Week15-4up - 337 STA 2023 c D.Wackerly - Lecture 25 STA...

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Unformatted text preview: 337 STA 2023 c D.Wackerly - Lecture 25 STA 2023 c D.Wackerly - Lecture 25 338 STA 2023 Final Exam Locations Tuesday, 12/17/02, 10:00 am – 12:00 noon Thought: Start every day with a smile and get it over with. -W.C. Fields Final Exam Sections Donte Ford Salvador Gezan Saurabh Kumar Adam Meyers Antara Roy Sourish Saha Mathew Smeltzer Kelly Sodec Michael Thomas £¡¡ §£¡¡ §& ! %¤¢¢©¢¢¡ &£¡¡ §¥¡¡ §#¡ %¤¢¨©¢¡ £¢('¢©"¤¢¡ ¡¡ §¡£¡¡ §£¡ #¡¡ §!¡¡ §!¡ ¢©¢¡ ¥# §¥¡¡ §&¡ $¨¢¢¢¡ ¡¡ § £¡¡ §#£¡ ¢¢%¤©"¤¢¡ #£¥# § ¡¡ §¡ %$¢©¢¡ "¤¢¢¢©¢¢¡ £¡¡ §¡¡ § ! ¢¢¢¡ ¡¡ §¡¡ §¡¡ ¡¡ §¡¡ §¥£¡ ¢¢©¨¦¤¢¡ David Finlay Location CAR 100 CAR 100 CLB C130 Assignments : Today : Pages 506 –517 Tomorrow : Exer. 11.5, 10, 14, 18, 19, 21 TUR L007 TUR L007 CAR 100 Wednesday : Pages 523 – 532 Thursday : Exer. 11.27, 36, 37, 38, 39, 41 NRN 137 NRN 137 Monday : CLB C130 CAR 100 STA 2023 c D.Wackerly - Lecture 25 339 Instructor STA 2023 c D.Wackerly - Lecture 25 Graphing Relationships Between Two Variables: Scatter Diagrams, Section 2.9 Plot one variable on horiz axis, another on vert. axis Plotted the number of votes for Buchanan in the ‘00 Data set: Gainesville Sun, 12/27/01 election versus the number for Perot in the ‘96 election SAT UWF 3.3 1187.7 UCF 3.6 1148.8 3.2 1027.4 FIU 3.5 1109.8 USF 3.5 1085.0 UNF 3.5 1125.1 FAU 3.3 1051.0 FGCU 3.4 1024.6 SAT UF 3.9 1265.4 FSU 3.7 FAMU School (both candidates from the same “minority party”) 1077.5 GPA Graph ) GPA School Plot; select(double click) Y variable, select(double click) X variable, OK Palm Beach county corresponds to the “unusual” 0 point in the graph. ) Graph Plot; select(double click) Y variable, select(double click) X variable, OK 340 0 Suppose that increasing change in ? £¥ §¦ EXAMPLES : ¡ , and of by one “unit” produces a units. © ¨ £¡ ¤¢ an “independent” variable, 342 Mathematical Relationship (Formula)? Chapter 9 : Simple Linear Regression Relationship between a “response” variable, STA 2023 c D.Wackerly - Lecture 25 ¥ 341 STA 2023 c D.Wackerly - Lecture 25 – LINEAR relationship ¡ – Plot : Depend. var (dependent var.) on vertical axis, ¥ ¡ Indep. var (independent var.) on horizontal axis and get a ¥ SAT score H.S. GPA % Buchanan votes % Perot votes age in weeks height of plant time stored potency of antibiotic conc. of enzyme amount of mental activ. in blood exhibited by child fixed value observed response not random random because don’t straight line. ¥ !  ©  ¨  ¡     ¨ intercept slope always get same response ¥ for fixed -value Ex. Plants of same age not all same height 343 STA 2023 c D.Wackerly - Lecture 25 STA 2023 c D.Wackerly - Lecture 25 344 Deterministic Model ¥ ¥ !  , .  ¨ % !  $¡  ¨  0 0 Due to variation in environmental conditions, © 0 In practice, many relationships NOT deterministic. ¥ ¡  ¡ Probabilistic Model ! when  #¥  "¥  Intercept = value of   ¡  – If .   & 0 – ¡ Given , formula gives EXACT value of random error deterministic part Criteria for fitting line: individuals, etc. © ¨ !  % ¨  y pred. by line '¡ ) 0%0  is square of the distance between actual value of '¥ 5"¥  and the point on the line where . '¡ 1 act. obs.y 3 42 £ ' ¥ ¥ 42 Notation : each point § ¡ 40 0 23 distances from data points to the line '¥ ¦ 27 Want : Minimize the sum of squared vertical pairs ' (¡ 26 ¥ ¦ 34 § 16 £ ¤¡ 12 0 Fake data, four 0 STA 2023 c D.Wackerly - Lecture 25  Fake Example : 346 '¥ '¡ 1156 676 884 23 729 529 621 42 1600 1764 3629 3225 3' ¡ 3' ¥ £  £  !#! ¥ #¢¡   ¦   ¨¦ ¦ 1 '¡     ¡! & #!   ¦ ££ ¢& ! ¨ ©  ¨ 1 ¡ 1 ¢ £'  ¦ ¨¦  ¡ ' (¡ )  © ¨ 0   ¦ ¨¦ ¨ © ¨ Where: 3377 !  ¢ "¡ !  1 ! © ¥ ¨ ©¦¦ ¨§¥42 £ ' ¥ ¤3  %£ ¨ © Solutions: 26 27 to MINIMIZE 16 40 and 12 34 Will choose ' ¡' ¥  345 ¡ STA 2023 c D.Wackerly - Lecture 25  ¡ ' ¡' ¥  1  3£' ¥ © ¨    1 '¥ 1  £ ¤¡ 1 ¡ 3' ¥ ¡ £ ' ¢%§¥  3 §¥ £ ¡ ¥ ' ¦ 1 ¡ ¥ ' ¦ 1  ¦ ¨¦   ¦ ¨¦ ¨ Thus the straight-line model that best fits the data is   ¡  #¥  ¢ Based on the fake data, if 347 ¡ the value of ? STA 2023 c D.Wackerly - Lecture 25 STA 2023 c D.Wackerly - Lecture 25 348 , how do we predict  £ ! "¢£ "¡   ¥ !  # ! ¡! ¢& #!  £ ££ ¢& ! !  ¡ ¡! ¢& #! Example : Concrete with cement content of 200lbs/cu yard. Strength  © ¨  !# ! ¢¢ $  ()# ¡! ¦¥ #& & '£ 1 # !  %£ 1 # &¡ ! (£ ##  #¥ © ¨ 1 ¡   ¨ Thus the straight-line model that best fits the data is ! #  !  ¥ # !  %£ 1 !# ! ¢ #  ¡ 8.7090 ¦ ¨¦ 6.5682 # ! ¢ %£ 12.9652  ¦ ¨¦ 6.148  8.74 1 1.2727 1 .5055 ¡! !  " ¥! ¨¡  # # 3.2041 ¦ 0.711  1.79  1.3009  ¦ .6448  2.6244 # ! ¢¢ $"£ 0.803 1 1.62 and ¡! ¦¥ #& 1.5184 ¡! ! ¢ # 1.0816 # 2.1316 £ 1.040 1.46 3£ 1.4536  3' ¡ ' ¡' ¥ 1.1257 ¦ 1.8769  ¦ 1.061  1.37  ! ¢¢¥ #& 1.5880 1 1.5154 £ 1.6641 1.231 ¡ ¥ #& ! 1.29 %£ 1.5754 &¡ ! %£ ## 1.6952 £ 1.4641  1.302 3' ¥ £ ' ¢ ¡ (100’/lb) 1 1.21 £' ¥ Rat. ¥! ¨¡  # Wat/Cem 349 STA 2023 c D.Wackerly - Lecture 25 STA 2023 c D.Wackerly - Lecture 26 350 STA 2023 Final Exam Locations Minitab? Tuesday, 12/17/02, 10:00 am – 12:00 noon 0 Stat 0 Select “response” variable, select “predictor” 0 Click OK Regression ) Regression ) Final Exam Instructor David Finlay The regression equation is Strength = 2.56 - 1.05 W/C rat Sections Donte Ford Predictor Constant W/C rat Coef 2.5606 -1.05439 StDev 0.1400 0.09527 T 18.28 -11.07 Salvador Gezan P 0.000 0.000 Saurabh Kumar Adam Meyers S = 0.04608 R-sq = 96.8% R-Sq(adj) = 96.0% Antara Roy Analysis of Variance Source Regression Error Total DF 1 4 5 Sourish Saha SS 0.26007 0.00849 0.26857 MS 0.26007 0.00212 F 122.48 P 0.000 Mathew Smeltzer Kelly Sodec Michael Thomas 351 CAR 100 CAR 100 CLB C130 TUR L007 TUR L007 CAR 100 NRN 137 NRN 137 CLB C130 CAR 100 STA 2023 c D.Wackerly - Lecture 26 Location £¡¡ §£¡¡ §& ! %¤¢¢©¢¢¡ &£¡¡ §¥¡¡ §#¡ %¤¢¨©¢¡ £¢('¢©"¤¢¡ ¡¡ §¡£¡¡ §£¡ #¡¡ §!¡¡ §!