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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 variancesx: random variablek: constantE(x)=expected value of xV(x)=variance of xE(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:
n
xn
x
96.1,96.1