This cumulative review problem uses material from Chapters 3, 5, and 10. Recall that the Poisson distribution deals with rare events. Death from the kick of a horse is a rare event, even in the Prussian army. The following data are a classic example of a Poisson application to rare events.
A reproduction of the original data can be found in C. P. Winsor, Human Biology, Vol. 19, pp. 154-161. The data represent the number of deaths from the kick of a horse per army corps per year for 10 Prussian army corps for 20 years (1875-1894). Let x represent the number of deaths and f the frequency of x deaths.
(a) First, we t the data to a Poisson distribution (see Section 5.4).
Poission distribution:
Where λ ≈ x̅ (sample mean of x values) From our study of weighted averages (see section 3.1),
Verify that x̅ ≈ 0.61 For the category 3 or more, use 3.
(b) Now we have
1, 2, 3. p
Find P(0), P(1), P(2), and P(3 ≤ x). Round to three places after the decimal.
(c) The total number of observations is ∑f = 200.
For a given x, the expected frequency of x deaths is 200P(x). The following table gives the observed frequencies O and the expected frequencies E = 200P(x).
(d) State the null and alternate hypotheses for a chi-square goodness-of- t test. Set the level of signifi cance to be a = 0.01. Find the P-value for a goodness-of- t test. Interpret your conclusion in the context of this application. Is there reason to believe that the Poisson distribution fits the raw data pro
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