实验设计与方差分析 Chap2_handout

Chapter 2 Basic Statistical Methods Jingyuan Liu SOE and WISE Xiamen University DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Outline of this Chapter Basic statistical concepts One population Inference about mean and variance 2 Two population Comparison between two means 1and 2 and two variances 2 1 and 2 2 Independent sample Paired sample only focus on mean comparison DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Basic Concepts Population vs Sample The population is the large group of units we are interested in from which the sample was selected E g We are interested in the height of students in Xiamen U A sample is the collection of units people animals cities fi elds whatever you study that is actually measured or surveyed in a study E g We measure the height of students in our classroom The sample a subset of the population is used to estimate characteristics of the population DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Sample v s Population DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Parameters for Population and Statistics for Sample We use diff erent notations for population Statistic point estimate Inference we wantInformation we have DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods One Population Inference about Mean and Variance 2 Inference usually contains tests and confi dence intervals Assumption Sample y1 ynindependently and identically distributed i i d from N 2 or n is large 30 for the central limit theorem CLT approximate normality Statistics Point estimate of y Pn i 1yi n Point estimate of 2 S2 Pn i 1 yi y 2 n 1 Distributions y n N 0 1 y S n t n 1 n 1 S2 2 2 n 1 DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods One Population Inference about Mean and Variance 2 One sample tests about when 2is known Hypotheses 1 One sided H0 0 or 0 vs H1 0 2 One sided H0 0 or 0 vs H1 z where z is the upper quantile of N 0 1 2 Z z 2 DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods One Population Inference about Mean and Variance 2 Confi dence interval CI of when 2is known Defi nition Find a and b s t P a b 1 Distribution y n N 0 1 1 CI P y n z 2 1 1 CI y z 2 n y z 2 n DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods One Population Inference about Mean and Variance 2 One sample tests about when 2is unknown Hypotheses 1 One sided H0 0 or 0 vs H1 0 2 One sided H0 0 or 0 vs H1 t n 1 where t n 1 is upper quantile of t n 1 2 T t 2 n 1 DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods One Population Inference about Mean and Variance 2 Confi dence interval CI of when 2is unknown Defi nition Find a and b s t P a b 1 Distribution y S n t n 1 1 CI P y S n t 2 n 1 1 y t 2 n 1 n y t 2 n 1 n DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods One Population Inference about Mean and Variance 2 One sample tests about 2 Hypotheses 1 One sided H0 2 2 0 or 2 2 0 versus H1 2 20 2 One sided H0 2 2 0 or 2 2 0 versus H1 2 2 n 1 the upper quantile of 2 n 1 2 U 2 1 n 1 3 U 2 2 n 1 DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods One Population Inference about Mean and Variance 2 Confi dence interval CI of 2 Defi nition Find a and b s t P a 2 b 1 Distribution n 1 S2 2 2 n 1 1 CI P 2 1 2 1 n n 1 S2 2 2 n 1 1 1 CI n 1 S2 2 n 1 2 n 1 S2 1 2 n 1 DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Populations Compare Two Means About this two population comparison problem we need to consider the following two scenarios Independent sample Population 1 and 2 are independent Paired sample Population 1 is paired with population 2 DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Independent Populations Compare Two Means E g For packing plants it is supposed that a new machine will pack faster on average than the machine currently used To test that hypothesis the times in seconds it takes each machine to pack ten cartons are recorded DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Independent Populations Compare Two Means Scenario 1 Independent sample Assumptions y11 y1n1i i d N 1 2 1 y21 y2n2 i i d N 2 2 2 or n1 30 and n2 30 for the CLT approximate normality y1jindependent of y2j0 j 1 n1 j0 1 n2 Statistics Point estimate of 1and 2 y1and y2 Point estimate of 1 2 y1 y2 Point estimate of 2 1 and 2 2 S21 and S2 2 DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Independent Populations Compare Two Means Scenario 1 Independent sample 2 1 and 2 2 known Distribution y1 y2 1 2 q 2 1 n1 2 2 n2 N 0 1 Hypotheses 1 One sided H0 1 2 or 1 2 vs H1 1 2 2 One sided H0 1 2 or 1 2 vs H1 1 z 2 Z z 2 DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Independent Populations Compare Two Means Scenario 1 Independent sample 2 1 22 2unknown Distribution y1 y2 1 2 Sp 1 n1 1 n2 t n1 n2 2 where the pooled estimate of 2is S2 p n1 1 S2 1 n2 1 S 2 2 n1 n2 2 P2 i 1 Pni j 1 yij yi 2 n1 n2 2 Hypotheses 1 One sided H0 1 2 or 1 2 vs H1 1 2 2 One sided H0 1 2 or 1 2 vs H1 1 t n1 n2 2 2 T t 2 n1 n2 2 DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Independent