• Save
Interval Estimation & Estimation Of Proportion
Upcoming SlideShare
Loading in...5
×
 

Like this? Share it with your network

Share

Interval Estimation & Estimation Of Proportion

on

  • 3,670 views

Interval Estimation & Estimation Of Proportion

Interval Estimation & Estimation Of Proportion

Statistics

Views

Total Views
3,670
Views on SlideShare
3,663
Embed Views
7

Actions

Likes
0
Downloads
0
Comments
0

1 Embed 7

http://www.slideshare.net 7

Accessibility

Categories

Upload Details

Uploaded via as Microsoft PowerPoint

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

Interval Estimation & Estimation Of Proportion Presentation Transcript

  • 1. 2.2 Interval Estimation& Estimation of Proportion
  • 2. 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.
  • 3. 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.
  • 4. INTERVAL ESTIMATION
    Confidence limits:-
    The quantities L1 and L2 are called upper and lower
    confidence limits
    Degreeof confidence (confidence
    coefficient)
    1-α
  • 5. 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 - .
  • 6. 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
  • 7. 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 
  • 8. 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 
  • 9. 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
  • 10. 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
  • 11. 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
  • 12. Estimation of Proportion
    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
  • 13. Estimation of Proportion
    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
  • 14. Estimation of Proportion
    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.
  • 15. Confidence Interval for p
    Point Estimate = X / n
    Confidence Interval
  • 16. Estimation of Proportion
    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)