John E. Freunds Mathematical Statistics With Applications
This is a summary of chapters 11, 12, 13, 15 and 16 of the book Mathematical Statistics for the study Econometrics and Operations Research. It contains all the theory from the book and also notes of the lectures and a number of examples.
Definition 1
If 𝜃̂1 and 𝜃̂2 are values of the random variables 𝛩̂1and 𝛩̂2 such that
𝑃(𝛩̂1 < 𝜃 < 𝛩̂2 ) = 1 − 𝛼
for some specified probability 1 − 𝛼, we refer to the interval
𝛩̂1 < 𝜃 < 𝛩̂2
as a (1 − 𝛼)100% confidence interval for 𝜃. The probability 1 − 𝛼 is called the degree of
confidence, and the endpoints of the interval are called the lower and upper confidence
limits.
Like point estimates, interval estimates of a given parameter are not unique.
It is desirable to have the length of a (1 − 𝛼)100% confidence interval as short as possible
and to have the expected length, 𝐸(𝛩̂2 − 𝛩̂1 ) as small as possible.
Theorem 1
If 𝑋̅, the mean of a random sample of size n from a normal population with the known
variance 𝜎 2 , is to be used as an estimator of the mean of the population, the probability is
𝜎
1 − 𝛼 that the error will be less than 𝑧𝛼/2 ∙
√𝑛
In general, we make probability statements about the potential error of an estimate and
confidence statements once the data have been obtained.
Theorem 2
If 𝑥̅ is the value of the mean of a random sample of size n from a normal population with the
known variance 𝜎 2 , then
𝜎 𝜎
𝑥̅ − 𝑧𝛼 ∙ < 𝜇 < 𝑥̅ + 𝑧𝛼 ∙
2 √𝑛 2 √𝑛
is a (1 − 𝛼)100% confidence interval for the mean of the population.
Rule of thumb: it is reasonable to assume that the true parameter value lies within two
standard deviations of the estimate.
𝜎 𝜎
Confidence-interval formulas are not unique. The formulas 𝑥̅ − 𝑧2𝛼 ∙ < 𝜇 < 𝑥̅ + 𝑧𝛼 ∙
3 √𝑛 3 √𝑛
𝜎
Or the one-sided (1 − 𝛼)100% confidence interval 𝜇 < 𝑥̅ + 𝑧𝛼 ∙ generate the same range
√𝑛
for the interval. Also notice that Theorem 1 and Theorem 2, by the central limit theorem
(CLT) can also be used for random samples from non-normal populations when n ≥ 30, in
that case we may also substitute for 𝜎 the value of the sample standard deviation s.
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