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

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