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6 Lecture Outline Thur. Jan. 29 Review from Lecture 5 Sampling with vs. without replacement Confidence Intervals Case Study 2.1.1 Two sample t-test and confidence intervals (Chapter 2.3) Levene's test for equality of variances (Chapter 4.5.3) Terminology Review A statistic is any quantity computed from the sample, e.g., sample mean, sample standard deviation, minimum of sample. The sampling distribution of a statistic for a sample of size n is the probability distribution for the statistic over repeated random samples of size n. The sampling distribution of the value for a sample of size 1 is called the population distribution. Standard Deviations and Standard Errors The standard deviation of a statistic is the standard deviation of the statistic's probability distribution, i.e., the square root of the average squared distance of the statistic from its mean over repeated samples. The standard error of a statistic is an estimate of the statistics' standard deviation. Example: For sampling with replacement, s SD(Y ) = , SE (Y ) = n n Sampling with vs. without replacement For a sample of size n from a population of size N without replacement, N -n s N -n SD(Y ) = , SE (Y ) = N -1 N -1 n n The factor N - n is called the finite population N-1 correction (FPC). Note that the FPC is near 1 if N/n>50 so that we regard sampling with replacement and sampling without replacement as essentially equivalent if N/ n>50. One-sample t-tools and paired t-test Testing hypotheses about the mean difference in pairs is equivalent to testing hypotheses about the mean of a single population Probability model: Simple random sample with replacement from population. H 0 : = *, H1 : * Test statistic: t = Y - * = Y - * SE (Y ) s/ n p-value Fact: If H0 is true, then t has the Student's t-distribution with n-1 degrees of freedom Can look up quantiles of t-distribution in Table A.2. The (2-sided) p-value is the proportion of random samples with absolute value of t >= observed test statistic |to| if H0 is true. Schizophrenia example: to=3.23, p-value = Prob>|t| = . 0061. The reliability of the p-value (as the probability of observing as extreme a test statistic as the one actually observed if H0 is true) is only guaranteed if the probability model of random sampling is correct if the data is collected haphazardly rather than through random p-value animation 8 7 6 5 4 3 2 1 0 -0.4 -0.3 -0.2 -0.1 .0 X .1 .2 .3 .4 Y Estim Mean 0.1986666667 Hypoth Mean 0 T Ratio 3.2289280811 P Value 0.0060615436 Sample Size = 15 Matched pairs t-test in JMP Click Analyze, Matched Pairs, put two columns (e.g., affected and unaffected) into Y, Paired Response. Can also use one-sample t-test. Click Analyze, Distribution, put difference into Y, columns. Then click red triangle under difference and click test mean. Confidence Intervals Point estimate: a single number used as the best estimate Y of a population parameter, e.g., for . Interval estimate (confidence interval): range of values used as an estimate of a population parameter. Uses of a confidence interval: Provides a range of values that is "likely" to contain the true parameter. Confidence interval can be thought of as the range of values for the parameter that are "plausible" given the data. Conveys precision of point estimate as an estimate of population parameter. Confidence interval construction A confidence interval typically takes the form: point estimate margin of error The margin of error depends on two factors: Standard error of the estimate Degree of "confidence" we want. Margin of error = Multiplier for degree of confidence * SE of estimate For a 95% confidence interval, the multiplier for degree of confidence is about 2 in most cases. CI for population mean If the population distribution of Y is normal and the sample is a random sample, 100(1 - )% CI for mean of single population: Y tn-1 (1 - / 2) * SE (Y ) = s Y tn-1 (1 - / 2) * n For schizophrenia data, 95% CI: .199cm3 2.145 0.615cm3 = 0.067cm3 to 0.331cm3 Interpretation of CIs A 95% confidence interval will contain the true parameter (e.g., the population mean) 95% of the time if repeated random samples are taken. It is impossible to say whether it is successful or not in any particular case, i.e., we know that the CI will usually contain the true mean under random sampling but we do not know for the schizophrenia data if the CI (0.067cm3 ,0.331cm3) contains the true mean difference. Confidence interval will only have guaranteed coverage if the assumptions about the probability model