1
Types of estimatio: i. Poit estimatio: Example (1) Cosider the sample observatios 17,3,5,1,18,6,16,10 X 8 X i i1 8 17 3 5 118 6 16 10 8 116 8 14.5 14.5 is a poit estimate for usig the estimator X ad the give sample observatios. ii. Iterval estimatio: Costructig cofidece iterval The geeral form of a iterval estimate of a populatio parameter: Poit Estimate ± Criticalvalue *Stadard error This formula geerates two values called the cofidece limits; - Lower cofidece limit (LCL). - Upper cofidece limit (UCL). Aother way to fid the cofidece iterval we used the cofidece
Case1: Cofidece Iterval for Populatio Mea with kow Stadard Deviatio (ormal case): The cofidece limits are: Steps for calculatig: 1. Obtai Z,from the table of the area uder the ormal curve.. Calculate 3. L= X Z U= X Z Z. X :The mea estimator σ : The stadard deviatio of the populatio. : The stadard error of the mea x. : Critical value. Z 3
Example () A sample of 49 observatios is take from a ormal populatio with a stadard deviatio of 10.the sample mea is 55,determie the 99 percet cofidece iterval for the populatio mea Solutio: X ~ N, X ~ N,,Cofidece level = 0.99, σ = 10, = 49, The cofidece limits are: 10 X Z 55.58 51.3143 58.6857 51.3143, 58.6857 49 55 3.6857 Example (3) - IF you have (51.3143, 58.6857). Based o this iformatio, you kow that the best poit estimate of the populatio mea ˆ is: upper lower ˆ 58.6857 51.3143 110 55 4
Case: Cofidece Iterval for a Populatio Mea with ukow Stadard Deviatio S ˆ X t 1; Example (4) The ower of Britte's Egg Farm wats to estimate the mea umber of eggs laid per chicke. A sample of 0 chickes shows they laid a average of 0 eggs per moth with a stadard deviatio of 8 eggs per moth (a sample is take from a ormal populatio). i. What is the value of the populatio mea? What is the best estimate of this value? ii. Explai why we eed to use the t distributio. What assumptio do you eed to make? iii. For a 95 percet cofidece iterval, what is the value of t? iv. Develop the 95 percet cofidece iterval for the populatiomea. v. Would it be reasoable to coclude that the populatio mea is 1 eggs? What about 5 eggs? Solutio: i. the populatio mea is ukow, but the best estimate is 0,the sample mea ii. Use the t distributio as the stadard deviatio is ukow.however, assume the populatio is ormally distributed. 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 S 3.74 0.936 19.064, v. Yes, because the value of µ=1 is icluded withi the cofidece iterval estimate. No, because the value of µ=5 is ot icluded withi the cofidece iterval estimate. 5
Example (5) Fid a 90% cofidece iterval for a populatio mea for these values: 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 6
Whe the sample size is large 100, 0.05 0.95, 5, 1 5, the sample proportio, X P = P ~ 1 N, The cofidece iterval for a populatio proportio: P Z P 1 P P 1 P, The stadard error of the proportio Example (6) The ower of the West Ed credit Kwick Fill Gas Statio wishes to determie the proportio of customers who use a credit card or debit card to pay at the pump. He surveys 100 customers ad fids that 80 paid at the pump. a. Estimate the value of the populatio proportio. b. Develop a 95 percet cofidece iterval for the populatio proportio. c. Iterpret 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 0.8 0. 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.we are reasoably sure the populatio proportio is betwee 0.7 ad o.88 percet. 7 0.04 0.8 0. 0784
Example (7) The Fox TV etwork is cosiderig replacig oe of its prime-time crime ivestigatio shows with a ew family-orieted comedy show. Before a fial decisio is made, etwork executives commissio a sample of 400 viewers. After viewig the comedy, 0.63 percet idicated they would watch the ew show ad suggested it replace the crime ivestigatio show. d. Estimate the value of the populatio proportio. e. Develop a 99 percet cofidece iterval for the populatio proportio. f. Iterpret 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.we are reasoably sure the populatio proportio is betwee 0.57 ad o.69 percet. 8
Note: If the value of estimated proportio(p) ot metioed we substitute it by o.5( as studies ad reachears recommeded) 9
Or The legth of cofidece iterval= UCL LCL The legth of C.I= The sample size for estimatig the populatio mea: ( ) Example (8) A studet i public admiistratio wats to determie the mea amout members of city coucils i large cities ear per moth as remueratio for beig a coucil member. The error i estimatig the mea is to be less tha $100 with a 95 percet level of cofidece. The studet foud a report by the Departmet of Labor that estimated the stadard deviatio to be $1,000. What is the required sample size? Solutio: Give i the problem: E, the maximum allowable error, is $100 The value of z for a 95 percet level of cofidece is 1.96, The estimate of the stadard deviatio is $1,000. Z e 1.961000 100 384.16 385 11
Example (9) A populatio is estimated to have a stadard deviatio of 10.if a 95 percet cofidece iterval is used ad a iterval of is desired.how large a sample is required? Solutio:Give i the problem: E, the maximum allowable error, is The value of z for a 95 percet level of cofidece is 1.96, The estimate of the stadard deviatio is10. Z e Example (10) 1.96 10 96.04 97 If a simple radom sample of 36 people was used to make a 95% cofidece iterval of e? (0.57,0.67), what is the margi of error Solutio: upper lower e 0.67 0.57 0.1 0.05 Example (11) If =34, the stadard deviatio 4. What is the maximum allowable error E? Solutio:, 1 95% Z e e e e Z 1.96 4. 8.3 34 Z e 1.99 1.41 34 67.77 1.99 34 The maximum allowable error (e) = 1.4 11
The margi error for the cofidece iterval for a populatio proportio: Solvig "E" equatio for "" yields the followig result: ( ) Or ( ) Z 1 e Example (1) The estimate of the populatio proportio is to be withi plus or mius o.o5, with a 95 percet level of cofidece. The best estimatio of the populatio proportio is o.15.how large a sample is required? Solutio: Z 1 e 1.96 0.151 0.15 0.05 3.8416 0.175 0.4898 195.9 196 0.005 0.005 3.8416 0.15 0.85 0.005 1
Example (13) The estimate of the populatio proportio is to be withi plus or mius o.10, with a 99 percet level of cofidece. How large a sample is required? Solutio: Z 1 e.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 13
Z Table: Negative Values Z ad Tables 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 14
Z Table: Positive Values 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 15
T Table 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 16