Ch. 1: Data and Distributions

Transcription

1 Ch. 1: Data and Distributions Populations vs. Samples How to graphically display data Histograms, dot plots, stem plots, etc Helps to show how samples are distributed Distributions of both continuous and discrete variables Density functions and Mass functions Three basic properties Shows the distribution of the entire population or process Some important distributions and associated Probability Continuous: Exponential, Normal, Uniform Discrete: Binomial, Poisson 4/4/1 H.X. Lecture 30: Final Summary 1

2 Ch. : Numerical Summary Measures Measure of center of Data (Sample) Sample mean Sample median, midpoint Trimmed means x x1 + x xn 1 = = xi n n Measure of variability for Data (Sample) ( x1 x) + ( x x) ( xn Sample variance s = n 1 Sample Standard deviation 1 = n 1 ( x x) Quartiles; Five-number-Summary; IQR and Outliers Graphical Display: Boxplots; Modified Version; Side- By-Side Boxplots s = s x) i 4/4/1 H.X. Lecture 30: Final Summary

3 Ch. (Cont.): Numerical Summary Measures Measures of Center (Distributions) Continuous: µ X = x f ( x) dx Discrete: = x p(x) µ X Measure of variability (Distributions) Continuous: Discrete: ( ) X x µ X f ( x) = dx Normal Quantile (QQ) plot σ ( x X ) σ X = µ p( x) 4/4/1 H.X. Lecture 30: Final Summary 3

4 Ch.3: Bivariate Data Scatterplots: Visually Display Bivariate data, y vs. x Pearson s Correlation Coefficient (between X and Y, both quantitative), r : r measures the strength and direction of the linear relationship, other convenient formulas for Sxy, Sxx and Syy Takes values between -1 and 1, inclusive Sign indicates type/direction of relationship (positive, negative) Value indicates strength: farther from 0 is stronger If switch roles of X and Y à r doesn t change Unit free unaffected by linear transformations Affected by Outliers, Not a resistant measure Correlation Causaiton 4/4/1 H.X. Lecture 30: Final Summary 4

5 Ch. 3: LS (Least Square) Regression Line Estimated straight line Equation: y = a + b x a is the intercept (where it crosses the y-axis) b is the slope (rate) b = r s s y x Predicted value of y Residual from the fit (or regression line) Breaking up Sum of Squares: SSR, SSE, SST Coefficient of Determination: SSR = = 1 SST Percent of variation explained by the linear regression between Y and X r SSE SST 4/4/1 H.X. Lecture 30: Final Summary 5

6 Ch. 3 (Cont.): MSE and Residual Plot Mean Squared Error about the LS line: Standard Deviation about the LS line: Also called root MSE in SAS output. Residual: e ˆ i = yi yi A residual plot, plotting the residuals against x. The residual plot should not have any pattern but a random scattering of points If a pattern is observed, the linear regression model is probably not appropriate. 4/4/1 H.X. Lecture 30: Final Summary 6

7 Ch. 5: Probability and Sampling Distributions Chance Experiments: Simple Events: individual outcomes Events: collections of simple events Sample Space: Venn Diagrams Tree Diagrams Complex Events: Event A or B, Event A and B, Event A (Complement of A) Disjoint Events (Mutually Exclusive) Independent Events 4/4/1 H.X. Lecture 30: Final Summary 7

8 Probability Basic Rules Probability Axioms: 0 P(A) 1 for any event A P(S) = 1, where S is the sample space Addition Rule - For any disjoint events A and B, P(A or B) = P(A)+P(B) Complementary Events: P(A ) = 1 - P(A) General Addition Rule: (for any events A and B) P(A or B) = P(A)+P(B)-P(A and B) Independence Rule: P (A and B) = P(A) P(B) Conditional Probability: P(A B) = P (A and B) / P(B) Bayes Rule for Calculation of Conditional Probability, Tree Diagrams 4/4/1 H.X. Lecture 30: Final Summary 8

9 Random Variables and Sampling Distribution Random Variables Discrete Distribution Table, Prob. Histogram Continuous Distribution Curve, density function Independent R.V.s Sampling Distribution of a Sample Mean Sampling Distribution of a Sample Proportion (rule of thumb for Normal Appox.) Central Limit Theorem Continuity Correction (from Binomial to Normal Appox.) 4/4/1 H.X. Lecture 30: Final Summary 9

