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STATISTICS 211 HONORS PROF EMANUEL PARZEN Chapter 6A STATISTICAL INFERENCE CONFIDENCE INTERVALS HYPOTHESIS TESTS STATISTICAL INFERENCE seeks to learn from data values of parameters of the probability distribution obeyed by the random variable of which the data is a random sample. It is important to remember that one learns from a random sample ONLY about the population represented in the sample. POINT ESTIMATOR OF A PARAMETER: Learning starts with a point estimator of the parameter and an exact or approximate knowledge of the sampling distribution of the estimator given the parameter value. To answer Scientific questions we prefer interval estimators of the parameter (called confidence intervals) which provide information about the parameter (including precision (margins of error, error bars)) that fit the observed data. Formulas for statistic inference have a general theory but it is important to learn each parameter its own details which are important in practice. CONTINUOUS DATA SUMMARY: For a continuous or quantitative random variable Y a sample has summary Y name of variable Sample size sample mean standard deviation SE(mean) n M(Y) Sobserv, Sassume S/sqrt(n) QUANTILE SUMMARY: Quantiles MIN Q1 Q2 Q3 MAX, boxplot, and QQ plot of continuous data are increasingly being reported and diagnosed to identify probability distribution (especially Normal) to fit to data In this chapter we do not emphasize reporting and interpreting the quantile summary of continuous data. EXAMPLE: Y fill weight (in ounces) of box or can produced by a plant n mean M(Y) Sobserv SE(mean) 100 12.1 . 1 .1/10=.01 CONFIDENCE INTERVAL ENDPOINT FUNCTION: For large samples confidence interval endpoint function, denoted \mu(P)=Q(P;\mu), is given approximately by (applying Central Limit Theorem) \mu(P)=M(Y)+Q(P;Z) S/sqrt(n) It is derived from pivot Z(M(Y),\mu) defined below under hypothesis test. Note that for n<40 and Y is assumed to obey Normal distribution, confidence interval endpoint function of parameter \mu is exactly (and more accurately) mu(P)=M(Y)+Q(P;Student(n-1)) S/sqrt(n
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Confidence interval of confidence level 1-\alpha is an interval of values of \mu with endpoints \mu(lower limit)=\mu(\alpha/2) \mu(upper limit)=\mu(1-\alpha/2)) EXAMPLE: For Y fill weight of box, with sample statistics summarized above, 90% confidence interval has endpoints, using Q(.05;Z)=-1.645, Q(.95;Z)=1.645, \mu(.05)=12.1 – 1.645 (.01)= 12.08355, \mu(.95)=12.1+1.645 (.01)=12.11645 HYPOTHESIS TEST: Formulas for confidence intervals are derived by finding a pivot which is a function of estimator and parameter whose distribution does not depend on the parameter. For sample mean M(Y) and population mean \mu (parameter) define Z(M(Y), \mu)=(M(Y)-\mu)/SE(M(Y)) )=(M(Y)-\mu)/(S/sqrt(n)) For n large distribution of Z(M(Y),\mu) is approximately Normal(0,1) Z:
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This note was uploaded on 03/27/2008 for the course STAT 211 taught by Professor Parzen during the Fall '07 term at Texas A&M.

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