Page 18-23
to estimate is its mean value, µ. We will use as an estimator the mean value
of the sample, X, defined by (a rule):
∑
=
⋅=
n
i
i
X
n
X
1
.
1
For the sample under consideration, the estimate of µ is the sample statistic x
= (2.2+2.5+2.1+2.3+2.2)/5 = 2.36. This single value of X, namely x =
2.36, constitutes a point estimation of the population parameter µ.
Estimation of Confidence Intervals
The next level of inference from point estimation is interval estimation, i.e.,
instead of obtaining a single value of an estimator we provide two statistics, a
and b, which define an interval containing the parameter θ with a certain
level of probability. The end points of the interval are known as confidence
limits, and the interval (a,b) is known as the confidence interval.
Definitions
Let (C
l
,C
u
) be a confidence interval containing an unknown parameter θ.
• Confidence level or confidence coefficient is the quantity (1-α), where 0 <
α < 1, such that P[C
l
< θ < C
u
] = 1 - α, where P[ ] represents a
probability (see Chapter 17). The previous expression defines the so-
called two-sided confidence limits.
• A lower one-sided confidence interval is defined by Pr[C
l
< θ] = 1 - α.
• An upper one-sided confidence interval is defined by Pr[θ < C
u
] = 1 - α.
• The parameter α is known as the significance level. Typical values of α
are 0.01, 0.05, 0.1, corresponding to confidence levels of 0.99, 0.95,
and 0.90, respectively.
Confidence intervals for the population mean when the
population variance is known
Let X be the mean of a random sample of size n, drawn from an infinite
population with known standard deviation σ. The 100(1-α) % [i.e., 99%,
95%, 90%, etc.], central, two-sided confidence interval for the population
mean µ is (X−z
α
/2
⋅σ/√n , X+z
α
/2
⋅σ/√n ), where z
α
/2
is a standard normal
variate that is exceeded with a probability of α /2. The standard error of
the sample mean, X, is ⋅σ/√n.
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