10 Ch 7: Estimation and Statistical Inference by C.I. s (Unbiased, Consistent) Point Estimation Large-Sample C.I.s for a Population Mean (Normality Assumption) s X ± (z critical value) one-sided C.I.s: Upper or Lower bound C.I. Interpretation of Confidence Level. Necessary sample size for a desired Bound (round up): ZCrits n = B Small-Sample C.I. s X ± (t critical value) n t-crit is associated with d.f. = n -1 Normailty Assumption still holds. 4/4/1 H.X. Lecture 30: Final Summary 10 n

11 C.I. for a Population Proportion Point Estimation for a Population Proportion Large-Sample C.I.s for a Population Proportion pˆ (1 pˆ ) pˆ ± Zcrit n Necessary sample size for a desired Bound (round up for not-an-integer): z _ critical n= p*(1 p*) B p* = pˆ, or 0.5 if p-hat is unavailable. Small-Sample C.I. replaces z-crit by t-crit 4/4/1 H.X. Lecture 30: Final Summary 11

12 C.I. for two Population Means Difference Large-Sample C.I.s for Difference between two Population Means (Normality Assumption) 1 X 1 X ± Zcrit + n1 Small-Sample C.I., Zcrit replaced by t-crit, with (round down for non-integer) ( ) s1 n1 + s n df = ( ) ( ) s1 n1 s n + n n /4/1 H.X. Lecture 30: Final Summary 1 s s n

13 t C.I. for Paired Data 4/4/1 H.X. Lecture 30: Final Summary 13

14 Ch. 8: Hypotheses Testing State Hypotheses Both Null and Alternative (one or two-sided) Determine an appropriate α level. If not specified, use 5% Type I error; Significance Level. Understand it. Calculate the appropriate test statistic Find the P-value, the probability of the as extreme or more extreme than the test statistic Reject H 0, when the P-value is smaller than the significance level α. Otherwise: Fail to reject H 0 State a conclusion in layman s terms 4/4/1 H.X. Lecture 30: Final Summary 14

15 One-sample t Test for a Population Mean: The null hypothesis is H 0 : µ = µ 0 The alternative hypothesis could be: H a : µ µ 0 (two-sided) H a : µ > µ 0 (one-sided) H a : µ < µ 0 (one-sided) Test Statistic X t ~ Student s t-distribution df = n 1 s µ 0 t = n If n is large ( 30), CLT guarantees an approximate normal distribution and the t can be replaced with z, where z follows a standard normal distribution. 4/4/1 H.X. Lecture 30: Final Summary 15

16 P-value tied to H a Two-sided (both tails) H a : µ µ 0 One-sided (right tail) H a : µ > µ 0 One-sided (left tail) H a : µ < µ 0 4/4/1 H.X. Lecture 30: Final Summary 16

17 Other Tests or Remarks Two-Sample z (or t, depending on sample sizes) test for Two Population Means When using t, the d.f. calculation One-Sample t Test with (Matched) Paired Data Focus on two population means difference A two-sided significance test <-> A two-sided C.I. for the same parameter If the claimed value is in the CI à fail to reject H 0 If the claimed is not in the CI à reject H 0 NOTE: must have in H a! Statistical Significance Practical Sig. 4/4/1 H.X. Lecture 30: Final Summary 17

18 Cautions (for both C.I. and tests of significance): Data: assume SRS (random sampling) Population need to be If n < 30, have to check normality (by Normal QQ-plot) With n 30, CLT can give us approximate normality in most situations. 4/4/1 H.X. Lecture 30: Final Summary 18

19 Ch. 9: One Way ANOVA Hypotheses: H 0 : µ 1 = µ = = µ k vs. H a : At least one µ i is different F test statistic ANOVA table test statistic = between - samples variation within - samplesvariation Source DF SS MS Model (Between) Error (Within) k 1 n k SSM (formula) SSE (formula) Total n 1 SST = SSM + SSE SSM/k 1 SSE/n k P-value is always the upper tail of the F distribution with (k 1, n k) degrees of freedom. Tables of critical values for F distribution: (Table VIII) F statistic > F critical value <=> P-value < α => Reject H 0 4/4/1 H.X. Lecture 30: Final Summary 19

20 Assumptions (prior to Running one-way ANOVA) 1. Constant variance: The variances of the k populations are the same. Check this with the ratio of the largest and smallest standard deviations, the ratio must be <. Each of the k populations follows a normal distribution. Check this by looking at QQplots for each group Remark: statistical significance practical significance 4/4/1 H.X. Lecture 30: Final Summary 0