are correct, in particular the sample must be a sample. random Confidence Intervals in JMP For both methods of doing paired t-test (Analyze, Matched Pairs or Analyze, Distribution), the 95% confidence intervals for the mean are shown on the output. Factors determining width of confidence interval 100(1 - )% confidence interval for under random sampling with replacement from a normal population:Y tn-1 (1 - / 2) * SE (Y ) = s Y tn-1 (1 - / 2) * n Factors determining width of confidence interval: Population standard deviation Sample size n Degree of confidence 1 - Case Study 2.1.1 Background: During a severe winter storm in New England, 59 English sparrows were found freezing and brought to Bumpus' laboratory 24 died and 35 survived. Broad question: Did those that perish do so because they lacked physical characteristics enabling them to withstand the intensity of this episode of selective elimination? Specific questions: Do humerus (arm bone) lengths tend to be different for survivors than for those that perished? If so, how large is the difference? Structure of Data Two independent samples Observational study cannot infer a causal relationship between humerus length and survival Sparrows were not collected randomly. Fictitious probability model: Independent simple random samples with replacement from two populations (sparrows that died and sparrows that survived). See Display 2.7 Two-sample t-test Population parameters: 1, 1, 2 , 2 H0: 1 - 2 = 0 , H1: 1 - 2 0 Equal spread model: 1 = 2 (call it ) Statistics from samples of2 size n1 and n2 2 from pops. 1 and 2: Y1, Y2 , s1 , s2 For Bumpus' data: Y1 = .728, Y2 = .738, Y2 - Y1 = .010, s1 = .024, s2 = .020 Sampling Distribution of 1 1 SD (Y1 - Y2 ) = + n1 n2 Y1 - Y2 (equal spread model) SE (Y1 - Y2 ) = s p Pooled estimate of sp 2 1 1 + n1 n2 2 2 : 2 (n1 - 1) s1 + (n2 - 1) s2 = (n1 - 1) + (n2 - 1) See Display 2.8 Confidence Interval for - 1 2 Assume the population distributions of group 1 and group 2 are both normal. 100(1- )% confidence interval for 1 - 2 : (Y1 - Y2 ) tdf (1 - / 2) * SE (Y1 - Y2 ) = (Y1 - Y2 ) tn1 +n2 -2 (1 - / 2) * s p 1 1 + n1 n2 For 95% confidence interval, t (.975) 2 df Bumpus' data: 95% confidence interval: - 0.01008 2.009 * 0.00567 = (-0.02143,0.00127)inches Two sample t-test , H0: 1 - 2 = * H1: 1 - 2 * Test statistic: t = (Y1 - Y2 ) - * . Values of t that are farther SE (Y1 - Y2 ) from zero are more implausible under H0 If population distributions are normal with equal , then if H0 is true, the test statistic t has a Student's t distribution n1 + n2 - 2 with degrees of freedom. p-value equals probability that |t| would be greater than observed |t| under random sampling model if H0 is true; calculated from Student's t distribution. For Bumpus data, two-sided p-value = .0809, suggestive but inconclusive evidence of a difference Two sample tests and CIs in JMP Click on Analyze, Fit Y by X, put Group variable in X and response variable in Y, and click OK Click on red triangle next to Oneway Analysis and click Means/ANOVA/t-test (Means/ANOVA/pooled t in JMP version 5). To see the means and standard deviations themselves, click on Means and Std Dev under red triangle Oneway Analysis of Humerus By Group 0.8 0.78 0.76 Humerus 0.74 0.72 0.7 0.68 0.66 0.64 Perished Group Survived t Test Perished-Survived Assuming equal variances Difference Std Err Dif Upper CL Dif Lower CL Dif Confidence -0.01008 0.00567 0.00128 -0.02145 0.95 t Ratio DF Prob > |t| Prob > t Prob < t -1.777 57 0.0809 0.9595 0.0405 -0.020 -0.010 .000 .005.010 .015 Bumpus' Data Revisited Bumpus concluded that sparrows were subjected to stabilizing selection birds that were markedly different from the average were more likely to have died. Bumpus (1898): "The process of selective elimination is most severe with extremely variable individuals, no matter in what direction the variations may occur. It is quite as dangerous to be conspicuously above a certain standard of organic excellence as it is to be conspicuously below the standard. It is the type that nature favors." Bumpus' hypothesis is that the variance of physical characteristics in the survivor group should be smaller than the variance in the perished group Testing Equal Variances Two independent samples from populations with variances 12 and 2 2 H0: 2 = 2 vs. H1: 1 2 2 2 1 2 Levene's Test Section 4.5.3 In JMP, Fit Y by X, under red triangle next to Oneway Analysis of humerus by group, click Unequal Variances. Use Levene's test. p-value = .4548, no evidence that variances are not equal, thus no evidence for Bumpus' hypothesis.