Populations Compare Two Means Scenario 1 Independent sample 2 16 22 unknown Distribution y1 y2 1 2 S2 1 n1 S 2 2 n2 t where S 2 1 n1 S2 2 n2 2 S21 n1 2 n1 1 S2 2 n2 2 n2 1 Hypotheses 1 One sided H0 1 2 or 1 2 vs H1 1 2 2 One sided H0 1 2 or 1 2 vs H1 1 t 2 T t 2 DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Independent Populations Design Issue Scenario 1 Independent sample Sample size determination When designing the experiment for the two sample test how to determine the sample sizes to maximize the effi ciency MSE given n1 n2 n for fi xed n E g What if we claim that 2 1 2 2 E g What if we claim that 2 1 10 2 2 We need to fi nd n1and n2such that 1 n1 n2 n 2 var y1 y2 is minimized Why DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Independent Populations Compare Two Means Scenario 1 Independent sample Sample size determination var y1 y2 2 1 n1 2 2 n2 2 1 n1 2 2 n n1 Q n1 Want to fi nd n1such that Q n1 n1 0 If 2 1 2 2 Q n1 1 n1 1 n n1 Q n1 n1 0 n1 n2 n 2 If 2 1 10 2 2 Q n1 10 n1 1 n n1 Q n1 n1 0 n2 1 10n2 2 For more details refer to section 2 4 3 in the book 2 4 2 for the 5th edition DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Independent Populations Compare Two Means Scenario 1 Independent sample back to the machine example Whether the new machine has smaller mean packing time than the old machine DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Independent Populations Compare Two Means Scenario 1 Independent sample back to the machine example 1 Check assumptions 1 Are these independent samples Yes since the samples from the two machines are not related 2 Are the samples normal Or are they big samples n 30 for the CLT approximate normality They are normal according to the normality plot from minitab 3 Do the populations have equal variance Yes since S2 1 and S2 2 are not so diff erent need variance tests DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Independent Populations Compare Two Means DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Independent Populations Compare Two Means DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Independent Populations Compare Two Means Scenario 1 Independent sample back to the machine example 2 Hypothesis testing 1 Set up the hypothesis H0 1 2versus Ha 1 2 2 Signifi cance level We set the signifi cance level 0 05 3 Compute the test statistic Recall that y1 42 14 S1 0 683 y2 43 23 S2 0 750 Therefore Sp q 9 0 683 2 9 0 750 2 10 10 2 0 717 and the statistic T 42 14 43 23 0 717 p1 10 1 10 3 40 4 Critical value t0 05 10 10 2 1 734 5 Conclusion T 2 2 2 One sided H0 2 1 2 2 or 2 1 2 2 versus H1 21 F n1 1 n2 1 upper quantile of F n1 1 n2 1 2 F0 F1 n1 1 n2 1 3 F0 F 2 n1 1 n2 1 DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Paired Populations Compare Two Means Now let s consider a new testing problem Boys shoes DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Paired Populations Compare Two Means Now let s consider a new testing problem E g Two materials A and B are available for the making boys shoes To compare them 10 volunteer boys are gathered for the experiment How to design the experiment i e how to assign the two materials to the boys Which of the following designs is the best And why 1 Boys into two groups Group A A Group B B 2 For each boy Left foot A Right foot B 3 For each boy One foot A The other foot B Randomly choose which foot guarantee half A and half B are assigned DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Paired Populations Compare Two Means DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Paired Populations Compare Two Means DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Paired Populations Compare Two Means Scenario 2 Paired sample Assumptions y11 y1niid with mean 1 y21 y2niid with mean 2 y1iis related to paired with y2i i 1 n y1i y2iiid N d 2 d 1 2 or n is large for CLT note that if the number of pairs is 0 2 One sided H0 d 0 or d 0 vs H1 d t n 1 2 T t 2 n 1 DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Paired Populations Compare Two Means Scenario 2 Paired sample back to the boy s shoes example DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Paired Populations Compare Two Means DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Paired Populations Compare Two Means Scenario 2 Paired sample back to the boy s shoes example 1 Check assumptions 1 Are these paired samples Yes since the left and right foot of each boy are related 2 Are the diff erences normal Or are they big samples n 30 for the CLT approximate normality They can be treated as normal DOE and ANOVA Chapter2Chapter 2 Basic Statistical Methods Two Paired Populations Compare Two Means Scenario 2 Paired sample back to the boy s shoes example 2 Hypothesis testing 1 Set up the hypothesis H0 d 0 versus Ha d 0 2 Signifi cance level We set the signifi cance level 0 005 3 Compute the test statistic d 0 41 Sd pS2 d qP 10 i 1 di 0 41 2 10 1 0 386 Therefore the test statistic T 0 4