S-assignment-4-3502-S10 - STAT 3502 Solution for Assignment...

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STAT 3502 Solution for Assignment # 4 Total mark=25 Due: 19 July, 2010 prior to the start of class 1. Suppose that a random sample of size n is drawn from the Bernoulli distribution f ( x ; p ) = ( p x (1 - p ) 1 - x x = 0 , 1 0 otherwise. Find a maximum likelihood estimator of p .[3] Sol: f ( x 1 , ··· ,x n ,p ) = p x i (1 - p ) (1 - x i ) ln f ( x 1 , ··· ,x n ,p ) = x i (ln p ) + (1 - x i )(ln(1 - p )) ln f ( x 1 , ··· ,x n ,p ) ∂p = x i p - (1 - x i ) 1 - p = 0 x i p = n - x i 1 - p ˆ p = x i n = ¯ X 2. In a survey of 4720 American, 708 of them were overweight. calculate and interpret a 99% confidence interval for the proportion of all American who are overweight.[3] Sol: ˆ p = X n = 708 4720 = 0 . 15 Confidence interval for population proportion is ˆ p ± z α/ 2 q ˆ p ˆ q n = 0 . 15 ± 2 . 58 q 0 . 15(0 . 85) 4720 , so 0 . 1366 < p < 0 . 1634, this confidence interval means if we construct 100 CIs based on 100 random samples, 99 of them contain the true population proportion. 3. For a sample of 69 healthy trees, the sample mean of dye-layer density was 1.028 and
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S-assignment-4-3502-S10 - STAT 3502 Solution for Assignment...

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