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notes16.ppt
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Stat 112: Lecture 16 Notes Finish Chapter 6: Influential Points for Multiple Regression (Section 6.7) Assessing the Independence Assumptions and Remedies for Its Violation (Section 6.8) Homework 5 due next Thursday. I will email it tonight. Ple...
notes10.ppt
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Stat 112: Lecture 10 Notes Fitting Curvilinear Relationships Polynomial Regression (Ch. 5.2.1) Transformations (Ch. 5.2.2-5.2.4) Schedule: Homework 3 due on Thursday. Quiz 2 next Curvilinear Relationship Reconsider the simple regression prob...
notes9.ppt
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Stat 112: Lecture 9 Notes Homework 3: Due next Thursday Prediction Intervals for Multiple Regression (Chapter 4.5) Multicollinearity (Chapter 4.6). Summary of F tests Partial F tests are used to test whether a subset of the slopes in multiple re...
notes21.ppt
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Stat 112: Lecture 21 Notes Model Building (Brief Discussion) Chapter 9.1: One way Analysis of Variance. Homework 6 is due Friday, Dec. 1st. I will be e-mailing you tonight or tomorrow some comments on your project ideas. I will have the quizzes ...
notes22.ppt
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Stat 112: Lecture 22 Notes Chapter 9.1: One-way Analysis of Variance. Chapter 9.3: Two-way Analysis of Variance Homework 6 is due on Friday. Errors in Hypothesis Testing State of World Null Hypothesis True Accept Null Correct Hypothesis Decision ...
notes3.ppt
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Stat 112 Notes 3 Homework 1 is due at the beginning of class next Thursday. Relationship between Y=Time to Bed Last Night and X=Cups of Coffee Drunk Yesterday Time to bed last night (hours past 12 noon) 17 16 15 14 13 12 11 10 9 8 -1 0 1 2 3 4 5 6...
lect8.ppt
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Lecture 8 Resistance of two-sample t-tools and outliers (Chapters 3.3-3.4) Transformations of the Data (Chapter 3.5) Outliers and resistance Outliers are observations relatively far from their estimated means. Outliers may arise either (a) if t...
notes15.ppt
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Stat 112: Lecture 15 Notes Finish Chapter 6: Review on Checking Assumptions (Section 6.4-6.6) Outliers and Influential Points (Section 6.7) Homework 4 is due this Thursday. Please let me know of any ideas you want to discuss for the final proje...
notes19.ppt
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Stat 112: Lecture 19 Notes Chapter 7.2: Interaction Variables Thursday: Paragraph on Project Due Interaction Interaction is a three-variable concept. One of these is the response variable (Y) and the other two are explanatory variables (X1 and X2...
lect4.ppt
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Lecture 4 Outline Chapter 1.3, 1.5 Control in Experimental Design Causal Inference in Observational Studies Summarizing Data Numerical methods Graphical methods The meaning of the causal inference In the motivation-creativity study, we conclu...
notes23.ppt
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Stat 112: Lecture 23 Notes Chapter 9.3: Two-way Analysis of Variance Schedule: Homework 6 is due on Friday. Quiz 4 is next Tuesday. Final homework assignment will be e-mailed this weekend and due next Monday. Final Project due on Dec. 19th Two...
