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Exercise (18-33)
18.n=610 ;P̂x =50.7=.507
1­α =.99 α=.01, α/2=.005 , 1­α/2 =.995 100(1- α) % C.I for P̂x is given by: P̂x-Z α/2√ P̂(1-P̂x)n <Px <P̂x + Z α/2√ P̂(1-P̂x)n
=.507-2.575√.507(1-.507)610<Px <.507+2.575√.507(1-.507)610
=.4548< Px < .5591
19(a). n=189 ;X=132; P̂x =xn=132189=.6984
1­α =.90 α=.10, α/2=.05 , 1­α/2 =.95 100(1- α) % C.I for P̂x is given by: P̂x-Z α/2√ P̂(1-P̂x)n <Px <P̂x + Z α/2√ P̂(1-P̂x)n
=.6984-1.645√.6984(1-.6984)189<Px <.6984+1.645√.6984(1-.6984)189
=.6436< Px < .7533
(b)Wider
20. n=323; P̂x =.479
100(1- α) % C.I for P̂x is given by:
P̂x-Z α/2√ P̂(1-P̂x)n <Px <P̂x + Z α/2√ P̂(1-P̂x)n
P̂x-Z α/2√ P̂(1-P̂x)n =.458
=>.479--Z α/2 √ .479(1-.479)323 =.458
=> Z α/2 .24955323 =.479-.458
=> Z α/2 *0.027796 =0.021
=> Z α/2 =.755
=>P( Z <α/2) =.755
=>1- P( Z >α/2) =.755
=> P( Z >α/2) =1-.755
=> P( Z >α/2 ) =.245
=> α/2 =.245
=> α =.49
=>1- α =1-.49
=>1- α =.51
21. n=134; x=82 P̂x =Xn=82134=.6119
1­α =.95 α=.05, α/2=.025 , 1­α/2 =.975
100(1- α) % C.I for P̂x is given by:
P̂x-Z α/2√ P̂(1-P̂x)n <Px <P̂x + Z α/2√ P̂(1-P̂x)n
=.6119-1.96√.6119*.3881134 <Px<.6119+1.96√.6119*.3881134
=.5294 <Px<.6944
22.(a) n=95 ;X=29; P̂x =xn=2995=.3052
1­α =.99 α=.01, α/2=.005 , 1­α/2 =.995 100(1- α) % C.I for P̂x is given by: P̂x-Z α/2√ P̂(1-P̂x)n <Px <P̂x + Z α/2√ P̂(1-P̂x)n
=.3052-2.575√.3052(1-.3052)95<Px <.3052+2.575√.3052(1-.3052)95
=.183< Px < .426
(b) Narrower
23. n=96 ;X=32; P̂x =xn=3296=.3333
1­α =.80 α=.20, α/2=.10 , 1­α/2 =.90 100(1- α) % C.I for P̂x is given by: P̂x-Z α/2√ P̂(1-P̂x)n <Px <P̂x + Z α/2√ P̂(1-P̂x)n
=.3333-1.285√.3333(1-.3333)96<Px <.3333+1.285√.3333(1-.3333)96
=.2714< Px < .3951
24. n=198; X=98 P̂x =98198=.4949
100(1- α) % C.I for P̂x is given by:
P̂x-Z α/2√ P̂(1-P̂x)n <Px <P̂x + Z α/2√ P̂(1-P̂x)n
P̂x-Z α/2√ P̂(1-P̂x)n =.445
=>.4949--Z α/2 √ .4949(1-.4949)198 =.445
=> Z α/2 .4949*.506198 =.4949-.445
=> Z α/2 *0.0355 =0.0499
=> Z α/2 =1.40
=>P( Z <α/2) =.9192
=>1- P( Z >α/2) =.9192
=> P( Z >α/2) =1-.9192
=> P( Z >α/2 ) =.0808
=> α/2 =.0808*2
=> α =.1616
=>1- α =1-.1616
=>1- α =.8384
25.n=15; Sx=.880;V=n-1=15-1=14
1­α =.95 α=.05, α/2=.025 , 1­α/2 =.975 sx(n-1)Xv,α/2<σx< sx(n-1)Xv,1-α/2
=.880(15-1)X14,.025 <σx< .880(15-1)X14,.975
=.7744*1426.12 <σx<.7744*145.63
=.4150 <σx< 1.925
26.n=7;V=6
∑Xi=79+73+68+77+86+71+69=523
X̅=∑Xin=5237=74.714
∑Xi=79+73+68+77+86+71+69 =39321
Sx=∑Xi-nx̅n-1=39321-7*(74.714)6=40.954
1­α =.80 α=.20, α/2=.10 , 1­α/2 =.90
100(1- α) % C.I for σx is given by: sx(n-1)Xv,α/2<σx< sx(n-1)Xv,1-α/2 =40.954(7-1)X6,.10 <σx< 40.954(7-1)X6,.90
=245.72410.64 <σx< 245.7242.20
=23.09 <σx< 111.692

