Final Exam Practice A Solution

1 Stat 421: Final Exam Formula Sheet z .005 = 2.576 z .025 = 1.96 z .05 = 1.645 1. ( N n ) = N ! n! ( N − n ) ! s 2 = 1 n − 1 ∑ i = 1 n ( y i − ¯ y ) 2 2. ¯ y = 1 n ∑ i = 1 n y i ^ V  ( ¯ y ) = s 2 n ( 1 − n N ) 3. ^ t = N  ¯ y ^ V  ( ^ t ) = N 2  ^ V  ( ¯ y ) 4. ^ p = ¯ y ^ V ( ^ p ) = ^ p  ( 1 − ^ p ) n − 1 ( 1 − n N ) 5. n = z α / 2 2 s 2 e 2 + z α / 2 2 s 2 N n = z α / 2 2 N 2 s 2 e 2 + z α / 2 2 Ns 2 6. ^ t ψ = 1 n ∑ i ∈ A Q i y i ψ i ^ V ( ^ t ψ )= 1 n ( n − 1 ) ∑ i ∈ A Q i [ y i ψ i − ^ t ψ ] 2 ¯ ^ y ψ = ^ t ψ N ^ V ( ¯ y ψ )= ^ V ( ^ t ψ ) N 2 7. ^ t str = ∑ h = 1 H ^ t h = ∑ h = 1 H N h ¯ y h ^ V ( ^ t str )= ∑ h = 1 H ( 1 − n h N h ) N h 2 s h 2 n h ¯ y h = ∑ j = 1 n h y hj n h ^ t h = N h n h ∑ j = 1 n h y hj = N h ¯ y h s h 2 = ∑ j = 1 n h ( y hj −¯ y h ) 2 n h − 1

2 8. ^ p str = ∑ h = 1 H N h N ^ p h ^ V ( ^ p str )= ∑ h = 1 H ( 1 − n h N h ) ( N h N ) 2 ^ p h ( 1 − ^ p h ) n h − 1 9. ^ B = ¯ y / ¯ x ^ V ( ^ B ) = ( 1 − n N ) s e 2 n ¯ x U 2 e i = y i − x i ^ B s e 2 = 1 n − 1 ∑ i = 1 n ( e i −¯ e ) 2 10. ¯ ^ y r = ^ B ¯ x U ^ V ( ¯ ^ y r ) = ¯ x U 2  ^ V ( ^ B ) 11. ^ B 1 = s xy s x 2 = rs y s x ^ B 0 =¯ y − ^ B 1 ¯ x s xy = 1 n − 1 ∑ i = 1 n ( x i − ¯ x )( y i − ¯ y )  r = s xy / s x s y ¯ ^ y reg  =¯ y + ^ B 1 (¯ x U −¯ x )  = ^ B 0 + ^ B 1 ¯ x U ^ V ( ¯ ^ y reg )= ( 1 − n N ) s e 2 n e i = y i − ^ B 0 − x i ^ B 1 s e 2 = 1 n − 1 ∑ i = 1 n e i 2 12. n c S N c S N n H l l l l h h h h    1 / /    H l l l l h h h h c S N c S N n n 1 / / 2 2 2 / 2 1 2 2 2 2 / e z e S N n n z n H h h h h             

3 13. ¯ y d = ^ B = ¯ u ¯ x ^ V ( ¯ y d )= ( 1 − n N ) 1 n ¯ x U 2 ∑ i = 1 n ( u i − ^ B x i ) 2 n − 1 ¯ y d = 1 n d ∑ i ∈ S d y i ^ V ( ¯ y d )≃ ( 1 − n N ) s yd 2 n d s yd 2 = 1 n d − 1 ∑ i ∈ S d ( y i − ¯ y d ) 2 ^ t d = N d ¯ y d ^ t d = N ¯ u 14. ¯ ^ y post = ∑ h = 1 H N h N ¯ y hR ^ V ( ¯ ^ y post ) ≃ ∑ h = 1 H ( N h N ) 2 ( 1 − n hR N h ) ( s hR 2 n hR ) 15. ^ t unb = N n ∑ i = 1 n t i ^ V ( ^ t unb ) = N 2 ( 1 − n N ) s t 2 n s t 2 = ∑ i = 1 n ( t i − ^ t unb N ) n − 1 2 ¯ ^ y unb = ^ t unb K ^ V ( ¯ ^ y unb ) = ^ V ( ^ t unb ) K 2 ¯ ^ y r = ∑ i = 1 n t i ∑ i = 1 n M i ^ V ( ¯ ^ y r ) = ( 1 − n N ) 1 n ¯ M U 2 ∑ i = 1 n ( t i − ¯ ^ y r M i ) n − 1 2 =

4 ( 1 − n N ) 1 n ¯ M U 2 ∑ i = 1 n M i 2 ( ¯ y i − ¯ ^ y r ) n − 1 2 ^ t r = K ¯ ^ y r ^ V ( ^ t r ) = K 2 ^ V ( ¯ ^ y r ) 16. ^ t i = M i ¯ y i ¯ y i = 1 m i ∑ j = 1 m i y ij ^ t unb = N n ∑ i = 1 n ^ t i ^ V ( ^ t unb ) = N 2 ( 1 − n N ) s t 2 n + N n ∑ i = 1 n ( 1 − m i M i ) M i 2 s i 2 m i s t 2 = ∑ i = 1 n ( ^ t i − ^ t unb N ) n − 1 2 ¯ ^ y r = ∑ i = 1 n M i ¯ y i ∑ i = 1 n M i ^ V ( ¯ ^ y r ) = ( 1 ¯ M 2 ) [ ( 1 − n N ) s r 2 n + 1 nN ∑ i = 1 n ( 1 − m i M i ) M i 2 s i 2 m i ] s r 2 = ∑ i = 1 n ( M i ¯ y i − M i ¯ ^ y r ) 2 n − 1 = ∑ i = 1 n M i 2 ( ¯ y i − ¯ ^ y r ) n − 1 2

5 Stat 421 – Spring 2016 Name: _____________________________ Final Exam, May 5, 2016 This midterm has 3 “short answer” questions, each with multiple parts, and 5 True/False questions. The questions cover pages 5-16 of this packet. Please show your work, and remember to include units where relevant. For the short answer computations, if you include a complete formula with numbers, you do not need to complete the calculation. 1. A real estate company wants to understand the composition of houses in a community with a total of 40 houses. The real estate company selects an SRSWOR of 4 houses. For the sampled houses, the real estate company collects information on the size of the garage and the number bedrooms in the house. The table below contains the collected data. [30 points] Sample House ID #s Garage size (square feet) Number of Bedrooms 1 384 2 2 308 1 3 484 2 4 576 4 a. First, the real estate company wants to estimate the average number of bedrooms among houses with a garage size of at least 400 square feet. (i) Define the domain U d of interest in words. [3 points] Houses with a garage size of at least 400 feet. (ii) Provide a mathematical expression for the domain population parameter of interest. Define the meanings of the symbols used in your expression, including the definition of y i . [3 points] ´ y U d = 1 N d ∑ i ∈ U d y i , where N d is the number of elements in U d , and y i is the number of bedrooms in house i