lect9.ppt
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Lecture 9 Today: Log transformation: interpretation for population inference (3.5) Rank sum test (4.2) Wilcoxon signed-rank test (4.4.2) Thursday: Welchs t-test (4.3.2) Practical vs. statistical significance (4.5.1) Presentation of statisti...
lect24.ppt
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Lecture 24: Thurs., April 8th Inference for Multiple Regression Types of inferences: Confidence intervals/hypothesis tests for regression coefficients Confidence intervals for mean response, prediction intervals Overall usefulness of predictors (...
notes9.doc
Path: UPenn >> STAT >> 512 Fall, 2008
Description: Statistics 512 Notes 9: The Monte Carlo Method Continued The Monte Carlo method: Consider a function g ( X ) of a random vector X where X has density f ( X ) . Consider the expected value of g ( X ) : E[ g ( X )] = g ( x ) f ( x )dx . Suppose we tak...
notes15.doc
Path: UPenn >> STAT >> 512 Fall, 2008
Description: Statistics 512 Notes 15: Properties of Maximum Likelihood Estimates Continued Computation of maximum likelihood estimates Example 2: Logistic distribution. Let X 1 ,K , X n be iid with density exp{( x )} f ( x; ) = 2 , < x < , < < . (1 + exp{( ...
notes10.doc
Path: UPenn >> STAT >> 512 Fall, 2008
Description: Statistics 512 Notes 10: Bootstrap Procedures Bootstrap standard errors X 1 ,K , X n iid with CDF F and variance 2 . 2 X1 + L + X n 1 Var = n 2 Var ( X 1 + L + X n ) = n . n SD( X ) = . n s SE ( X ) = We estimate SD( X ) by where s is the n ...
notes6.doc
Path: UPenn >> STAT >> 512 Fall, 2008
Description: Statistics 512 Notes 6 Hypothesis Testing Continued Quick Review on Hypothesis Testing Goal: Decide between two hypotheses about a parameter of interest H 0 : 0 H1 : 1 , where 0 U1 = . Null vs. Alternative Hypothesis: The alternative hypothe...
obsstudies_syllabus.doc
Path: UPenn >> STAT >> 921 Fall, 2009
Description: Statistics 921: Design and Analysis of Experiments and Observational Studies Syllabus Professor: Dylan Small E-mail: dsmall@wharton.upenn.edu Office: 464 Huntsman Hall Office Hours: Tuesday, 4:45-5:45; Wednesday, 9-10; Thursday, 4:45-5:45; by appoint...
notes19.doc
Path: UPenn >> STAT >> 512 Fall, 2008
Description: Statistics 512 Notes 19: Example 2: Gamma distribution: 1 x 1e x / , 0 < x < f ( x; , ) = ( ) 0, elsewhere log f ( X ; , ) \'( ) = log + log X ( ) log f ( X ; , ) X = + 2 I ( , ) = E , \'( )( ) + ( \'( ) ) 2 1 2 ( ( ) ...
notes11.doc
Path: UPenn >> STAT >> 512 Fall, 2008
Description: Statistics 512 Notes 11: Motivation for percentile bootstrap confidence intervals: ^ Suppose there exists a monotone transformation U = m( ) such that U ~ N ( , c 2 ) where = m( ) . We do not suppose that we know the transformation, only that one ^...
notes25.doc
Path: UPenn >> STAT >> 512 Fall, 2008
Description: Statistics 512 Notes 25: Decision Theory Decision Theoretic Approach to Statistics: Views statistics as a mathematical theory for making decisions in the face of uncertainty. The Decision Theory Paradigm: The decision maker chooses an action a from a...
notes8.doc
Path: UPenn >> STAT >> 512 Fall, 2008
Description: Statistics 512 Notes 8: The Monte Carlo Method The t-test Let X 1 ,K , X n be iid with mean and unknown distribution. Consider the hypotheses H 0 : = 0 vs. H1 : > 0 If the distribution of the X i is normal (with unknown variance), then a test with...
notes4.doc
Path: UPenn >> STAT >> 512 Fall, 2008
Description: Statistics 512 Notes 4 Confidence Intervals Continued Role of Asymptotic (Large Sample) Approximations in Statistics: It is often difficult to find the finite sample sampling distribution of an estimator or statistic. Review of Limiting Distributions...
