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4 a the random variable y is the number of tigers

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4. (a) The random variable, Y is the number of tigers with the bacteria present. A rea- sonable model (given the available infor- mation) is Y B (15 , p ). The hypothe- ses can be written: H 0 : p = 0 . 05 vs H A : p > 0 . 05. Evidence against the null is obtained for ’large’ observed values of Y . Using the basic definition of p-values, p - val = P ( Y 2). This can be written as 1 - ( P ( Y = 0) + P ( Y = 1)). Using the binomial formula results in p - val = 0 . 171. This means that there is no meaningful ev- idence against the claim that the rate of occurrence of this bacteria is 0.05 or less. (b) The random variable Y is the number of cats with bacteria present. A reasonable model is Y B (60 , p ). H 0 : p = 0 . 15 vs H A : p 0 . 15. Since np (= 9) and n (1 - p )(= 51) are both larger than 5, the nor- mal approx may be used. Let ˆ p = Y/ 60. Then, under H 0 , ˆ p NA N ( . 15 , ( . 0461) 2 ). α = P p 0 . 25) = P ( Z 2 . 17) = . 015. 5. (a) The general form for a CI for the difference between two proportions is: ( ˆ p A - ˆ p B ) ± Z α/ 2 * p A * (1 - p A ) n A + p B * (1 - p B ) n B . Because p A = 0 . 4 and p B = 0 . 6, both expressions of the form p (1 - p ) are the same. Since Z α/ 2 affects all CIs equally (given the same 1 - α
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