1. Goodness-of-Fit Test: Rare Events 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.

x 0 1 2 3 or more f 109 65 22 4

(a) First, we fit the data to a Poisson distribution

(see Section 5.4).

Poission distribution: P1x2 5 ellx x!

where l < x (sample mean of x values)

From our study of weighted averages (see Section 3.1), x 5

©xf

©f Verify that x < 0.61 Hint: For the category 3 or more, use 3.

(b) Now we have P1x2 5 e0.6110.612x x!

for x 5 0, 1, 2, 3. p Find P(0), P(1), P(2), and P13  x2. Round to three places after the decimal.

(c) The total number of observations is ©f 5 200.

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 5 200P1x2.
x O 5 f E  200P(x)
0 109 20010.5432 5 108.6 1 65 20010.3312 5 66.2 2 22 20010.1012 5 20.2 3 or more 4 20010.0252 5 5 Compute x2 5 ©
1O  E22 E

(d) State the null and alternate hypotheses for a chisquare goodness-of-fit test. Set the level of significance to be a 5 0.01. Find the P-value for a goodness-of-fit test. Interpret your conclusion in the context of this application. Is there reason to believe that the Poisson distribution fits the raw data provided by the Prussian army? Explain.