¡ ¢©¢¡ ¥# §¥¡¡ §&¡ $¨¢¢¢¡ ¢¢%¤©"¤¢¡ ¡¡ § £¡¡ §#£¡ #£¥# § ¡¡ §¡ %$¢©¢¡ "¤¢¢¢©¢¢¡ £¡¡ §¡¡ § ! ¢¢¢¡ ¡¡ §¡¡ §¡¡ ¡¡ §¡¡ §¥£¡ ¢¢©¨¦¤¢¡ Regression Analysis STA 2023 c D.Wackerly - Lecture 26 352 Last Time: Probabilistic Model   & ¥ ! random error !  $¡  ¨  ¨ Use data to estimate (intercept) and ¨  ¨ % deterministic part you realize you haven’t fallen asleep yet. (slope). © © Thought: Stress is when you wake up screaming and Assignments : ¡ 1 © ¨     ¨¦ ¦ ¨  © ¨ Where: '¡  '¥   1 ¡ ' ¡' ¥ 3£' ¥    1 3' ¥ £ ¤¡ ¡ 1 ¡ £ ' ¢%§¥  3 §¥ £ ¡ ¥ ' ¦ 1 1 ¥ ' ¦ ¡   ¦ ¨¦  ¦ ¨¦ Tuesday : Exer. 11.47, 48, 49, 51, 55 ¦ ¨¦ Monday : Pages 537 – 542  Thursday : Exer. 11.27, 36, 37, 38, 39, 41 ¥ Solutions:  Today : Pages 523 – 532 353 STA 2023 c D.Wackerly - Lecture 26 STA 2023 c D.Wackerly - Lecture 26 354 What about this “random error” part of the model? Question : How do we decide whether the independent variable is useful in helping us predict the value of the dependent variable? 0 Could use two prediction equations:  ¡ ¨ describes the variability of ! £ – The “parameter” Does not use © ¨   ¡ INDEPENDENT. ¥ – The errors for different observations are Uses ¥ £ . ¡ distribution with mean 0 and standard deviation,  each have a Normal ¥ ¡ £ ¢ & – The errors,  0 Assumptions: linearly data points around the linear relationship. ¥ 0 The first does not use at all, it always predicts a typical (average) y-value. Regardless of the value of ¡ value will be . ¡ 1 ¥ ALWAYS predict that 0 The second uses a (least squares) linear function of value. Different 1 ¥ ¥ vales of ¡ to predict the corresponding will give different predictors for the ¡ corresponding values of . STA 2023 c D.Wackerly - Lecture 26 356 ¡ ¥ Does using 355 STA 2023 c D.Wackerly - Lecture 26 help in predicting ? Inferences about the Slope ¡  & , (independent £ is the standard ). (p. 529) where ©© ¨ ¥  § ¨¤ © ¨  0  ¦    £3 £ £ ¦ ¨¦ §     ¡ 0 0   £ model. £¥ §¦  ¦ 1  0 © ¨ ¥ ¥ ¤ § ¨¤  0 0  ¥ 0 § ¨¤  ¥  ¤ © ¨ ¦ ¥ 0 ¡ That is, tend  ¦ © ¨ © ¨ ¥ £¡ ¤¢ £¥ §¦ ! "£ © ¨ ¥ § ¨¤ £¡ ¤¢ versus ? has a NORMAL distribution. (p. 529)  ¤ © ¨ ¦ ¥ – Do antibiotics stored for a longer time © ¨   ¡ deviation of the “error term” in the probabilistic tends to produce a larger to have lower potencies ¨  tend to value of .   © ¨ ¦ ¥ ¤ HOW? and ? versus Or, whether increasing (p. 529) Assumptions about error terms tends to produce a larger value of . – Do students with higher HS GPA’s © 0 is helpful in predicting have higher SAT’S  Least squares estimator does not help. Might also be interested deciding whether increasing  is true, pairs of observations ' ¥ If is true, Have: § If versus £ (¡ ' 0 Test has a STANDARD NORMAL distribution. 