Chaptr (8) Estimatio ad Cofdc Itrvals Exampls Typs of stimatio: i. Poit stimatio: Exampl (1): Cosidr th sampl obsrvatios, 17,3,5,1,18,6,16,10 8 X i i1 17 3 5 118 6 16 10 116 X 14.5 8 8 8 14.5 is a poit stimat for usig th stimator X ad th giv sampl obsrvatios. ii. Itrval stimatio: Costructig cofidc itrval Th gral form of a itrval stimat of a populatio paramtr: Poit Estimat ± Criticalvalu *Stadard rror This formula grats two valus calld th cofidc limits; - Lowr cofidc limit (LCL). - Uppr cofidc limit (UCL). Aothr way to fid th cofidc itrval w usd th cofidc 1
Cas1: Cofidc Itrval for Populatio Ma with kow Stadard Dviatio (ormal cas): Th cofidc limits ar: Stps for calculatig: 1. Obtai,from th tabl of th ara udr th ormal curv. Z. Calculat Z 3. L= X Z U= X Z X :Th ma stimator σ : Th stadard dviatio of th populatio. Z : Th stadard rror of th ma : Critical valu.. x Exampl (): A sampl of 49 obsrvatios is tak from a ormal populatio with a stadard dviatio of 10.th sampl ma is 55,dtrmi th 99 prct cofidc itrval for th populatio ma Solutio: X ~ N, X ~ N, σ = 10, = 49,X = 55,Cofidc lvl = 0.99, α = 1 0.99 = 0.01 = Z 0.005 =.58 Z0.01 Th cofidc limits ar: X Z 55.58 51.3143 58.6857 51.3143, 58.6857 10 55 3.6857 49.
Exampl (3): - IF you hav (51.3143, 58.6857). Basd o this iformatio, you kow that th bst poit stimat is: ˆ of th populatio ma uppr lowr 58.6857 51.3143 110 ˆ 55 Cas: Cofidc Itrval for a Populatio Ma with ukow Stadard Dviatio S ˆ X t 1; Exampl (4): Th owr of Britt's Egg Farm wats to stimat th ma umbr of ggs laid pr chick. A sampl of 0 chicks shows thy laid a avrag of 0 ggs pr moth with a stadard dviatio of 8 ggs pr moth (a sampl is tak from a ormal populatio). i. What is th valu of th populatio ma? What is th bst stimat of this valu? ii. Explai why w d to us th t distributio. What assumptio do you d to mak? iii. For a 95 prct cofidc itrval, what is th valu of t? iv. Dvlop th 95 prct cofidc itrval for th populatioma. v. Would it b rasoabl to coclud that th populatio ma is 1 ggs? What about 5 ggs? Solutio: i. th populatio ma is ukow, but th bst stimat is 0,th sampl ma 3
ii. Us th t distributio as th stadard dviatio is ukow.howvr, assum th populatio is ormally distributd. iii. t t t. 093 1; 0.05 01, 19,0.05 8 0 iv. X t 0.093 0 3. 74 1; 16.6 16.6, S 3.74 3.74 v. Ys, bcaus th valu of µ=1 is icludd withi th cofidc itrval stimat. No, bcaus th valu of µ=5 is ot icludd withi th cofidc itrval stimat. Exampl (5): Fid a 90% cofidc itrval for a populatio ma for ths valus: 14, 158, x s 45796, X ~ N, Solutio: 1 0.90 0.10 t t t 1; 141, ˆ X t 0.10 1; S 13,0.05 14 158 1.771 14 158 101.9 1156.71 1359.9 1156.71, 1359.9 1.771 4
