interval. Consequently the
timate the population parameters (Table 10.1). calculated limit is closer to the x statistic than to the
As n increases, however, the curve resembles corresponding limit for a two-sided interval. In the above
example the upper limit could have been calculated instead
that of the normal case and for all practical pur- of the lower.
poses for more than 50 samples the curves are 3. See Table 10.1 for t values and Fig. 10.4 for a graphical
identical. Examples of the use of this formula representation.
are given in Table 10.2 and Fig. 10.4. NA, not appropriate.
The width (i.e. spread) of a two-sided interval
estimate about the sample arithmetic mean
depends on the term:
S
=
t
n
Two-sided
Frequency
TABLE 10.1 Critical values of t for different sample 2.5% 95% of values 2.5%
numbers and confidence levels.
9.90% X 11.20% % Zinc
Number of Confidence levels 10.55%
samples (n)
80% 90% 95% 99%
One-sided
1 3.08 6.3 12.71 31.82
5 1.48 2.02 2.57 3.37 Frequency
10 1.37 1.81 2.23 2.76 5%
25 1.32 1.71 2.06 2.49 95% of values
40 1.30 1.68 2.02 2.42
60 1.30 1.67 2.00 2.39 10.01% X % Zinc
120 1.29 1.66 1.98 2.36 10.55%
>120 1.28 1.65 1.96 2.33
FIG. 10.4 Graphical representation of a two-sided
The number of samples (n) is referred to as “the degrees of and one-sided confidence interval at 95% confidence
freedom.” level. See Table 10.2 for calculations.