21 Ch. 9: Multiple Comparison If insignificant in one-way ANOVA, we don t have to try further steps Otherwise, run Multiple Comparison to see which explicitly means are different. Tukey s Mehtod ( cldiff or lines format) Dunnett s Method (only if there s a control group) 4/4/1 H.X. Lecture 30: Final Summary 1

22 9.4: Randomized Complete Block Design RCBD (both treatment and block factor must be categorical) Source DF SS MS Factor A (treatment) Factor B (block) a 1 SSA MSA b 1 SSB MSB Error (a 1)(b 1) SSE MSE Total ab 1 SST In RCBD, we are only interested in the treatment factor The block factor might affect response but that s not of interest. Two F tests Blocking Effect? Use test statistic and P-value to conclude Treatment Effect? Use test statistic and P-value to conclude 4/4/1 H.X. Lecture 30: Final Summary

23 Necessary Assumptions for RCBD Similar to one-way ANOVA 1. Constant variance. Each of the k populations follows a normal distribution One additional assumption 3. There is no interaction between the treatment and blocking variables Can assess just using common sense (Just ask: Do/ should they interact?) OR check by a Two-way ANOVA model Interaction Plot 4/4/1 H.X. Lecture 30: Final Summary 3

24 Ch. 10: Two-Way ANOVA Testing Two factors and their interaction s effect to the response variable Source DF SS MS Factor A a 1 SSA MSA Factor B b 1 SSB MSB AB interaction (a 1)(b 1) SSAB MSAB Error ab(r 1) SSE MSE Total abr 1 SST Test First, Interaction (of the most interest). Then Factor A and B, respectively. If Interaction significant, still run slicing for Factor A and B. If Interaction insignificant while a single Factor significant, run one-way ANOVA and multiple comparison. 4/4/1 H.X. Lecture 30: Final Summary 4

25 Ch. 10 (Cont.): Two-Way ANOVA Interaction plot Roughly speaking, there s no Interaction effect if all lines are parallel to each other In summary, for Ch. 9 and 10 we should know: All of One-way ANOVA (Ch. 9) By hand and/or using SAS Most of randomized Blocking design (Sec 9.4), Two-way ANOVA (Ch. 10, Section ) For both: Complete ANOVA tables, calculate DFs and F test statistic Perform F tests using F table Interpret SAS output Know the general concept of a higher order (multi-way) ANOVA model. 4/4/1 H.X. Lecture 30: Final Summary 5

26 Ch. 11: Inferential Methods in Regression and Slopes (Correlations) Normal Error Regression Model Error Item (3 assumptions: Independence, Normality and Constant Variance) SSE, MSE, and Root MSE Coefficient of Determination, R^ % of variation explained by the regression model Simply by squaring r Statistical Inference about the slope in SLR Model: C.I. for β (the slope): b ± (t crit) * s b Hypotheses Testing w.r.t. the slope, i.e. test of Linear Relationship Remark: t~student s t-distribution with d.f. = n 4/4/1 H.X. Lecture 30: Final Summary 6

27 Using ANOVA table to test SLR Source DF SS MS Model (Regression) 1 SSM (or SSR) SSM/1 = MSM (or MSR) Error n SSE (or SSResid) Total n 1 SST = SSM + SSE SSE/n = MSE Remark: d.f. of F test statistic = (1, n ) 4/4/1 H.X. Lecture 30: Final Summary 7

28 Multiple Linear Regression Model MLR Model: Y X X X e i = α + β1 1+ β βp p + i Test the above linear relationship H 0 : All β i s = 0 vs. H α : At least one β i 0 A rejection of the null indicates that collectively the Xs do well at explaining Y; otherwise don t have to run the following step But it doesn t show which explicit Xi s are doing the explaining Model Selection, especially Backward Elimination The Estimated Line, from SAS output Use it to Predict Yi; Get residual by Actual Y_i Predicted Value 4/4/1 H.X. Lecture 30: Final Summary 8

29 After Class Review Notes, practices, Hw, Labs and previous tests. Wed, Lab#8 (optional) Final Exam (Close book, Close notes) Next Wed, 8-10am Student ID, a calculator (SAT policy, NO QWERTY keyboard) and pencils, two-page crib sheet (8 by 11 ) handwritten by yourself, two-sided. SEE CALCULATOR POLICY and crib sheet (on Syllabus) from course website. No electronics except a calculator. Not allowed to exchange calculator or crib sheet during the exam. Not allowed to type/print your crib sheet. 4/4/1 H.X. Lecture 30: Final Summary 9