notes18.doc
Path: UPenn >> STAT >> 512 Fall, 2008
Description: Statistics 512 Notes 18: Multiparameter maximum likelihood estimation We consider X 1 ,K , X n iid with pdf f ( x; ), where = (1 ,K , p ) is p-dimensional. As before, L( ) = f ( xi ;1 ,K , p ) i =1 n The maximum likelihood estimate is MLE = ...
notes7.doc
Path: UPenn >> STAT >> 512 Fall, 2008
Description: Statistics 512 Notes 7 Hypothesis Testing Continued Testing a normal mean Example: A highway patrol officer believes that the average speed of cars traveling over a certain stretch of highway exceeds the posted limit of 55 mph. The speeds of a random...
notes22.doc
Path: UPenn >> STAT >> 512 Fall, 2008
Description: Statistics 512 Notes 22: Wrap up of Sufficiency, Most Powerful Tests Rao-Blackwell Theorem: Theorem 7.3.1 (stated a little differently): Let X 1 ,K , X n be an iid sample from the pdf or pmf f ( x; ) , . Let u ( X 1 ,K , X n ) be a sufficient stat...
notes16.doc
Path: UPenn >> STAT >> 512 Fall, 2008
Description: Statistics 512 Notes 16: Efficiency of Estimators and the Asymptotic Efficiency of the MLE Method of moments estimator X 1 ,K , X n iid f ( x; ), . Find E ( X i ) = h( ) . ^ Method of moments estimator MOM = h -1 ( X ) . Examples: (1) X 1 ,K , X ...
actdata.txt
Path: UPenn >> STAT >> 475 Fall, 2009
Description: scores 5 13 7 20 11 16 18 4 7 17 15 9 14 6 5 11 11 14 3 2 18 15 12 18 16 12 11 12 25 8 9 13 10 3 15 17 15 6 11 15 14 10 14 7 12 6 3 15 8 9 5 5 4 12 12 10 4 18 8 1 14 0 8 6 5 0 8 7 9 4 12 10 12 11 4 8 12 15 13 10 3 9 13 3 13 3 7 7 14 11 14 13 13 11 11...
hospitals.txt
Path: UPenn >> STAT >> 475 Fall, 2009
Description: discharges beds 57 10 35 16 23 20 120 24 92 25 98 26 118 30 66 34 95 38 87 40 141 43 260 50 229 50 247 57 91 62 240 64 255 67 233 69 315 70 200 73 266 81 120 91 228 96 362 100 414 100 518 103 389 110 273 111 440 116 431 120 534 122 535 127 426 130 50...
golf.txt
Path: UPenn >> STAT >> 112 Fall, 2008
Description: [LEXIS(R)-NEXIS(R)] [Main Menu] [Help] [Sources] [Results List] [Return to Search] [Previous Document] [Next Document] [Full View] [Kwic View] -- ...
xvector.txt
Path: UPenn >> STAT >> 512 Fall, 2008
Description: xvector=c(-0.228, 0.415, 0.181, 0.012, 0.727, 0.046, 0.302, -0.036, 0.462, 0.619, 0.366, 0.303)...
cholesterol.txt
Path: UPenn >> STAT >> 512 Fall, 2008
Description: cholesterol=c(95, 108, 108, 114, 115, 124, 129, 129, 131, 131, 135, 136, 136, 139, 140,142, 142, 143, 143, 144, 144, 145, 145, 148, 152, 152, 155, 157, 158, 158, 162, 165, 166, 171, 172, 173, 174, 175, 180, 181, 189, 192, 194, 197, 204, 220, 223, 226...
Lecture15.txt
Path: UPenn >> STAT >> 541 Fall, 2008
Description: - LECTURE 15: * ORG: - Writing Homework: \"Style\", exercise 5.2, 3. part; hand in your solution ON PAPER. Due: Tue, Nov 4, in class. Be prepared to discuss this piece in class. - Homework 7 to be posted end of the week. *...
nya90-97.txt
Path: UPenn >> STAT >> 712 Fall, 2009
Description: DATE COMPOSITE INDUSTRIAL TRANS. UTILITY FINANCE (yyyymmdd) 19900102 198.00 236.68 182.25 102.92 158.17 19900103 197...
