Interval Estimation & Estimation Of Proportion
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This document discusses interval estimation and estimation of proportions from sample data. It defines an interval estimate as a range of values (L1 to L2) that is likely to contain the true population parameter. A 100(1-α)% confidence interval is an interval where the probability of the parameter falling within the interval is 1-α. For large samples where the population variance is known, the confidence interval for the mean is calculated. For small samples or unknown variance, approximations are used. Estimation of a population proportion p is discussed, where the sample proportion p=X/n is an unbiased estimator of p. For large samples, a normal approximation can be used to construct a confidence interval for p. Sample size determination is
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- Given a hypothesis h that makes r errors on a test set of n examples, the sample error r/n provides an unbiased estimate of the true error. The variance of this estimate depends on r and n based on the binomial distribution.
- For large n, the binomial distribution can be approximated by the normal distribution. Confidence intervals for the true error can then be determined based on the sample error and standard deviation
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This document discusses methods for estimating population parameters from sample data, including point estimation, bias, confidence intervals, sample size determination, and hypothesis testing. Key points include defining point estimates as single values representing plausible population values based on sample data, describing how to calculate confidence intervals for population proportions and means using z-tests and t-tests, and outlining how to determine necessary sample sizes to achieve a desired level of accuracy and confidence.
Estimation in statistics
This document discusses key concepts in statistical estimation including:
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- A point estimate is a single value that estimates an unknown population parameter, while an interval estimate provides a range of plausible values for the parameter.
- A confidence interval gives the probability that the interval calculated from the sample data contains the true population parameter. Common confidence intervals are 95% confidence intervals.
- Formulas for confidence intervals depend on whether the population standard deviation is known or unknown, and the sample size.
Lesson04_Static11
This document discusses statistical confidence interval estimation. It covers:
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2) Confidence interval estimation for the proportion.
3) Factors that affect confidence interval width like data variation, sample size, and confidence level.
4) How to estimate sample sizes needed to estimate a population mean or proportion within a given level of precision and confidence.
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This document provides information on estimating population characteristics from sample data, including:
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- Confidence intervals provide a range of plausible values for population characteristics with a specified degree of confidence.
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- Guidelines for determining necessary sample sizes to estimate population values within a specified margin of error are also outlined.
Estimating a Population Proportion
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Chapter 7: Estimating Parameters and Determining Sample Sizes
7.1: Estimating a Population Proportion
Estimating a Population Proportion
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Chapter 7: Estimating Parameters and Determining Sample Sizes
7.1: Estimating a Population Proportion
Lesson04_new
This document discusses various methods for constructing confidence intervals to estimate population parameters using sample statistics. It covers confidence interval estimation for the mean when the population standard deviation is known and unknown, estimation for the proportion, and addresses situations involving finite populations. Factors that influence confidence interval width and formulas for determining necessary sample sizes are also presented. Examples are provided to illustrate how to set up confidence intervals and calculate required sample sizes.
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This document discusses confidence intervals for estimating population parameters. It provides examples of constructing point and interval estimates for the population mean and proportion from sample data. Confidence intervals allow us to estimate a range of plausible values for the true population parameter based on the sample results and desired confidence level, rather than just a single point value. The width of the confidence interval depends on the sample size and confidence level, with larger samples and lower confidence levels producing narrower intervals.
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Categorical data analysis full lecture note PPT.pptx
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A simple planning agent uses percepts from the environment to build a model of the current state and calls a planning algorithm to generate a plan to achieve its goal. Practical planning involves restricting the language used to define problems, using specialized planners rather than general theorem provers, and adopting hierarchical decomposition to store and retrieve abstract plans from a library. A solution is a fully instantiated, totally ordered plan that guarantees achieving the goal.
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AI: Belief Networks
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Interval Estimation & Estimation Of Proportion
- 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)