Suppose that Y1 Y2 Y3 and Y4 have a multinomial distribution

Suppose that Y1, Y2, Y3, and Y4 have a multinomial distribution with n trials and probabilities p1, p2, p3, and p4 for the four cells. Just as in the binomial case, any linear combination of Y1, Y2, Y3, and Y4 will be approximately normally distributed for large n.

a) Determine the variance of Y1-Y2. [Hint: Recall that the random variables Yi are dependent.]

b) A study of attitudes among residents of Florida with regard to policies for handling nuisance alligators in urban areas showed the following. Among 500 people sampled and presented with four management choices, 6% said the alligators should be completely protected, 16% said they should be destroyed by wildlife officers, 52% said they should be relocated live, and 26% said that a regulated commercial harvest should be allowed. Estimate the difference between the population proportion favoring complete protection and the population proportion favoring destruction by wildlife officers. Use a confidence coefficient of .95.

Solution

E(Y1-Y2) = E(Y1)-E(Y2) = np1-np2

E(Y1²-Y2²) = E(Y1²)-E(Yz²) = np1q1+n²p1²-np2q2-n²p2²

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Favouring complete protection -- 6%

destroyed by wild life -- 16%

p1 = 0.06

p2 = 0.16

n1 = n2 = 500

SE = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] }

p = 22% =0.22

SE = 0.0185

p1-p2 = -0.10

z= p1-p2/SE = -5.405

95% z value is 1.96

1.96(se) = 1.96(0.0185) = 0.0363

Conf interval = (-0.10-0.0363, -0.10+0.0363)

= (- 0.1363,-0.0637)