Wh th sampl siz is larg 100, 0.05 0.95, 5, 1 5, th sampl proportio, P P ~ X = 1 N, Total umbr of succsss Total umbr of trials Th cofidc itrval for a populatio proportio: P Z P 1 P P 1 P, Th stadard rror of th proportio Exampl (6): Th owr of th Wst Ed crdit Kwick Fill Gas Statio wishs to dtrmi th proportio of customrs who us a crdit card or dbit card to pay at th pump. H survys 100 customrs ad fids that 80 paid at th pump. a. Estimat th valu of th populatio proportio. b. Dvlop a 95 prct cofidc itrval for th populatio proportio. c. Itrprt your fidigs. Solutio: a. P X 80 100 0.8 b. Z Z Z 1.96 Z Z 1. 96 0.05 0.05 0.9750 0.05 1 1 P 0.80. 0.9750 P P Z 0.8 1.96 100 0.8 1.96 0.0016 0.8 1.96 0.7 0.88 0.7, 0.88 c. W ar rasoably sur th populatio proportio is btw 0.7 ad o.88 prct. 0.04 0.8 0. 0784 5
Exampl (7): Th Fox TV twork is cosidrig rplacig o of its prim-tim crim ivstigatio shows with a w family-oritd comdy show. Bfor a fial dcisio is mad, twork xcutivs commissio a sampl of 400 viwrs. Aftr viwig th comdy, 0.63 prct idicatd thy would watch th w show ad suggstd it rplac th crim ivstigatio show. a. Estimat th valu of th populatio proportio. b. Dvlop a 99 prct cofidc itrval for th populatio proportio. c. Itrprt your fidigs. Solutio: a. P 0.63 b. Z Z 0.01 Z0.005 0.01 1 Z 1 0.005.58 Z 0.9950.58 P1 P 0.630.37 P Z 0.63.58 0.63.58 400 0.63.580.0414 0.63 0. 063 0.57 0.69 0.57, 0.69 0.0005875 c. W ar rasoably sur th populatio proportio is btw 0.57 ad o.69 prct. 6
Not: If th valu of stimatd proportio(p) ot mtiod w substitut it by o.5( as studis ad rachars rcommdd) 7
= ±Z σ Or = UCL LCL Th lgth of cofidc itrval= UCL LCL Th lgth of C.I= σ = Z α Th sampl siz for stimatig th populatio ma: σ = ( Z α Exampl (8): A studt i public admiistratio wats to dtrmi th ma amout mmbrs of city coucils i larg citis ar pr moth as rmuratio for big a coucil mmbr. Th rror i stimatig th ma is to b lss tha $100 with a 95 prct lvl of cofidc. Th studt foud a rport by th Dpartmt of Labor that stimatd th stadard dviatio to b $1,000. What is th rquird sampl siz? Solutio: Giv i th problm: E, th maximum allowabl rror, is $100 Th valu of z for a 95 prct lvl of cofidc is 1.96, Th stimat of th stadard dviatio is $1,000. Z 1.961000 100 384.16 385 ) 8
Exampl (9): A populatio is stimatd to hav a stadard dviatio of 10.if a 95 prct cofidc itrval is usd ad a itrval of is dsird.how larg a sampl is rquird? Solutio: Giv i th problm: E, th maximum allowabl rror, is Th valu of z for a 95 prct lvl of cofidc is 1.96, Th stimat of th stadard dviatio is10. Z 1.96 10 96.04 97 Exampl (10): If a simpl radom sampl of 36 popl was usd to mak a 95% cofidc itrval of (0.57,0.67), what is th margi of rror? Solutio: uppr lowr 0.67 0.57 0.1 0.05 Exampl (11): If =34, th stadard dviatio 4. What is th maximum allowabl rror Solutio: Z 4. 1.96 1.96 0.703 1. 41 Th maximum allowabl rror () = 1.41 E?, 1 95% 9
Th margi rror for th cofidc itrval for a populatio proportio: π(1 π) = Zα Solvig "E" quatio for "" yilds th followig rsult: Or = ( Zα π(1 π) ) Zα = π(1 π) ( ) Z 1 Exampl (1): Th stimat of th populatio proportio is to b withi plus or mius o.o5, with a 95 prct lvl of cofidc. Th bst stimatio of th populatio proportio is o.15.how larg a sampl is rquird? Solutio: Z 1 3.8416 0.175 0.005 1.96 0.151 0.15 0.05 0.4898 0.005 195.9 196 3.8416 0.15 0.85 0.005 10
