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