Lecture19.txt
Path: UPenn >> STAT >> 541 Fall, 2008
Description: = LECTURE 19: * Quiz: - to what type of null hypothesis do permutation tests apply? . or . - list all the examples you can think of where permutation tests apply: . - what is the theory underlying permutation tests...
Lecture09.txt
Path: UPenn >> STAT >> 541 Fall, 2008
Description: = LECTURE 9 * RECAP: Linear model theory - V[b] - Meaning of (X^T X)/(N-1) and its limiting behavior as N->Inf - V[y], V[yhat], V[r] - Meaning of H_ij = xi^T (X^T X)^{-1} xj = \'<xi, xj>\' - Total variance of y, yhat, r - Connection ...
Lecture17.txt
Path: UPenn >> STAT >> 541 Fall, 2008
Description: - LECTURE 17: * ORG: HW 7 * Q: Name for the null-sufficient sampling method for model diagnostics? precedents: bootstrap, ACE, MARS, PRIM, Jackknife, . * IN THE NEWS: Elections www.fivethirtyeight.com: They seem to use a sampling/simula...
Lecture21.txt
Path: UPenn >> STAT >> 541 Fall, 2008
Description: = LECTURE 21 * Style: Please, hand in your re-writes of 6.2.4 and 6.2.5. Discussion. 6.2.4: Mucosal and vascular permeability altered by a toxin elaborated by the vibrio is a current hypothesis to explain this k...
Lecture14.txt
Path: UPenn >> STAT >> 541 Fall, 2008
Description: - LECTURE 14: * Quiz: - What is yhat(xx)? . - How do you find a stderr and stderr.est for yhat(xx)? compute var(yhat(xx), estimate sigma with s - What guarantees does a CI for yhat(xx) make? P[ <xx,beta> in CI ] = .95 -...
Lecture22.txt
Path: UPenn >> STAT >> 541 Fall, 2008
Description: - LECTURE 22: * ORG: - Homework 8 due Tue, Dec 2, before class (not 5pm) - New Style exercise, NOT from book, due Tue, Nov 25 in class * Style: \'topic\' and \'stress\' location in sentences - Establish the topic/actor in the beginning of the se...
Lecture13.txt
Path: UPenn >> STAT >> 541 Fall, 2008
Description: - LECTURE 13: * HW 6 due Thu * Degrees of freedom: . Def: dfs = dimension of space considered . Ex.: dfs due to model = dim space of fits - dim(mean space) dfs due to residuals = dim residual space * F-tests for c...
Lecture18.txt
Path: UPenn >> STAT >> 541 Fall, 2008
Description: - LECTURE 18 * ORG: - New HW to be posted by Tue * RECAP: - Collinearity analysis: general tool? - Collinearity effects on individual coefficients? - Extreme collinearity effects on interpretation of coefficients? - Adjusted RSquare?...
Lecture20.txt
Path: UPenn >> STAT >> 541 Fall, 2008
Description: - LECTURE 20: * Style: Read chapter 6 Do Excercises 6.2.4. and 6.2.5. (Do not consult solutions if the book has them.) Bring your re-writes to class and hand in on paper. Due Tue, Nov 18 * Quiz: - What typ...
final.txt
Path: UPenn >> STAT >> 540 Fall, 2008
Description: FINAL EXAM This is the evolving page of the \"Final Exam\" . New Rules: 1) You must work individually for the rest of the semester. 2) You can consult with me, but not with each other. 3) You must declare whatever external resource...
syllabus_Q3_2004.pdf
Path: UPenn >> STAT >> 622 Fall, 2008
Description: Statistics 622 Advanced Quantitative Modeling Professor Robert Stine 444 Huntsman Hall stine@Wharton.upenn.edu Overview This half-semester course extends regression modeling beyond the scope of Statistics 621. These extensions include methods for ...
