Homework 10 Solution : STAT 400 - Spring 20-

Ha Khanh Nguyen

① ECI ) -- ECE ) -= E Var LE ) = 6,'

var ( E) -- ECov ( E

.E ) = G iz

E,= aE .

+ ( I - a) Ez

MSE ( Es ) F Var ( Es ) = Var [ a E t ( I- a) OI ]

unbiased= a'

Var ( E ) t- 2all -a) Cov CE,Ea) t ( t -a)

'

Var CE)= a' 6

,

'

t La ( I - a) f iz t C l - a)- 65

¥ MSE (Est = 2a 6,2 t 2¥ - 4a f.a - 2622 t 4a 65 Eet O

⇒ a ( 26 ,

-- 40 iz t 465 ) ⇐ 26: - 20,2

⇒a=t÷Check 2nd derivative to make sure this is the minimum.

② Xi . . . . . Xn"nd f Cx) =

3×2

F = ITgs

'O S X C p

MSE (B) = Bias (B)'

t Var (B)Bias (B) = ECB) - p = ECI) - p = E CX) - B

E ( x) = f."

x - SFI dx ⇐ If, Jo"x3dx = / ! = f- p⇒ Bias (B) =. f- p - p = - I p ①

Var ( B ) = Var CI ) ⇐ Varden

F- ( XD = !"

ni . 3¥-

dx = ¥, f."

x"dx = Is . / ! = Epa

z3

⇒ var CPT -- I [ Ipa - ( q p)'

] = gotHence

,MSE ( B ) = ( - I, p)

-

t got p- =(÷t-n)③ X

, . Xz , X ,"id N ( pi , 6

? I )

MSEl pit ) = Bias ( pi )'

t Var ( pi, )= FE ( IX ,

t I Xattz Xs) - µ]-

t- Var ( } X, t IX, * I XD= [ 3 . tf - µ ]

-

t GE Cl ) = IMSE ( Az) = Bias ( pic)

'

t Var ( pi )= FE ( at X ,

t f Xat f- Xs) - gu ]-

t- Var ( f- X, t tax, * f- XD= [ 3 - Ig - µ ]

-

t -411) = GI pi + ITSo, for MSE ( fu) S MS El pet )⇒ Imitate 's⇒ µ

-

s I

⇒ -fEsmf

②

④ n = 11 I = 7.9 s = 0.12

Normal population(a) A 95% CI for the overall average weight of beef in a

" Yz pound"

burger at Burger Queen :

I ± tf , df = n . ,= 7.9 ± 2.228 .

-

£0.025, df - IO = 2.228 =(7.8/94,79806*16)

If we repeat this process many time, 95% of the calculated CI's

from the same sample size) will capture the true overall average weight

of beef in a" 42 pound

"

burger at Burger Queen .

OR we are 95% confident that the true overall average weightof beef in a

" 42 pound"

burger at Burger Queen is between 7.8194oz

and F . 9806oz .

⑤ n = GO,I -= 10.2

,6--4.7

(a) A 99% CI for the overall average weekend studying time :

I ± Zak - Fn = 10.2 I 2.575 .⇐ 186376,11762€

in

£0.005 = 2 . 575

( b) E = 0.5

n = [ Z4g )"

= [ £575l4 ]-

¥ 585.882

0.5

⇒③