For the variable PHE, we reject the hypothesis of equal variances; therefore, we
look to the
t
test results in the bottom row, which are based on formulas 4.14 and
4.15. The null hypothesis of equal means is also rejected, now with higher
significance since
p
= 0.002. Note that the means of the two groups are more than
three times the standard error apart.
±
Figure 4.10.
a) Window of STATISTICA Power Analysis module used for the
specifications of Example 4.10; b) Results window for the previous specifications.
Example 4.10
Q: Compute the power for the ASP variable (aspartame content) of the previous
Example 4.9, for a one-sided test at 5% level, assuming that as an alternative
hypothesis white wines have more aspartame content than red wines. Determine
what is the minimum distance between the population means that guarantees a
power above 90% under the same conditions as the studied samples.
A: The one-sided test for this RS situation (see section 4.2) is formalised as:
H
0
:
µ
1
≤
µ
2
;
H
1
:
µ
1
>
µ
2
.
(White wines have more aspartame than red wines.)
The observed level of significance is half of the value shown in Table 4.6, i.e.,
p
= 0.011; therefore, the null hypothesis is rejected at the 5% level. When the data
analyst investigated the ASP variable, he wanted to draw conclusions with
protection against a Type II Error, i.e., he wanted a low probability of wrongly not
detecting the alternative hypothesis when true. Figure 4.10a shows the

4.4 Inference on Two Populations
137
STATISTICA specification window needed for the power computation. Note the
specification of the one-sided hypothesis. Figure 4.10b shows that the power is
very high when the alternative hypothesis is formalised with population means
having the same values as the sample means; i.e., in this case the probability of
erroneously deciding H
0
is negligible. Note the computed value of the standardised
effect (
µ
1
–
µ
2
)/s
= 2.27, which is very large (see section 4.2).
Figure 4.11 shows the power curve depending on the standardised effect, from
where we see that in order to have at least 90% power we need
E
s
= 0.75, i.e., we
are guaranteed to detect aspartame differences of about 2 mg/l apart (precisely,
0.75
×
2.64 = 1.98).
±
0.0
0.5
1.0
1.5
2.0
2.5
.3
.4
.5
.6
.7
.8
.9
1.0
Power
Standardized Effect (Es)
Power vs. Es (N1 = 30, N2 = 37, Alpha = 0.05)
Figure 4.11.
Power curve, obtained with STATISTICA, for the wine data
Example 4.10.
Commands 4.3.
SPSS, STATISTICA, MATLAB and R commands used to
perform the two independent samples
t
test.
SPSS
Analyze; Compare Means; Independent
Samples T Test
STATISTICA
Statistics; Basic Statistics and Tables;
t-test, independent, by groups
MATLAB
[h,sig,ci] = ttest2(x,y,alpha,tail]
R
t.test(formula, var.equal = FALSE)
The MATLAB function
ttest2
works in the same way as the function
ttest
described in 4.3.1, with
x
and
y
representing two independent sample vectors. The
function
ttest2
assumes that the variances of the samples are equal.

138
4 Parametric Tests of Hypotheses
The R function
t.test
, already mentioned in Commands 4.1, can also be used to
perform the two-sample
t
test. This function has several arguments the most
important of which are mentioned above. Let us illustrate its use with Example 4.9.

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- Hsin Chu
- Normal Distribution, Spss, The American, The Lottery, Probability theory