## 统计代写|统计计算代写Statistical calculation代考|Sampling distribution of the means

A sampling distribution is a distribution of a statistic (such as the mean) of all the possible samples of the same size selected from the population.

The sampling distribution theorem forms the foundation of inferential statistics. The theorem states that if you draw all the possible different samples of the same size $(n)$ from a population, you can calculate the mean of each sample and these means will most probably differ. If you list all these possible samples together with their means, it is known as the sampling distribution of the means.
Properties of the sampling distribution of the means

• If you calculate the mean of all the different sample means, it is equal to the population mean:
• The differences between the means are known as the variability in the sampling distribution of the means and can be measured by the standard error of the mean:
$$\sigma_x=\frac{\sigma}{\sqrt{n}}$$
The larger the sample size, the smaller the standard error of the mean and the better the estimate of the population mean because of the lesser dispersion.
• In most cases the population standard deviation $(s)$ is unknown and the sample standard deviation $(s)$ is used as an approximation to the population standard deviation $(s)$.
The standard error of the mean $s$ then becomes:
$$s_{\bar{x}}=\frac{s}{\sqrt{n}}$$
• If the population is normally distributed with a known population $s$, the sample distribution of the mean is also normal, regardless of the sample size.
• The central limit theorem states that if the population from which the sample is drawn is not normal, the distribution of the sample means will become more and more normal as the sample size increases. A sample of $n \geq 30$ is considered by most as being large enough to assume a normal distribution.
• In a population that is normal or close to normal with an unknown population $\sigma$, the distribution of sample means is referred to as the student’s $t$ distribution. The sample standard deviation (s) is used in the calculation of the standard error of estimate.

## 统计代写|统计计算代写Statistical calculation代考|Point estimation

A point estimate of a population parameter is done by making use of a single value of a statistic that is based on sample data.

• The best estimator for the population mean $\mu$ is the sample mean $\bar{x}$.
• The best estimator for the population standard deviation $\sigma$ is the sample standard deviation $s$.
• The sample proportion $\pi$ is the best estimator for the population proportion $p$.
• The discrepancy between a sample statistic and its population parameter
• is called sampling error. Measuring sampling error forms a large part of inferential statistics.
• These point estimates will almost never provide the exact value of the population parameter, because the sample mean $(\bar{x})$ depends on which sample, out of all the possible samples, was selected. Each possible sample mean $(\bar{x})$ will result in a different estimate for the population parameter.
• Because of this sampling variability, a point estimate is much more useful if it is accompanied by a probability to measure how close the sample mean is to the population mean. This can be done by making use of a confidence interval estimate. A confidence interval estimate is a range of values within which the population parameter probably lies.

# 统计计算代写

## 统计代写|统计计算代写Statistical calculation代考|Sampling distribution of the means

• 如果计算所有不同样本均值的均值，则它等于总体均值:
• 均值之间的差异被称为均值抽样分布的可变性，可以通过均值的标准误差来衡量:
$$\sigma_x=\frac{\sigma}{\sqrt{n}}$$
样本量越大，均值的标准误差越小，由于离差越小，总体均值的估计值就越好。
• 在大多数情况下，总体标准差 $(s)$ 末知且样本标准差 $(s)$ 用作总体标准差的近似值 $(s)$. 平均值的标准误差 $s$ 然后变成:
$$s_{\bar{x}}=\frac{s}{\sqrt{n}}$$
• 如果人口服从已知人口的正态分布 $s$ ，无论样本大小如何，均值的样本分布也是正态 的。
• 中心极限定理指出，如果抽取样本的总体不正态, 则样本均值的分布将随着样本量 的增加而变得越来越正态。的样本 $n \geq 30$ 被大多数人认为足够大，可以假设正态分 布。
• 在正常或接近正常且末知人群的人群中 $\sigma$, 样本均值的分布称为学生的 $t$ 分配。样本 标准偏差 (s) 用于计算估计的标准误差。

## 统计代写|统计计算代写Statistical calculation代考|Point estimation

• 总体均值的最佳估计量米是样本均值X¯.
• 总体标准差的最佳估计量p是样本标准差秒.
• 样本比例π是人口比例的最佳估计量p.
• 样本统计量与其总体参数之间的差异
• 称为抽样误差。测量抽样误差构成了推论统计的很大一部分。
• 这些点估计几乎永远不会提供总体参数的准确值，因为样本均值(X¯)取决于从所有可能的样本中选择了哪个样本。每个可能的样本均值(X¯)将导致对总体参数的不同估计。
• 由于这种抽样可变性，如果点估计伴随着一个概率来衡量样本均值与总体均值的接近程度，它就会更有用。这可以通过使用置信区间估计来完成。置信区间估计值是总体参数可能位于的值范围。

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