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Point and Confidence Interval Estimates for a Population Mean
A confidence interval (CI) is used to indicate the reliability of an estimate. There are two types of CIs, that is, interval estimate and point estimate, and both of these can be used to estimate the mean for a population sample. These two forms of confidence intervals are computed from observations and differ from dataset to dataset, but frequently include the parameter of interest, and should be obtained within specified errors of margin if the experiment is repeated. An error of margin is a parameter that is used to indicate the random sampling error in an experimental survey. The larger the value, the less reliable the estimate for the parameter is, in this case the mean.
Interval estimates for a sample mean can be contrasted from point estimates for the population mean, for example, the value could be given as a 95% confidence interval. A point estimate is a single figure that is given as an approximation of the for a population mean of some observation or statistical experiment. In contrast, an interval estimate specifies a range within which the approximated mean of the population lies. Both of these estimates are used to indicate the reliability of a population.
Since the estimate for the population mean is applicable to the whole population, it must have certain characteristics, i.e. it must be consistent within the whole population, it must be unbiased (must be centered on the actual population mean), and it must have a small standard error (the lower the error, the more reliable it is). For example, if we have a population sample (x1, x2… xn), a point estimate for the mean, say θ, would be one that can be used in place of the individual observations. The estimator can be rejected if it cannot be used in place many observations (Groebner, 2007).
An interval estimate gives more information about a population characteristic than a point estimate. Confidence intervals are sometimes referred to as confidence intervals. The general formula for computing the confidence value is:
- Point Estimate ± (Critical Value)(Standard Error)
In order to obtain the confidence interval for the mean when δ is known, we use the following equation:
Consider a 95% confidence interval, in this case,
Determining the Required Sample Size for Estimating a Population Mean
In order to find an estimator for the population mean, the surveyor must use a minimum population size that is based on certain parameters of the population. Before a surveyor picks a sample size, he must know all or part of the following: the standard deviation, δ, of the population, the maximum acceptable difference (margin of error), the mean, and the desired confidence level. In most sample surveys, the surveyor will not have the exact value of δ. Generally, the standard deviation of the population will be approximated from the following: findings of a previous study, the value from a pilot survey, from secondary data, or from the surveyor’s assessment of the dataset. The maximum acceptable difference is the maximum error that the researcher can accept, that is, the maximum deviation of the estimate from the true mean that is permitted. The desired confidence level is the degree of confidence of the surveyor that the estimate of the mean does not deviate from the true mean by a significant value, or by more than the maximum acceptable deviation. Generally, most studies use a 95% confidence level.
The margin of error (e), is added or subtracted from the point estimate to find the value at any point.
Given the margin of error, e, and level of confidence, (1 – α), the required sample size can be obtained using the following formula,
For example, if the standard deviation is 53, the level of confidence is 95% and the margin of error is ±5, then the required sample size, n, is found as:
Hence, the required sample size, which must be a whole number, is estimated as 304.
If δ is unknown, it can be approximated through the sample size method, i.e. use a δ that is thought to be as close as possible to the true δ, or, select a value of δ that is derived from a pilot survey, as outlined above (Groebner, 2007).
Estimating a Population Proportion
An interval estimate for the proportion of a population that follows a normal distribution, p, can be computed by adding the margin of uncertainty to the sample proportion, Ρ. Recalling that a population is normally distributed if the sample size is large, and the standard deviation is given as:
We can approximate this with the sample data:
The upper and lower confidence limits for the population proportion are computed using the following formula:
Where,
z is the standard normal value for the level of confidence desired
is the sample proportion, and n is the sample size.
The maximum error of the population proportion is obtained by adding or subtracting the margin or error from both sides, given by the formula:
Hence, P lies between – e < P< + e
Work Cited
Groebner, David F. Business Statistics: A Decision-Making Approach. New Jersey: Prentice-Hall, Inc., 2007.
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