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Standard error of estimate & Confidence interval Two results of probability theory Central limit theorem Sum of random variables tends to be normally distributed as the number of variables increases Law of large numbers Larger sample size -> the relative frequency in the sample approaches that of a population -> the sample average is closer to population mean Calculating expected values and variances x: random variable k: constant E(x)=expected value of x V(x)=variance of x E(x+x)=E(x)+E(x) V(x+x)=V(x)+V(x) (if independent) E(k*x)=k*E(x) V(k*x)=k2 V(x) V(x/k)=V(x)/ k2 Standard error of an estimator Before knowing the value: “Standard deviation of the estimates in repeated sampling IF the true value of the parameter was known” After knowing the observed value: “Standard deviation of the estimates in repeated sampling IF the true value of the parameter is the observed one” Not a statement of uncertainty about the parameter, but a statement of uncertainty about the hypothetical values of the estimator Confidence interval 95% CI: Intervals calculated like this one include the true value of the parameter in 95% of the cases within infinitely repeated sampling Interval is random, it depends on the randomly sampled data Wrong interpretation: “The true value of the parameter lies in this interval with probability 0.95” 95% Confidence interval for the mean Interval that contains the true mean in 95% of the cases in infinitely repeated sampling Sample averages are approximately normally distributed Assume known standard deviation of the population: , x 1.96 x 1.96 n n