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Estimating Population Mean with Unknown Standard Deviation

Estimating Population Mean with Unknown Standard Deviation

In practice, we rarely know the population standard deviation. In the past, when the sample size was large, this did not present a problem to statisticians. They used the sample standard deviation s as an estimate for σ and proceeded as before to calculate a confidence interval with close enough results. However, statisticians ran into problems when the sample size was small. A small sample size caused inaccuracies in the confidence interval.
William S. Gosset (1876–1937) of the Guinness...

Estimating Population Standard Deviation

Estimating Population Standard Deviation

When the population standard deviation is unknown and the sample size is large, the sample standard deviation s is commonly used as a point estimate of σ. However, it can sometimes under or overestimate the population standard deviation. To overcome this drawback, confidence intervals are determined to estimate population parameters and eliminate any calculation bias accurately. However, this only applies to random samples from normally distributed populations. Knowing the sample mean and...

Estimating Population Mean with Known Standard Deviation

Estimating Population Mean with Known Standard Deviation

To construct a confidence interval for a single unknown population mean μ, where the population standard deviation is known, we need sample mean as an estimate for μ and we need the margin of error. Here, the margin of error (EBM) is called the error bound for a population mean (abbreviated EBM). The sample mean is the point estimate of the unknown population mean μ.
The confidence interval estimate will have the form as follows:
(point estimate - error bound, point estimate + error bound)
The...

Distributions to Estimate Population Parameter

Distributions to Estimate Population Parameter

The accurate values of population parameters such as population proportion, population mean, and population standard deviation (or variance) are usually unknown. These are fixed values that can only be estimated from the data collected from the samples. The estimates of each of these parameters are sample proportion, the sample mean, and sample standard deviation (or variance). To obtain the values of these sample statistics, data are required that have particular distribution and central...

Prediction Intervals

Prediction Intervals

The interval estimate of any variable is known as the prediction interval. It helps decide if a point estimate is dependable.
However, the point estimate is most likely not the exact value of the population parameter, but close to it. After calculating point estimates, we construct interval estimates, called confidence intervals or prediction intervals. This prediction interval comprises a range of values unlike the point estimate and is a better predictor of the observed sample value, y.
The...

Uncertainty: Confidence Intervals

Uncertainty: Confidence Intervals

The confidence interval is the range of values around the mean that contains the true mean. It is expressed as a probability percentage. The interpretation of a 95% confidence interval, for instance, is that the statistician is 95% confident that the true mean falls within the interval. The upper and lower limits of this range are known as confidence limits. The confidence limits for the true mean are estimated from the sample's mean, the standard deviation, and the statistical factor 't,' or...

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Related Experiment Videos

Stochastic population forecasts based on conditional expert opinions.

F C Billari¹, R Graziani, E Melilli

¹Bocconi University Milan, Italy.

Journal of the Royal Statistical Society. Series A, (Statistics in Society)

|August 11, 2012

Summary

This summary is machine-generated.

This study introduces an expert-based stochastic population forecasting method to generate probabilistic population projections. The approach utilizes expert opinions on demographic components, applied to Italian population data.

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Area of Science:

Demography
Statistical Modeling
Population Studies

Background:

Official population forecasts often rely on deterministic scenarios.
There is a need for probabilistic forecasting methods that incorporate expert knowledge.
Understanding future population dynamics requires robust forecasting tools.

Purpose of the Study:

To develop and apply an expert-based stochastic population forecasting methodology.
To enable the creation of probabilistic versions of official scenario-based forecasts.
To provide a framework for incorporating expert judgment into demographic projections.

Main Methods:

Elicitation of expert opinions on demographic components.
Conditional scenario-based expert opinion gathering (two-step or multi-step).
Specification of the full probability distribution for population forecasts.

Main Results:

A novel expert-based stochastic population forecasting method was developed.
The method was successfully applied to generate a stochastic forecast for the Italian population.
Probabilistic population projections were derived from official scenarios.

Conclusions:

Expert-based stochastic forecasting offers a valuable alternative to traditional methods.
The developed method enhances the probabilistic nature of official population forecasts.
This approach provides a more comprehensive understanding of future population trends.