Question

2. The Prussian horse-kick data: The derivation of the Poisson distribution that we did in class is due to Poisson. However, this distribution did not see much application until a text by Bortkiewicz in 1898. One famous example from that text is the use of the “Prussian horse-kick data" to illustrate how the Poisson distribution may help evaluate whether rare events are really occurring independently or randomly. Bortkiewicz studied the distribution of 122 men kicked to death by horses among ten Prussian army corps over 20 years. (for a total of 200 groups in the sample): Deaths: 0 1 # of groups with this many deaths: 109 65 22 3 1 Relative frequency/observed percentage of groups with this many deaths: 0.545 0.325 0.11 0.015 0.005 2 3 4 (a) Let X = "# of deaths in a group that occur by horse-kick”, and use the values from the first and third columns of the table to calculate a weighted average/mean for X. Then, using this mean for u, calculate the probability according to the Poisson distribution of having exactly x many such deaths for x = 0,1,2,3, and 4. Finally, multiply each of these values of p(x) by 200 to to get an expected number of groups to have each number of deaths. (b) Without seeing all the data, suppose we are told that one of the corps, in one of the years, had 4 men die of horse-kicks. Could that really just be something that randomly happened? Or is it more likely that something else was wrong with that group that caused it? Since the Poisson distribution assumes each occurrence of the rare event is independent from the others, the observed data matching the predictions of the Poisson distribution is a good indication that something seemingly remarkable was indeed just a random outcome. So, what do you think? Was there something wrong with the corps who had 4 men die from horse-kicks that year? Or were they just an unlucky group?

Any help?

Answer 1

ANSWER::

Deaths (x)	# of groups with this many deaths (f)	xf	P(X<x)	Expected (E )=P*200	(f-E)^2/E
0	109	0	0.5434	108.6702	0.0010
1	65	65	0.3314	66.2888	0.0251
2	22	44	0.1011	20.2181	0.1570
3	3	9	0.0206	4.1110	0.3003
4	1	4	0.0031	0.6269	0.2220
	200	122			0.7054

The mean of the geven data, n = Eniti = 122 = = Edi 200 =0.61 PXEA) = è ? 720 20 X =0 The chir square valice = is = 0.7054 Cher square = 0.7054 with 5124 The degru of p-value for freedom is 0,9506.

Here, the P-value is 0.9506 and greater than 0.05 level of significance. Therefore, the given data follows a Poisson distribution. Hence, we can say that without seeing all the data, 4 men die of horse-kicks in one of the years is randomly happened. So, there was wrong with the crops who had 4 men from horse-kicks that years because some year may not happen 4 men die due to horse-kicks.

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