finalPractice.pdf
Path: UPenn >> STAT >> 431 Fall, 2008
Description: Statistics 431, Fall 2007 Practice Final Exam 1 Practice Final Exam Instructions Open books and open notes. Calculators may be used for numerical computations. No laptops are allowed. On the actual exam you will be asked to write your answers on t...
syllabus553.pdf
Path: UPenn >> STAT >> 553 Fall, 2009
Description: Statistics 553: Machine Learning Mikhail Traskin University of Pennsylvania The Wharton School Department of Statistics January 13, 2009 Instructor Instructor: Mikhail Traskin E-mail: mtraskin@wharton Oce: 466 Jon M. Huntsman Hall Recommended Text [...
final2007.pdf
Path: UPenn >> STAT >> 431 Fall, 2008
Description: Stat 431: Statistical Inference Final Exam December 12, 2007 Name: Instructions Open books and open notes. Calculators may be used for numerical computations. No laptops are allowed. Write your answers on the test pages along with your work in the pr...
midterm2007.pdf
Path: UPenn >> STAT >> 431 Fall, 2008
Description: Stat 431: Statistical Inference Midterm Exam October 24, 2007 Name: Instructions Open notes (no textbooks). Calculators may be used for numerical computations. No laptops are allowed. Write your answers on the test pages along with your work in the p...
homework04.pdf
Path: UPenn >> STAT >> 431 Fall, 2008
Description: Statistics 431, Hwk #4 Due 1:30pm Wed Oct 15 1 All the problems are from J. L. Devore, Probability and Statistics for Engineering and the Sciences, 7th ed. If you use software (e.g. JMP) provide explanation of the output. Problems: 10.2, 10.6, 10....
homework02.pdf
Path: UPenn >> STAT >> 431 Fall, 2008
Description: Statistics 431, Hwk #2 Due 1:30pm Mon Sep 29 1 All the problems are from J. L. Devore, Probability and Statistics for Engineering and the Sciences, 7th ed. Majority of the problems require at most a calculator to solve, however for the problem #56...
homework10.pdf
Path: UPenn >> STAT >> 431 Fall, 2008
Description: Statistics 431, Hwk #10 No due date 1 This homework will not be collected. Solutions to the following problems will be provided on Monday, December 8. Final exam may include one question related to topics of this homework assignment. The following...
homework03.pdf
Path: UPenn >> STAT >> 431 Fall, 2008
Description: Statistics 431, Hwk #3 Due 1:30pm Mon Oct 6 1 All the problems are from J. L. Devore, Probability and Statistics for Engineering and the Sciences, 7th ed. Problems: 9.4, 9.14, 9.24, 9.34, 9.40, 9.46, 9.50, 9.56, 9.72, 9.74, 9.76, 9.88. For problem...
homework05.pdf
Path: UPenn >> STAT >> 431 Fall, 2008
Description: Statistics 431, Hwk #5 Due 1:30pm Mon Oct 27 1 All the problems are from J. L. Devore, Probability and Statistics for Engineering and the Sciences, 7th ed. Problems: 11.2, 11.8, 11.18, 11.24, 11.50, 11.52, 11.56, 11.60. ...
homework09.pdf
Path: UPenn >> STAT >> 431 Fall, 2008
Description: Statistics 431, Hwk #9 Due 1:30pm Wed Dec 3 1 All problems marked with (*) are from The Statistical Sleuth by Fred L. Ramsey and Daniel W. Schafer. You might want to wait until Monday, December 1 class for problem parts 4c and 6d. Mandatory proble...
homework08.pdf
Path: UPenn >> STAT >> 431 Fall, 2008
Description: Statistics 431, Hwk #8 Due 1:30pm Mon Nov 24 1 All the problems are from J. L. Devore, Probability and Statistics for Engineering and the Sciences, 7th ed. Problems: 13.66, 13.70, 13.74, 13.76, 13.78, 13.82. In addition, you may also do the follow...
practiceMidterm.pdf
Path: UPenn >> STAT >> 431 Fall, 2008
Description: Statistics 431, Fall 2007 Practice Midterm Exam 1 Practice Midterm Exam Instructions Open notes (no textbooks). Calculators may be used for numerical computations. No laptops are allowed. On the actual exam you will be asked to write your answers ...