2.(a) δx=.12 ; n= 16; X̅̅= 4.07 ; 1­α =.99 1­α =.99
99 % C.I for µx is given by: α=.01
X̅ - Z α/2 δx/n< µx < X̅ + - Z α/2 δx/n α/2=.005
=4.07-(2.57)*.12/√16< µx < 4.07+ (2.57)*.12/16 1­α/2 =.995
=3.992< µx < 4.147
(b) Narrower
(c) ) Narrower
(d) Wider
3. .(a) δx=3.8 ; n= 9; 1­α =.90; µx=?
1­α =.99, α=.01, α/2=.005 , 1­α/2 =.995
Xi=18.2+13.7+15.9+17.4+21.8+16.6+12.3+18.8+16.2=150.9
X̅ = 150.9/9=16.77
90 % C.I for µx is given by:
X̅ - Z α/2 δx/n< µx < X̅ + - Z α/2 δx/n
=16.77-(1.645)*3.8/√9< µx < 16.77+(1.645)*3.8/√9
=14.686< µx < 18.854
(b) Wider
4.(a) δx=32.4 ; n= 9; X̅=187.9; 1­α =.80; µx=?
1­α =.80 α=.20, α/2=.10 , 1­α/2 =.90
80 % C.I for µx is given by:
X̅ - Z α/2 δx/n< µx < X̅ + - Z α/2 δx/n
=187.9-(1.28)*32.4/√9< µx < 187.9+(1.28)*32.4/√9
=174.07< µx < 201.724

5.(a) Sx=1.57 ; n= 1562; X̅=3.92; 1­α =.95; µx=?
1­α =.95 α=.05, α/2=.025 , 1­α/2 =.90
95 % C.I for µx is given by:
X̅ - Z α/2 sx/n< µx < X̅ + - Z α/2 sx/n
=3.92-(1.96)*1.57/√1562< µx < 3.92+(1.96)*1.57/√1562
=3.842< µx < 3.997
6.(a) Sx=1.34 ; n= 541; X̅=3.81; 1­α =.90; µx=?
1­α =.90 α=.10, α/2=.05 , 1­α/2 =.95
90 % C.I for µx is given by:
X̅ - Z α/2 sx/n< µx < X̅ + - Z α/2 sx/n
=3.81-(1.645)*1.34/√541< µx < 3.81+(1.645)*1.34/√541
=3.72< µx < 3.90
(b) narrower

7. ) Sx=1.045 ; n= 457; X̅=3.59; 3.49< µx < 3.69 1­α =?
100(1- α ) % C.I for µx is given by: X̅ - Z α/2 sx/n< µx < X̅ + - Z α/2 sx/n
=3.59- Z α/2 *1.045/√457< µx < 3.59 + Z α/2 *1.045/√457
3.59- Z α/2 *1.045457=3.49 3 .59 + Z α/2 *1.045/457= 3.69
3.59- Z α/2 *1.045457=3.49
=>- Z α/2 *1.045457=3.59-3.49
=> Z α/2 *1.045=.1*457
=> Z α/2=2.1377/1.045
=> Fz (Z α/2)=Fz(2.045)
=>P(z< α/2)=0.97955
=>1-p(z> α/2)= 0.97955
=> p(z> α/2)=1-0.97955
=> p(z> α/2)=0.02045
=> α/2=0.02045
=> α=0.02045*2
=> α=0.0409
=>1­α =1-0.0409
=>1­α =0.9591 8. Sx=11.28 ; n= 352; X̅=60.41; 1­α =.90; µx=?
1­α =.90 α=.10, α/2=.05 , 1­α/2 =.95
90 % C.I for µx is given by:
X̅ - Z α/2 sx/n< µx < X̅ + - Z α/2 sx/n
=60.41-(1.645)*11.28/√352< µx < 60.41+(1.645)*11.28/√352
=59.42< µx < 61.399

10. Sx=38.89 ; n= 9; X̅=157.82; 1­α =.95;V=n-1=9-1=8; µx=?
1­α =.95 α=.05, α/2=.025
90 % C.I for µx is given by:
X̅ - tv, α/2 sx/n< µx < X̅ + - tv, α/2 sx/n
=157.82-(2.306)*38.89/√9< µx < 157.82+(2.306)*38.89/√9
=127.92< µx < 187.71

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