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M344Spring2009Midterm1sk

# M344Spring2009Midterm1sk - BU Department of Mathematics...

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BU Department of Mathematics Math 344 First Midterm Examination Solution Key Date: April 8, 2009 Time: 17:00-18:00 Full Name: Student Number: Signature: Q1 Q2 Q3 Q4 Total pts pts pts pts 100 pts 1) (5pts each) An urn contains m balls numbered 1 , 2 , . . . , m . We draw a random sample of the balls, X 1 , X 2 , . . . X n with replacement so that the distribution of each X i is P ( X i = x ) = 1 /m for x = 1 , 2 , . . . , m . a) Find E ( X ). Solution: We have E ( X i ) = m j =1 1 m j = 1 m m ( m +1) 2 = m +1 2 . So, E ( X ) = E ( 1 n n i =1 X i ) = 1 n n i =1 E ( X i ) = 1 n n m + 1 2 = m + 1 2 where we have just used the linearity of expectation. b) Show that 2 X - 1 is an unbiased estimator of m . Solution: We have E (2 X - 1) = 2( E ( X )) - 1 = 2 ( m +1 2 ) - 1 = m and so 2 X - 1 is an unbiased estimator for m . 1

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c) If the sample variance is S 2 and n is big, find a 95% confidence interval for ( m + 1) / 2. Solution: We know that μ = E ( X i ) = m +1 2 . The confidence interval for ( m + 1) / 2 turns out to be CI = ( X - z α/ 2 s n , X + z α/ 2 s n ) = ( X - 1 . 96 s n , X + 1 . 96 s n ) .
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M344Spring2009Midterm1sk - BU Department of Mathematics...

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