[...]... average, variance, and t statistic and plot them in the form of Figure 2 .11 1 © 2002 By CRC Press LLC 2 3 4 5 6 7 8 9 .1 9.5 10 .1 11. 9 9.6 9 .1 9.0 10 .4 9.7 9.4 8.9 9.2 11 .2 10 .3 10 .6 12 .1 7.8 10 .4 8.6 11 .6 11 .7 11 .1 10.4 11 .3 10 .6 11 .7 9.0 10 .6 9.2 10 .4 8.4 10 .9 12 .1 11. 2 10 .0 10 .4 9.7 9.3 8.7 9 .1 L1592_Frame_C02 Page 24 Tuesday, December 18 , 20 01 1:40 PM 2 .14 Sampling Distributions II Below are ten groups... of Freedom n α = 0 .1 2 4 6 10 20 25 26 27 40 ∞ 1. 886 1. 533 1. 440 1. 372 1. 325 1. 316 1. 315 1. 314 1. 303 1. 282 Tail Area Probability 0.05 0.025 0. 01 0.005 2.920 2 .13 2 1. 943 1. 812 1. 725 1. 708 1. 706 1. 703 1. 684 1. 645 6.965 3.747 3 .14 3 2.764 2.528 2.485 2.479 2.473 2.423 2.326 9.925 4.604 3.707 3 .16 9 2.845 2.787 2.779 2.7 71 2.704 2.576 4.303 2.776 2.447 2.228 2.086 2.060 2.056 2.052 2.0 21 1.960 Sampling Distribution... distribution For each sample of five (i.e., each column), calculate the average and variance and plot them in the form of Figure 2 .11 Does the distribution of the averages seem to be approximately normal? If so, explain why 1 2 3 4 5 6 7 8 9 10 2.3 24.4 12 .6 28.3 15 .1 24.4 12 .0 12 .3 3.0 4.8 62 .1 6.4 4 .1 4.2 17 .5 37 .1 25.2 38.5 10 .8 16 .2 10 .4 11 1.8 9.3 0.4 32.4 34.6 3.0 2.7 1. 3 14 .9 14 .7 56.4 2.5 3.4 8.8 7 .1 3.3... L1592_Frame_C02 Page 15 Tuesday, December 18 , 20 01 1:40 PM Nitrate Observation i -1 11 10 9 8 7 6 5 4 4 5 6 7 8 9 10 11 Nitrate Observation i FIGURE 2.9 Plot of measurement yi vs measurement yi 1 shows a lack of serial correlation between adjacent measurements σ 1 4 σ σ σ σ σ α2 α3 6 8 Nitrate (mg/L) 10 α4 12 FIGURE 2 .10 A normal distribution centered at mean η = 8 Because of symmetry, the areas 1. .. December 18 , 20 01 1:40 PM α z TABLE 2 .1 Tail Area Probability (α ) of the Unit Normal Distribution z 0.00 0. 01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 … 1. 5 1. 6 1. 7 1. 8 1. 9 2.0 0.5000 … 0.0668 0.0548 0.0446 0.0359 0.0287 0.0228 0.4960 … 0.0655 0.0537 0.0436 0.03 51 0.02 81 0.0222 0.4920 … 0.0643 0.0526 0.0427 0.0344 0.0274 0.0 217 0.4880 … 0.0630 0.0 516 0.0 418 0.0366 0.0268 0.0 212 0.4840 … 0.0 618 0.0505... randomly select 20 per day for analysis Explain how you would select the sample of n = 20 water specimens 2.4 Experimental Errors The measured concentration of phosphorus (P) for n = 20 identical specimens of wastewater with known concentration of 2 mg/L are: 1. 8 2.2 2 .1 2.3 2 .1 2.2 2 .1 2 .1 1.8 1. 9 2.4 2.0 1. 9 1. 9 2.2 2.3 2.2 2.3 2 .1 2.2 2.5 2.6 2.7 2.8 2.9 2 .10 2 .11 2 .12 2 .13 Calculate the experimental... L1592_Frame_C02 Page 19 Tuesday, December 18 , 20 01 1:40 PM 40 random samples of n = 4 14 Parent distribution N (10 ,1) y 10 6 y= Sampling distribution of the mean is normal 12 Σy 10 n 8 6 s2 = 4 Σ( y − y ) 2 2 n 1 0 Sampling distribution of the variance 2 y −η t= 0 s n Sampling distribution of t -2 FIGURE 2 .11 Forty random samples of n = 4 from a N (10 ,1) normal distribution to produce the sampling distributions... 0.0329 0.0262 0.0207 0.48 01 … 0.0606 0.0495 0.04 01 0.0322 0.0256 0.0202 0.47 61 … 0.0594 0.0485 0.0392 0.0 314 0.0250 0. 019 7 0.47 21 … 0.0582 0.0475 0.0384 0.0307 0.0244 0. 019 2 0.46 81 … 0.05 71 0.0465 0.0375 0.03 01 0.0239 0. 018 8 0.46 41 … 0.0559 0.0455 0.0367 0.0294 0.0233 0. 018 3 It is convenient to work with standardized normal deviates, z = (y − η)րσ, where z has the distribution N(0, 1) , because the areas... 14 .7 56.4 2.5 3.4 8.8 7 .1 3.3 28.0 17 .8 13 .9 15 .4 9.7 11 .3 2.4 3.3 26.9 8.4 5.9 20.7 11 .4 2 .15 Standard Error I Calculate the standard error of the mean for a sample of size n = 16 that has a variance of 9 2 .16 Standard Error II For the following sample of n = 6 data values, calculate the standard error of the mean, s y 3.9, 4.4, 4.2, 3.9, 4.2, 4.0 2 .17 t Statistic II For the phosphorus data in Exercise... Hunter (19 78) Statistics for Experimenters: An Introduction to Design, Data Analysis, and Model Building, New York, Wiley Interscience Devore, J (2000) Probability and Statistics for Engineers, 5th ed., Duxbury Johnson, R A (2000) Probability and Statistics for Engineers, 6th ed., Englewood Cliffs, NJ, Prentice-Hall © 2002 By CRC Press LLC L1592_Frame_C02 Page 23 Tuesday, December 18 , 20 01 1:40 PM . function of bias and good precision. y y 6.9 7.8 … 8 .1 7.9++++ 27 7. 51 mg/L== s 2 6.9 7. 51 () 2 … 7.9 7. 51 () 2 ++ 27 1 1. 913 8 (mg/L) 2 == s 1. 913 8 1. 38 mg/L== • • • • • • • ••• •••• • •• • • • Analyst A B C D 7.5. publica- tions in statistics and engineering. L1592_frame_CH- 01 Page 5 Tuesday, December 18 , 20 01 1:39 PM © 2002 By CRC Press LLC References Berthouex, P. M., W. G. Hunter, and L. Pallesen (19 81) . “Wastewater. any η ∑y i N = y 1 n y i ∑ = σ 2 ∑ y i η –() 2 N = σ ∑ y i η –() 2 N = s 2 ∑ y i y–() 2 n 1 = y s ∑ y i y–() 2 n 1 = L1592_Frame_C02 Page 10 Tuesday, December 18 , 20 01 1:40 PM © 2002