211 STATISTICS HONORS 2007 PROF EMANUEL PARZEN KEY CONCEPTS ONE SAMPLE STATISTICAL INFERENCE 10/31 0.Population Parameters \mu, p Estimators from sample \mu\hat, \p\hat Denote standard error by S.E.; derive formulas from SONG OF SUMS for mean, variance of sum of random variables 1.Continuous data summary of quantitative variable Y n M(Y) S SE(sample mean) MIN Q1 Q2 Q3 MAX 2. Continuous data quantile diagnostics symmetry outliers 3. 0-1 data summary of "success-failure" variable N K p\hat=K/n SE(p\hat) for C.I. SE(p_0) to test H_0 4. Population quantile Q(P;Y), 0<P<1, of continuous variable Y Pr[Y<Q(P)]=P, Quantile is Prediction interval endpoint function since Pr[ Q(\alpha/2;Y)<Y<Q(1-(\alpha/2);Y)]=1-\alpha 5. Standard Distributions notation Z=Normal(0,1) ; W Exponential(1) Normal(mean \mu, standard deviation \sigma)=\mu+\sigma Z CHISQ(n-1) Chi-squared; CHIAV(n-1)=CHISQ(n-1)/(n-1) STUDENT(n-1)=Z/sqrt(CHIAV(n-1)); F=CHIAV(m-1)/CHIAV(n-1) Binomial(n,p); Normal approximation Central Limit Theorem for large sample distribution of sum 6. Action strategy for comprehensive analysis of sample steps S I E V 7. Hypothesis tests null hypothesis H_0:parameter=parameter_0 Theoretical basis of confidence intervals Apply to Scientific questions posed in terms of test H_0 START with defining PIVOT=Test statistic for parameters. For \mu, p Z(parameter par\hat, estimated parameter population par_0)= (par\hat-par_0)/SE(parameter estimator) 8. Distribution of statistic Z interpreted Normal(0,1) Z or Student(n-1); WHY? 9.Rejection region of null hypothesis (values of statistic Z that reject H_0) choose value \alpha for probability of type I error (reject H_0 when H_0 is true ) 10. Pvalue (actually Pextreme-value) of observed value of test statistic Z is Pr[Z distribution has value more extreme than observed value Z statistic] Decide REJECT or NOT TO REJECT H_0: compare P value to alpha or observed value of Z to critical value of Z (computed from table of quantile function of Z) 11. Quick test to reject \mu_0 using interval from Q1 to Q3 of data summary; reject if \mu_0 is outside interval from Q1 to Q3 12. Confidence interval endpoint function for parameter \mu Q(P;\mu)=M(Y) + Q(P; Z statistic) S/sqrt(n) 95% confidence C.I. lower limit Q(.025;\mu) upper limit Q(.975;\mu) 99% confidence C.I. lower limit Q(.005;\mu) upper limit Q(.995;\mu) 13. Large sample traditional confidence interval endpoint function for p Q(P;p)=p\hat + Q(P; Z statistic) sqrt( p\hat(1-p\hat)/n) 14. Sample size determination for parameters \mu and p 15. Prediction interval endpoint function ASSUME Y is NORMAL Q(P;Y(future))=M(Y)+Q(P;Student(n-1)) S sqrt(1+(1/n)) 16. Paired data (X,Y) parameter is difference E[X]=E[Y] of population means