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Unformatted text preview: n and t he popul at ion is nor mal. This is St udent t dist r ibut ed with v = n – 1 degr ees of fr eedom. x = xin FinOp 250 s2= i2x-(xi)2nn-1 Chapter 12 Book Notes I NFERENCE ABOUT A POPULATION MEAN WHEN THE STANDARD DEVIATION IS UNKNOWN
The confidence interval estimator and the test statistic were derived from the sampling d istribution of the sample mean with σ known, expressed as z=x - μσ/n Now, we take the approach that if the population mean in unknown, so is the population standard deviation. We substitute the sample standard deviation “s” in the place of the u nknown population standard deviation ”σ ”. Test Statistic for µ when σ is unknown: t= x-μs/n Confidence Interval Estimator of µ w hen σ is unknown: x ±tα/2 sn Checking the requi red conditions The mathematical process that derived the Student t distribution is robust, which means t hat if the population is nonnormal, the results of the t-test and confidence interval estimate are still valid provided that the population is extremely nonnormal. Estimating the Totals to F inite Populations Large populations are defined to be at least 20x the sample size. Finite populations allow us to use the confidence interval estimator of a mean to produce a confidence interval estimator of the population total. To estimate the total, we multiply the lower and upper confidence limits of the estimate of the mean by the population size. Confidence Interval Estimator of the total: N(x ±t∝/2sn) Developing an Understanding of Statistical Concepts 1 The Student t distribution is based on using the sample variance to estimate the unknown population variance. Sample Variance: s2= (xi- x )2n-1 v=n-1 ...
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- Spring '08