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as092-soln

# as092-soln - STAT 854/454 Short Solutions to Assignment 2 1...

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STAT 854/454 Short Solutions to Assignment 2 1. Let N = 7 , 800 be the number of public schools in Canada. We wish to estimate p = M/N , where M is the number of schools that have Wireless Internet Access (WIA). (a) Find the required sample size n such that the estimate is accurate within 2 percentage points, 19 times out of 20 under SRSWOR, assuming prior information on p is not available. Soln: Use n = { Z 2 α/ 2 p (1 - p ) } / { e 2 + Z 2 α/ 2 p (1 - p ) /N } with e = 0 . 02, α = 0 . 05 and p = 0 . 5. (b) Re-do (a) assuming p = 5% for the purpose of sample size calculation. Soln: Omitted. (c) If we do inverse sampling with m = 10 under selection without replacement, what is the expected number of schools we have to sample? Assume p = 5% for the purpose of planning. Soln: The probability function for the random sample size n is given by P ( n = j ) = ( M m - 1 )( N - M j - m ) ( N j - 1 ) M - m + 1 N - j + 1 , j = m, m + 1 , · · · , N - M + m . Note that N - M + m j = m P ( n = j ) = 1 for any N , M and m < M , and also note that 1 j ( N j - 1 ) = 1 N + 1 ( N +1 j ) and ( M m - 1 ) = m M + 1 ( M +1 m ) , we can show that E ( n ) = jP ( n = j ) = m ( N + 1) / ( M + 1). Use M = pN and p = 0 . 05 to get the final answer.

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as092-soln - STAT 854/454 Short Solutions to Assignment 2 1...

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