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4.3 The Normal Distribution
Let the random variable X denote the number of would-be passengers who
show up for a flight. Since travelers are sometimes with their families, not every
ticket holder constitutes an independent event. Still, we can get a useful approximation to the probability that the flight is overbooked by assuming that X is binomial
with n = 178 and p = 0.9. What we are looking for is P(169 ≤ X ≤ 178), the probability that more ticket holders show up than there are seats on the plane. According to
Theorem 4.3.1 (and using the continuity correction),
P(Flight is overbooked) = P(169 ≤ X ≤ 178)
169 − 0.5 − np
X − np
178 + 0.5 − np
=P √
≤ √
np(1 − p)
np(1 − p)
np(1 − p)
168.5 − 178(0.9)
X − 178(0.9)
178.5 − 178(0.9)
≤ √
=P √
˙ P(2.07 ≤ Z ≤ 4.57) = Fz (4.57) − Fz (2.07)
From Appendix A.1, FZ (4.57) = P(Z ≤ 4.57) is equal to 1, for all practical
purposes, and the area under f Z (z) to the left of 2.07 is 0.9808. Therefore,
P(Flight is overbooked) = 1.0000 − 0.9808
= 0.0192
implying that the chances are about one in fifty that not every ticket holder will have
a seat.
Case Study 4.3.1
Research in extrasensory perception has ranged from the slightly unconventional to the downright bizarre. Toward the latter part of the nineteenth century
and even well into the twentieth century, much of what was done involved spiritualists and mediums. But beginning around 1910, experimenters moved out
of the seance parlors and into the laboratory, where they began setting up controlled studies that could be analyzed statistically. In 1938, Pratt and Woodruff,
working out of Duke University, did an experiment that became a prototype for
an entire generation of ESP research (71).
The investigator and a subject sat at opposite ends of a table. Between them
was a screen with a large gap at the bottom. Five blank cards, visible to both
participants, were placed side by side on the table beneath the screen. On the
subject’s side of the screen one of the standard ESP symbols (see Figure 4.3.4)
was hung over each of the blank cards.
Figure 4.3.4
(Continued on next page)