lec01.pdf
Path: UPenn >> STAT >> 431 Fall, 2008
Description: Statistics 431 Statistical Inference Lecture I: Introduction Mikhail Traskin University of Pennsylvania The Wharton School Department of Statistics Stat431: Lecture I p. 1 Course Information I Homepage at http:/www-stat.wharton.upenn.edu/ mtraskin...
homework01.pdf
Path: UPenn >> STAT >> 431 Fall, 2008
Description: Statistics 431, Hwk #1 Due 1:30pm Mon Sep 22 1 All the problems are from J. L. Devore, Probability and Statistics for Engineering and the Sciences, 7th ed. Majority of the problems require at most a calculator to solve, however for the problem #56...
sphere.txt
Path: UPenn >> STAT >> 553 Fall, 2009
Description: 1.70577823171127 -2.17414285477918 -0.50943812882322 4.86281328281118 -2.56515177178407 -4.64906809085007 -3.41501570628519 3.34539706213774 1.06365179154800 -2.86139123522112 1.25780357414342 2.35623919894695 -0.220029219918470 -1.35254613317098 1.7...
basic-tests.pdf
Path: UPenn >> STAT >> 102 Fall, 2004
Description: Stat 102, Spring 2000 1 Review of One and Two Sample Tests One Sample Tests: Normality Assume that the sample of n observations is from a normal population with mean and variance 2 (abbreviated N (, 2 ). Tests of one or sided hypotheses count th...
one-way.102.pdf
Path: UPenn >> STAT >> 102 Fall, 2004
Description: Statistics 102 Spring, 2000 One-way Anova -1- One-Way Analysis of Variance Administrative Items Midterm Grades. Make-up exams, in general. Getting help See me today 3-5:30 or Wednesday from 4-5:30. Send an e-mail to stine@wharton. Visit TAs, p...
multregr-summ.102.pdf
Path: UPenn >> STAT >> 102 Fall, 2004
Description: Statistics 102 Spring, 2000 Regression Summary -1- Regression Summary Project Analysis for Today First multiple regression Interpreting the location and wiring coefficient estimates Interpreting interaction terms Measuring significance Second mu...
DiceSimulation430.pdf
Path: UPenn >> STAT >> 430 Fall, 2008
Description: Statistics 430 Spring, 2003 1 Variance and the Volatility of Investments Overview Variance, often called volatility when speaking of returns on financial investments, is an important characteristic of investments like stocks. When we consider inves...
notes20.ppt
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Stat 112: Lecture 20 Notes Chapter 7.2: Interaction Variables. Chapter 8: Model Building. I will e-mail Homework 6 by Friday. It will be due on Friday, Dec. 1st (the Friday after Thanksgiving) Interaction Interaction is a three-variable concept....
hw5.pdf
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Homework 5, Statistics 112, Spring 2004 This homework is due Thursday, March 4th at the start of class. Late homework will not be accepted except for medical emergencies (with proof). Note that if a question says to explain your answer, you will get ...
hw6.pdf
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Homework 6, Statistics 112, Fall 2003 This homework is due Thursday, March 18th at the start of class. Late homework will not be accepted except for medical emergencies (with proof). 1. For the handicap study (Case Study 6.1.1, in Handicaps.JMP), con...
hw4.pdf
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Homework 4, Statistics 112, Spring 2004 This homework is due Thursday, February 12th at the start of class. Note that if a question says to explain your answer, you will get no credit without some explanation. 1. The Statistical Sleuth, Chapter 3, Pr...
hw7.pdf
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Homework 7, Statistics 112, Spring 2004 This homework is due Thursday, March 25th at the start of class. Late homework will not be accepted except for medical emergencies (with proof). 1. How well does the number of beers a students drinks predict hi...
hw9.pdf
Path: UPenn >> STAT >> 112 Fall, 2008
Description: Homework 9, Statistics 112, Spring 2004 This homework is due Friday, April 16th by 5 p.m. If you dont hand it in in class on Thursday, hand it in to my mailbox in the statistics department on the fourth oor of Huntsman Hall. Late homework will not be...
