8.3interval estimation

Why settle for a confidence level of 95% when a level of 99% is achievable? Answer. The confidence interval for the higher confidence level is wider. Since the 95% interval extends to each side of , the width of the interval is . Similarly, the width of the 99% interval is . That is, we have more confidence in the 99% interval precisely because it is wider. The higher the desired degree of confidence, the wider the resulting interval.
After the values of the random variables in random sample have been observed, the values of A=a and B=b are computed, and the interval ( a, b ) is called the observed value of the confidence interval.
§8.3
Confidence Intervals
Example. Suppose that X ~ N ( ,1),
The unbiased and effective estimator of is
constant
r.v.
Generally we have two kinds of forms to estimate a parameter, One is by a point, which is introduced in Cha 8.1, the other is by an interval.
(2) Variance 2 is unknown , the CI for
Proof. Random variable By Thus we get the CI for μ: , to find μ,
(3)The mean is given, CI for the variance 2
Interpretation of confidence intervals
One way to think of the random interval (A , B) is to imagine that the sample that we observed is one of many possible samples that we could have observed (or may yet observe in the future). Each such sample would allow us to compute an observed interval. Prior to observing the samples, we would exact 95% of the intervals to contain μ. Even if we observed many such intervals, we won’t know which ones contain μ and which ones don’t. If there are a plot of 100 observed values of confidence intervals, Each is computed from a sample of size n=5 from the normal distribution with mean μ=25.2, and standard deviation σ=1 . then about 95 of the 100 intervals contain the value of μ.
r.v. By
Then we get the CI for 2 ( confidence coefficient ):
，
(4) is unknown, CI for 2 , then by
0.15 0.125 0.1

2
We get the CI for 2 :
0.075 0.05 0.025
(I) Let the population X ~N ( 2) (1) The variance 2 is given, CI for
Proof. By
, we know
then
Solve To get the CI for
(confidence coefficient ):
For example,
For any between 0 and 1, constants a and b
can be found to satisfy
By
Then we get the CI for θ: As the above example:
, to get
Formula for CI

-2
•
2
4
68Βιβλιοθήκη •102Ex1 Consider the following sample of fat content (in percentage ) of randomly selected hot dogs:
(assume that fat content has a normal distribution N( 2))
Solution. (1)
i.e.,
By the observed values,
then we get the CI for by formula (1):
(2) Let
In fact, the 100% CI for μ is (-∞，∞), which is not terribly informative because, even before sampling, we knew that this interval covers μ . If we think of the width of the interval as specifying its precision or accuracy, then the confidence level (or reliability) of the interval is inversely related to its precision. A highly reliable interval estimate may be imprecise in that the endpoints of the interval may be far apart, whereas a precise interval may entail relatively low reliability. Thus, it cannot be said unequivocally that a 99% interval is to be preferred to a 95% interval; the gain in reliability entails a loss in precision. As appealing strategy is to specify both the desired confidence level and interval width, and then determine the necessary sample size.
For example, we estimate the age of a person is between 35 and 40 years; we called this kind of estimator interval estimator.
As the above example，we look for an interval so that the probability that the interval contains is 0.95. ( Let n = 5 )
-1
1
3
z 22
3
Deriving a confidence interval
Let denote a random sample on which the CI for a parameter θ is to be based. Suppose a random variable satisfying the following two properties can be found: 1. The variable depends functionally on both and θ. 2. The probability distribution of the variable does not depend on θ or on any other unknown parameters.
Let we get
then i.e., We call
is the 95% confidence interval for .
If , then the 95% confidence interval for is (24.12, 25.88).
Def. Let be a random sample from a distribution that depends on a parameter (or parameter vector) . Let A≤B be two statistics that have the property that P( A<θ<B ) = γ then the random interval (A,B) is called a coefficient γ confidence interval for θ.