# Estimation of a population mean

The most cardinal point and time interval estimate process involves the estimate of a population average. Suppose it is of matter to to estimate the population base, μ, for a quantitative variable. Data collected from a bare random sample can be used to compute the sample mean, x̄, where the value of x̄ provides a point estimate of μ.

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When the sample distribution mean is used as a sharpen estimate of the population beggarly, some error can be expected owing to the fact that a sample, or subset of the population, is used to compute the point estimate. The absolute value of the difference between the sample mean, x̄, and the population hateful, μ, written |x̄ − μ|, is called the sampling mistake. time interval estimate incorporates a probability statement about the magnitude of the sampling error. The sampling distribution of x̄ provides the footing for such a statement. Statisticians have shown that the hateful of the sampling distribution of x̄ is equal to the population base, μ, and that the standard diversion is given by σ/Square root of√n, where σ is the population standard diversion. The standard deviation of a sample distribution is called the standard error. For big sample sizes, the central limit theorem indicates that the sampling distribution of x̄ can be approximated by a normal probability distribution. As a matter of practice, statisticians normally consider samples of size 30 or more to be large. In the large-sample character, a 95 % confidence interval estimate for the population mean is given by x̄ ± 1.96σ/Square root of√n. When the population standard deviation, σ, is stranger, the sample criterion deviation is used to estimate σ in the confidence interval formula. The quantity 1.96σ/Square etymon of√n is frequently called the margin of error for the calculate. The quantity σ/Square root of√n is the standard error, and 1.96 is the number of criterion errors from the hateful necessity to include 95 % of the values in a normal distribution. The rendition of a 95 % confidence interval is that 95 % of the intervals constructed in this manner will contain the population think of. thus, any interval computed in this manner has a 95 % assurance of containing the population mean. By changing the constant from 1.96 to 1.645, a 90 % assurance interval can be obtained. It should be noted from the formula for an interval appraisal that a 90 % confidence interval is narrower than a 95 % confidence interval and as such has a slenderly smaller confidence of including the population mean. Lower levels of confidence lead to even more narrow intervals. In exercise, a 95 % confidence interval is the most wide used.

Owing to the presence of the n1/2 term in the convention for an interval estimate, the sample size affects the allowance of error. Larger sample sizes lead to smaller margins of error. This observation forms the footing for procedures used to select the sample size. sample distribution sizes can be chosen such that the confidence interval satisfies any desire requirements about the size of the margin of error. The procedure just described for developing interval estimates of a population mean is based on the use of a large sample distribution. In the small-sample case—i.e., where the sample distribution size nitrogen is less than 30—the triiodothyronine distribution is used when specifying the margin of erroneousness and constructing a confidence interval estimate. For exercise, at a 95 % level of confidence, a value from the deoxythymidine monophosphate distribution, determined by the value of newton, would replace the 1.96 value obtained from the normal distribution. The triiodothyronine values will always be larger, leading to wider confidence intervals, but, as the sample size becomes larger, the metric ton values get closer to the corresponding values from a convention distribution. With a sample size of 25, the metric ton value used would be 2.064, as compared with the normal probability distribution value of 1.96 in the large-sample shell.