Page 146 - Applied Statistics Using SPSS, STATISTICA, MATLAB and R

126 4 Parametric Tests of Hypotheses

2
Let the sample variance, computed in the n-sized sample, be s . The test of a
single variance is based on Property 5 of B.2.7, which states a chi-square sampling
distribution for the ratio of the sample variance, s 2 X ≡ s 2 (X ) , and the hypothesised
variance:

n
s 2 X /σ 2 ~ χ n 2 − 1 /( − ) 1 . 4.5

Example 4.4
Q: Consider the meteorological dataset and assume that a typical standard
deviation for the yearly maximum temperature in the Portuguese territory is
σ = 2.2º. This standard deviation reflects the spatial dispersion of maximum
temperature in that territory. Also, consider the variable T81, representing the 1981
sample of 25 measurements of maximum temperature. Is there enough evidence,
supported by the 1981 sample, leading to the conclusion that the standard deviation
in 1981 was atypically high?
A: The test is formalised as:

H 0:σ T 2 81 ≤ . 4 84 .
H 1:σ T 2 81 > . 4 84 .

2
The sample variance in 1981 is s = 7.5. Since the sample size of the example is
n = 25, for a 5% level of significance we determine the percentile:

χ 2 24 . 0 , 95 = 36 . 42 .

Thus, χ 2 24 . 0 , 95 / 24 = . 1 52 .
This determination can be done in a variety of ways, as previously mentioned
(in Commands 3.3): using the probability calculators of SPSS and STATISTICA,
using MATLAB chi2inv function or R q chisq function, consulting tables (see
2
D.4 for P(χ > x) = 0.05), etc.
Since s 2 /σ 2 = 7.5 / 4.84 = 1. 55 lies in the critical region [1.52, +∞[, we
conclude that the test is significant, i.e., there is evidence supporting the rejection
of the null hypothesis at the 5% level of significance.

4.4 Inference on Two Populations

4.4.1 Testing a Correlation
When analysing two associated sample variables, one is often interested in
knowing whether the sample provides enough evidence that the respective random
variables are correlated. For instance, in data classification, when two variables are