Summary
Chapter 11:
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.
Chapter 11:
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.
