Question

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. The data represent the number of deaths from the kick of a horse per army corps per year for 10 Prussian army corps over a period of time. Let x represent the number of deaths and f the frequency of x deaths. x 0 1 2 3 or more f 101 67 28 4 (a) First, we fit the data to a Poisson distribution. The Poission distribution states P(x) = e−λλx x! , where λ ≈ x (sample mean of x values). From our study of weighted averages, we get the following. x = Σxf Σf Verify that x = 0.675. Hint: For the category 3 or more, use 3. x = (b) Now we have P(x) = e−0.675(0.675)x x! for x = 0, 1, 2, 3 . Find P(0), P(1), P(2), and P(3 ≤ x). Round to three places after the decimal. P(0) = P(1) = P(2) = P(3 ≤ x) = (c) The total number of observations is Σf = 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). x 0 = f E = 200P(x) 0 101 200(0.509) = 101.8 1 67 200(0.344) = 68.8 2 28 200(0.116) = 23.2 3 or more 4 200(0.031) = 6.2 Compute χ2 = Σ  (O − E)2 E using the values in the table. (Round your answer to two decimal places.)

Answer 1

(a) Sample mean of x=(101*0+67*1+28*2+4*3)/(101+67+28+4)=0.675

(b) P(0)=exp(-0.675)=0.509

P(1)=exp(-0.675)*(0.675)=0.344

P(2)=exp(-0.675)*(0.675)²/2!=0.116

P( x is greater than equal to 3 )=1-P(0)-P(1)-P(2)=1-0.509-0.344-0.116=0.031

(c) E(x)= Expected frequency for the value x=200*P(x); x=0,1,2,...

Expected frequency for x=0=E(0)=200*P(0)=200*0.509=101.8

Expected frequency for x=1=E(1)=200*P(1)=200*0.344=68.8

Expected frequency for x=2=E(2)=200*P(2)=200*0.116=23.2

Expected frequency for x=3 or more=E( x is greater than equal to 3)=200*P(3 ≤ x)=200*0.031=6.2

(d) Value of chi-square=(101-101.8)²/101.8+(67-68.8)²/68.8+(28-23.2)²/23.2+(4-6.2)²/6.2=1.83