Page 144 - Applied Statistics Using SPSS, STATISTICA, MATLAB and R

124 4 Parametric Tests of Hypotheses

Using the t percentile value and the standard error, the non-critical region is the
interval [37.5 – 2.06×0.548, 37.5 + 2.06×0.548] = [36.4, 38.6]. As the sample mean
x = 39.8 falls outside this interval, we also decide the rejection of the null
hypothesis at that level of significance.

Example 4.3
Q: Redo previous Example 4.2 in order to assess whether 1981 was a year with an
atypically large average maximum temperature.

A: We now perform a one-sided test, using the alternative hypothesis:

H 1: µ T 81 > 37 5 . .

The critical region for this one-sided test, expressed in terms of X , is:

C = [ µ 0 + t df 1 , −α / s , n ∞
[
.

Since t 24 . 0 , 95 = . 1 71, we have C = [37.5 + 1.71×0.548, ∞ [ = [38.4, ∞ [. Once
again, the sample mean falls into the critical region leading to the rejection of the
null hypothesis. Note that the alternative hypothesis µ T81 = 39.8 in this Example
4.3 corresponds to a large effect size, E s = 0.84, to which also corresponds a high
power (larger than 95%; see Exercise 4.2).

Commands 4.1. SPSS, STATISTICA, MATLAB and R commands used to
perform the single mean t test.

SPSS Analyze; Compare Means; One-Sample T Test

STATISTICA Statistics; Basic Statistics and Tables;
t-test, single sample

MATLAB [h,sig,ci]=ttest(x,m,alpha,tail)

R t.test(x, alternative = c("two.sided",
"less", "greater"), mu, conf.level)

When using a statistical software product one obtains the probability of observing a
*
value at least as large as the computed test statistic t n(x) ≡ t , assuming the null
hypothesis. This probability is the so-called observed significance. The test
decision is made comparing this observed significance with the chosen level of
significance. Note that the published value of p corresponds to the two-sided
observed significance. For instance, in the case of Table 4.3, the observed level of
significance for the one-sided test is half of the published value, i.e., p = 0.00015.