Exampl (13): Th stimat of th populatio proportio is to b withi plus or mius o.10, with a 99 prct lvl of cofidc. How larg a sampl is rquird? Solutio: Z 1.58 0.51 0.5 0.10 6.6564 0.5 1.6641 166.41 167 0.01 0.01 6.6564 0.5 0.5 0.01 11
Z Tabl: Ngativ Valus Z ad Tabls z.00.01.0.03.04.05.06.07.08.09-3.80.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001-3.70.0001.0001.0001.0001.0001.0001.0001.0001.0001.0001-3.60.000.000.0001.0001.0001.0001.0001.0001.0001.0001-3.50.000.000.000.000.000.000.000.000.000.000-3.40.0003.0003.0003.0003.0003.0003.0003.0003.0003.000-3.30.0005.0005.0005.0004.0004.0004.0004.0004.0004.0003-3.0.0007.0007.0006.0006.0006.0006.0006.0005.0005.0005-3.10.0010.0009.0009.0009.0008.0008.0008.0008.0007.0007-3.00.0013.0013.0013.001.001.0011.0011.0011.0010.0010 -.90.0019.0018.0018.0017.0016.0016.0015.0015.0014.0014 -.80.006.005.004.003.003.00.001.001.000.0019 -.70.0035.0034.0033.003.0031.0030.009.008.007.006 -.60.0047.0045.0044.0043.0041.0040.0039.0038.0037.0036 -.50.006.0060.0059.0057.0055.0054.005.0051.0049.0048 -.40.008.0080.0078.0075.0073.0071.0069.0068.0066.0064 -.30.0107.0104.010.0099.0096.0094.0091.0089.0087.0084 -.0.0139.0136.013.019.015.01.0119.0116.0113.0110 -.10.0179.0174.0170.0166.016.0158.0154.0150.0146.0143 -.00.08.0.017.01.007.00.0197.019.0188.0183-1.90.087.081.074.068.06.056.050.044.039.033-1.80.0359.0351.0344.0336.039.03.0314.0307.0301.094-1.70.0446.0436.047.0418.0409.0401.039.0384.0375.0367-1.60.0548.0537.056.0516.0505.0495.0485.0475.0465.0455-1.50.0668.0655.0643.0630.0618.0606.0594.058.0571.0559-1.40.0808.0793.0778.0764.0749.0735.071.0708.0694.0681-1.30.0968.0951.0934.0918.0901.0885.0869.0853.0838.083-1.0.1151.1131.111.1093.1075.1056.1038.100.1003.0985-1.10.1357.1335.1314.19.171.151.130.110.1190.1170-1.00.1587.156.1539.1515.149.1469.1446.143.1401.1379-0.90.1841.1814.1788.176.1736.1711.1685.1660.1635.1611-0.80.119.090.061.033.005.1977.1949.19.1894.1867-0.70.40.389.358.37.96.66.36.06.177.148-0.60.743.709.676.643.611.578.546.514.483.451-0.50.3085.3050.3015.981.946.91.877.843.810.776-0.40.3446.3409.337.3336.3300.364.38.319.3156.311-0.30.381.3783.3745.3707.3669.363.3594.3557.350.3483-0.0.407.4168.419.4090.405.4013.3974.3936.3897.3859-0.10.460.456.45.4483.4443.4404.4364.435.486.447 0.00.5000.4960.490.4880.4840.4801.4761.471.4681.4641 1
Z Tabl: Positiv Valus z.00.01.0.03.04.05.06.07.08.09 0.00.5000.5040.5080.510.5160.5199.539.579.5319.5359 0.10.5398.5438.5478.5517.5557.5596.5636.5675.5714.5753 0.0.5793.583.5871.5910.5948.5987.606.6064.6103.6141 0.30.6179.617.655.693.6331.6368.6406.6443.6480.6517 0.40.6554.6591.668.6664.6700.6736.677.6808.6844.6879 0.50.6915.6950.6985.7019.7054.7088.713.7157.7190.74 0.60.757.791.734.7357.7389.74.7454.7486.7517.7549 0.70.7580.7611.764.7673.7704.7734.7764.7794.783.785 0.80.7881.7910.7939.7967.7995.803.8051.8078.8106.8133 0.90.8159.8186.81.838.864.889.8315.8340.8365.8389 1.00.8413.8438.8461.8485.8508.8531.8554.8577.8599.861 1.10.8643.8665.8686.8708.879.8749.8770.8790.8810.8830 1.0.8849.8869.8888.8907.895.8944.896.8980.8997.9015 