Transcript Standard error of estimate & Confidence interval

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
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n
n
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