0 3 £ Need : an estimate for 357 STA 2023 c D.Wackerly - Lecture 26 STA 2023 c D.Wackerly - Lecture 26 358 Example : Concrete with cement content of 200lbs/cu . yard.  3' ¡ ' ¡' ¥ 1.4641 1.6952 1.5754 1.231 1.6641 1.5154 1.5880 £' ¥ ! ¨ © 0  ¨ ¡ ' (¡ ) 1 © 1.37 1.061 1.8769 1.1257 1.4536 1.46 1.040 2.1316 1.0816 1.5184 1.62 0.803 2.6244 .6448 1.3009 1.79 0.711 3.2041 .5055 1.2727 8.74 ¡ ¢£  1  3' ¡    ¨¦ ¦ ©    3 ©  ¦ ¦ ¦ §  ¦ 1 ¨ : 6.148 12.9652 6.5682 8.7090 3' ¥  ¢ £' ¦ 1 £ © ¦ 0 ¦ 0 ¦ An unbiased estimator for 1.302 1.29 the data points to the fitted line. (100’/lb) £ ' ¢ ¡ ©¦¦ ¨§¥42 £ ' ¥ ¤3 1.21 Rat. SSE is the sum of squared vertical distances from Computing formula: if Strength Wat/Cem 0 3¡   – How many degrees of freedom associated with £ %£ &¡ ! %£ ##  ¡! !  # 3£ 1 1 # 1 ¡! !  # ¥¡ ! ¨ #  ¦ ¨¦  ! ¢¢¥ #& # £ ¡! ¦¥ #&  1 # !  %£ £ ¡ ¥ #& !  # ! ¢¢ $"£   ¦ ¨¦ © ¨ 360 ¨¥ © ¤  versus p-value © § ¨ ¨ ¦¤ £ RR £ ¤ Consider testing (p. 530)  § £ ¥ ¦¤ ¤ § ¥ £  ¤ ¨ © 3£' ¡ OR £ ¨ © § ¤ ¨ ¤ ¥¤ 1 §   ¤ or (tail area)   ¤ 3 ¥¤ £  ¥ ¤ 1 ¤  3  © ¨  ©  ¢ # #! © hypothesized value  standard error # $#  1  © 1 ¦   ¨ ! ¡  ¤ 1 ¤ freedom. distribution with  has a NULL HYPOTHESIS  ¦ 1 3' ¡ ¡    ¨¦ ¦    0 ©¦ ¨¦ #     ! ¤ Formula Sheet ! estimator !  ¨ TEST STATISTIC 1  OR ! "! £   ¦ ¨¦ STA 2023 c D.Wackerly - Lecture 26  3  ¦ ¨¦ 359  STA 2023 c D.Wackerly - Lecture 26 1 ? ¥! ¨¡  # estimator for degrees of ¡ ¢ !   3¡ ¡ 361  © § ¤ ¡ ¡ ¨¦¥£¢ 362 Ex. : Concrete example. Is there sufficient evidence to indicate that a increasing the water-cement ratio tends to decrease the strength of the concrete? Use .   !        3 ¥ ¤  ¦  % 3 ¦   ¡ 0 ¨ ©   formula sheet table  £ formula sheet standard errors ! estimator STA 2023 c D.Wackerly - Lecture 26 (p. 531)   CI for STA 2023 c D.Wackerly - Lecture 26 true change in average strength for a one unit   ¥ ¤ increase in water-cement ratio. © ¨ Example : Concrete with cement content of 200lbs/cu (1)  © ¤ § ¨¥ ©  !¤ ¨¥ level test, RR :   !  # ! ¢ %£  "0 # $   ! 0 1 % &0 1 value = % &0 Minitab? Regression Analysis The regression equation is Strength = 2.56 - 1.05 W/C rat Predictor Constant W/C rat S = 0.04608 Chef 2.5606 -1.05439 StDev 0.1400 0.09527 R-sq = 96.8% T 18.28 -11.07 P 0.000 0.000 R-Sq(adj) = 96.0% Analysis of Variance Source Regression Error Total DF 1 4 5 SS 0.26007 0.00849 0.26857 MS 0.26007 0.00212 F 122.48 P 0.000  value?  of concrete   ! water-cement ratio tends to decrease the strength ¤ confidence ) to indicate that increasing the  ¤ level of significance” ( or with ¤  ¤ the in favor of ? §  1  ¨¦ ¦ 0  £ !£ ! %¢£  ¢¢ # #! claim that there is sufficient evidence at  ¦   ¨¦ ¦ “ 1 ¦  1 ¥! ¨¡  # In terms of this problem: LEVEL!! ! ! "£ $ © 363 1  & ¢¢ ! ¦ THE ! ¢¢ ! # ! ¢ %£ ¡   ¦ reject    ¡! !  # ¨ © d.f. = 1 0 Conclusion : is strength for a 1 unit change in water/cement ratio. £ !£ ! %¢£  1 ¡  ! Test statistic : Want : 98% confidence interval for the change in STA 2023 c D.Wackerly - Lecture 26 (2) ! % yard. AT ...
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This note was uploaded on 07/28/2011 for the course STA 2023 taught by Professor Ripol during the Fall '08 term at University of Florida.

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