Interval Estimation & Estimation Of Proportion

2.2 Interval Estimation& Estimation of Proportion

INTERVAL ESTIMATION
By using point estimation ,we may not get desired
degree of accuracy in estimating a parameter.
Therefore ,it is better to replace point estimation
by interval estimation.

INTERVAL ESTIMATION
Interval estimate:-
An interval estimate of an unknown parameter is an interval of the form L1 ≤ θ≤ L2, where the end points L1 and L2 depend on the numerical value of
the statistic θ* for particular sample on the sampling distributon of θ* .
100(1-α)% Confidence Interval:-
A 100(1-α)% confidence interval for a parameter θ is an interval of the fprm
[L1 , L2] such that P(L1≤θ ≤L2) =1- α, 0< α <1regardless of the actual value of
θ.

INTERVAL ESTIMATION
Confidence limits:-
The quantities L1 and L2 are called upper and lower
confidence limits
Degreeof confidence (confidence
coefficient)
1-α

Interval Estimation
Suppose we have a large (n 30) random
sample from a population with the unknown
mean  and known variance 2. We know
inequality
will satisfy with probability 1 - .

Interval Estimation
This inequality we can rewrite
When the observed value become available, we obtain
Large sample
confidence interval
for  -  known
Thus when sample has been obtained and the value of has been
calculated, we can claim with probability (1 -  )100% confidence
That the interval from

Interval Estimation
Since  is unknown in most applications, we may have
to make the further approximation of substituting for 
the sample standard deviation s.
Large sample
confidence interval for 

Interval Estimation
For small samples (n < 30), we assume that we are sampling from normal population and proceed similarly as before we get the (1 - )100% confidence interval formula
Small sample confi-
dence interval for 

ESTIMATION OF PROPORTION
There are many problems in which we must Estimate
proportion
Proportion of Defectives
Proportion of objects or things having required attributes
The mortality rate of a disease.
Remark : In many of these problems it is reasonable to
assume that we are sampling a binomial population .hence
that our problem is to estimate the binomial parameter p .
The probability of success in a single trial of a binomial
experiment is p. This probability is a population proportion

ESTIMATION OF PROPORTION
Estimation of Proportion
Suppose that random sample of size n has been taken
from a population and that X( n)is the number of times
that an appropriate event occurs in n trials (observations).
THEN
Point estimator of the population proportion (p) is given by

Sample proportion is an Unbiased Estimator of population proportion
If the n trials satisfy the assumption underlying the binomial
distribution ,then
mean of number of successes is np
Variance of number of successes is np(1-p)
Expectation and variance of sample proportion
X denotes the number of successes in n trials

When n is large, we can construct approximate
confidence intervals for the binomial parameter p by
using the normal approximation to the binomial
distribution. Accordingly, we can assert with probability
1 -  that the inequality

will be satisfied. Solving this quadratic inequality for p
we can obtain a corresponding set of approximate
confidence limits for p in terms of the observed value of
x but since the necessary calculations are complex, we
shall make the further approximation of substituting x/n
for p in

Large sample confidence interval for p
where the degree of confidence is (1 - )100%.
Maximum error of estimate
With the observed value x/n substituted for p we obtain an
estimate of E.

Confidence Interval for p
Point Estimate = X / n
Confidence Interval

Sample size determination
But this formula cannot be used as it stands unless we have some information about the possible size of p. If no much information is available, we can make use of the fact that p(1 - p) is at most 1/4, corresponding to p = 1/2 ,as can be shown by the method of elementary calculus. If a range for p is known, the value closest to 1/2 should be used.
Sample size (p unknown)

Interval Estimation & Estimation Of Proportion

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