1.30.903.9049.9066.908.9099.9115.9131.9147.916.9177 1.40.919.907.9.936.951.965.979.99.9306.9319 1.50.933.9345.9357.9370.938.9394.9406.9418.949.9441 1.60.945.9463.9474.9484.9495.9505.9515.955.9535.9545 1.70.9554.9564.9573.958.9591.9599.9608.9616.965.9633 1.80.9641.9649.9656.9664.9671.9678.9686.9693.9699.9706 1.90.9713.9719.976.973.9738.9744.9750.9756.9761.9767.00.977.9778.9783.9788.9793.9798.9803.9808.981.9817.10.981.986.9830.9834.9838.984.9846.9850.9854.9857.0.9861.9864.9868.9871.9875.9878.9881.9884.9887.9890.30.9893.9896.9898.9901.9904.9906.9909.9911.9913.9916.40.9918.990.99.995.997.999.9931.993.9934.9936.50.9938.9940.9941.9943.9945.9946.9948.9949.9951.995.60.9953.9955.9956.9957.9959.9960.9961.996.9963.9964.70.9965.9966.9967.9968.9969.9970.9971.997.9973.9974.80.9974.9975.9976.9977.9977.9978.9979.9979.9980.9981.90.9981.998.998.9983.9984.9984.9985.9985.9986.9986 3.00.9987.9987.9987.9988.9988.9989.9989.9989.9990.9990 3.10.9990.9991.9991.9991.999.999.999.999.9993.9993 3.0.9993.9993.9994.9994.9994.9994.9994.9995.9995.9995 3.30.9995.9995.9995.9996.9996.9996.9996.9996.9996.9997 3.40.9997.9997.9997.9997.9997.9997.9997.9997.9997.9998 3.50.9998.9998.9998.9998.9998.9998.9998.9998.9998.9998 3.60.9998.9998.9999.9999.9999.9999.9999.9999.9999.9999 3.70.9999.9999.9999.9999.9999.9999.9999.9999.9999.9999 3.80.9999.9999.9999.9999.9999.9999.9999.9999.9999.9999 13
T Tabl 0.75 0.8 0.85 0.90 0.95 0.975 0.98 0.99 0.995 Df 0.5 0. 0.15 0.1 0.05 0.05 0.0 0.01 0.005 1 1.000 1.376 1.963 3.078 6.31 1.70 15.90 31.8 63.65 0.817 1.061 1.386 1.886.90 4.303 4.849 6.965 9.95 3 0.765 0.979 1.50 1.638.353 3.18 3.48 4.541 5.841 4 0.741 0.941 1.190 1.533.13.776.999 3.747 4.604 5 0.77 0.90 1.156 1.476.015.571.757 3.365 4.03 6 0.718 0.906 1.134 1.440 1.943.447.61 3.143 3.707 7 0.711 0.896 1.119 1.415 1.895.365.517.998 3.499 8 0.706 0.889 1.108 1.397 1.860.306.449.896 3.355 9 0.703 0.883 1.100 1.383 1.833.6.398.81 3.50 10 0.700 0.879 1.093 1.37 1.81.8.359.764 3.169 11 0.697 0.876 1.088 1.363 1.796.01.38.718 3.106 1 0.696 0.873 1.083 1.356 1.78.179.303.681 3.055 13 0.694 0.870 1.079 1.350 1.771.160.8.650 3.01 14 0.69 0.868 1.076 1.345 1.761.145.64.64.977 15 0.691 0.866 1.074 1.341 1.753.131.49.60.947 16 0.690 0.865 1.071 1.337 1.746.10.35.583.91 17 0.689 0.863 1.069 1.333 1.740.110.4.567.898 18 0.688 0.86 1.067 1.330 1.734.101.14.55.878 19 0.688 0.861 1.066 1.38 1.79.093.05.539.861 0 0.687 0.860 1.064 1.35 1.75.086.197.58.845 1 0.686 0.859 1.063 1.33 1.71.080.189.518.831 0.686 0.858 1.061 1.31 1.717.074.183.508.819 3 0.685 0.858 1.060 1.319 1.714.069.177.500.807 4 0.685 0.857 1.059 1.318 1.711.064.17.49.797 5 0.684 0.856 1.058 1.316 1.708.060.167.485.787 6 0.684 0.856 1.058 1.315 1.706.056.16.479.779 7 0.684 0.855 1.057 1.314 1.703.05.158.473.771 8 0.683 0.855 1.056 1.313 1.701.048.154.467.763 9 0.683 0.854 1.055 1.311 1.699.045.150.46.756 30 0.683 0.854 1.055 1.310 1.697.04.147.457.750 40 0.681 0.851 1.050 1.303 1.684.01.13.43.704 50 0.679 0.849 1.047 1.99 1.676.009.109.403.678 14