Download STA 3033 Notes on Interval Estimation – 7.1

STA 3033
Notes on Interval Estimation – 7.1 - 7.3
The simplest form of estimation is a point estimate, where a single number
(statistic) is calculated from a sample and is used to estimate a population
parameter.
e.g.
x
estimates the population mean, µ
s2
estimates the population variance σ2 .
^
In general, θ estimates, or is a point estimate of θ.
DESIRABLE PROPERTIES OF POINT ESTIMATORS
Unbiased:
Small Variance
CONFIDENCE INTERVALS
Note that usually all estimators are continuous random variables and for
^
continuous data, we have that P( θ = θ) = 0. As an alternative, we can use
interval estimation, where we estimate a parameter by an interval generated from
the sample data. These intervals can be constructed so that we have a known
confidence (probability) that the parameter is "captured" or surrounded by the
interval. This probability is called the confidence coefficient and denoted by (1 α). The resulting interval is called a (1 - α )100% confidence intervals or a (1
- α )100% C.I.
7.3.1 Confidence Interval for µ - large sample or σ known.
Note that x ~ N(µ, σ2/n)
population is normal.
if n is sufficiently large or if the underlying
Now let us look at the construction of a large sample confidence interval for µ:
Therfore, a (1-α)100% confidence interval for µ when σ is known is :
(
x - z (α/2)
σ
σ
, x + z (α/2)
n
n
)
Note that this can be written as
−
x ± z (α/ 2)
σ
n
Note that z (α/2) is often called the reliability coefficient, and
σ
n
is the standard
error.
If n is large enough and σ is unknown, substitute with s.
***
Whenever the estimate follows a normal (or approximately normal) distribution,
the confidence interval will be of the form:
estimate ± [(reliability coefficient) (standard error)]
Confidence Interval for µ - small sample, σ unknown.
Recall that we have said that for large n, the variable
x−µ
s/ n
has an
x−µ
s/ n
does not have a normal distribution even when the original population is normally
distributed.
approximately normal distribution. However, when n is not large enough,
In this situation, it can be shown that if the original population is normal, then
x−µ
has what is called a t distribution with n-1 degrees of freedom, df. The t
s/ n
distribution is symmetric about 0, is bell shaped but has somewhat "fatter" tails
than the normal distribution. The exact shape of the t distribution depends on its
df.
As before, define t(n-1), α to be the point on the t(n-1) curve with area α above it.
These values appear in the attached table.
Note: 1) If the correct df does not appear in the table, usually
round down to the closest df.
2) t(∞) = z. Thus Zc values can be obtained from the
bottom of this table.
Using a probability statement similar to the one before, and using algebra, we
have that a (1-α)100% confidence interval for µ when σ is unknown is :
−
x ± t ( n −1, α / 2 )
s
n