Statistical Procedures for Fuzzy Data in Medical Research - Fuzziness and Medicine - page 280

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Happiness

Fig. 19.4 Hypothetical fuzzy responses obtained from hospital 1 (show in gray) and from
hospital 2 (shown in black)

We ask a group of patients at the hospital to report their degrees of happiness us-
ing fuzzy sets. We may obtain fuzzy responses similar to the ones plotted with gray
lines in Figure 19.3. By applying a one-sample t test to the data, we can determine
whether to reject H 0 with a predetermined significance level. For mathematical de-
tails, see Körner [10], Montenegro et al. [13], and González-Rodríguez et al. [8].

19.3.2 Two-Sample t Te s t s

Montenegro et al. [12] and González-Rodríguez et al. [7] established a two-indepen-
dent-sample test of equality of fuzzy means, and González-Rodríguez et al. [7] de-
veloped a paired-sample test of the same type. These are considered extensions of
classical two-sample t tests to fuzzy data, and they assess whether the means of two
populations of fuzzy sets are different.
We will explain how to perform an independent-sample t test. Suppose that there
are two hospitals, call them hospital 1 and hospital 2, and that we want to know
whether the mean degree of happiness at hospital 1 is the same as that at hospital
2. Let μ 2 denote the mean degrees of happiness at hospitals 1 and 2, re-

spectively. The null hypothesis H 0 is that

μ 1 and

μ 1 = μ 2 . The alternative hypothesis H A is

that μ 1 = μ 2 . We collect fuzzy data from the two hospitals. Suppose that we obtain

fuzzy responses plotted in Figure 19.4. Assume that responses obtained from hospi-
tal 1 are shown in gray and that those obtained from hospital 2 are shown in black.
By applying an independent-sample t test to the data, we can determine whether to
reject H 0 with a predetermined significance level. See Montenegro et al. [12] and
González-Rodríguez et al. [7] for mathematical details.