Week15-2up - Instructor David Finlay Michael Thomas Kelly...

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