### 统计代写|生物统计学作业代写Biostatistics代考|ESTIMATION OF PARAMETERS

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|生物统计学作业代写Biostatistics代考|BASIC CONCEPTS

A class of measurements or a characteristic on which individual observations or measurements are made is called a variable or random variable. The value of a random variable varies from subject to subject; examples include weight, height, blood pressure, or the presence or absence of a certain habit or practice, such as smoking or use of drugs. The distribution of a random variable is often assumed to belong to a certain family of distributions, such as binomial, Poisson, or normal. This assumed family of distributions is specified or indexed by one or several parameters, such as a population mean $\mu$ or a population pro-portion $\pi$. It is usually either impossible, too costly, or too time consuming to obtain the entire population data on any variable in order to learn about a parameter involved in its distribution. Decisions in health science are thus often made using a small sample of a population. The problem for a decision maker is to decide on the basis of data the estimated value of a parameter, such as the population mean, as well as to provide certain ideas concerning errors associated with that estimate.

## 统计代写|生物统计学作业代写Biostatistics代考|Introduction to Confidence Estimation

Statistical inference is a procedure whereby inferences about a population are made on the basis of the results obtained from a sample drawn from that population. Professionals in health science are often interested in a parameter of a certain population. For example, a health professional may be interested in knowing what proportion of a certain type of person, treated with a particular drug, suffers undesirable side effects. The process of estimation entails calculating, from the data of a sample, some statistic that is offered as an estimate of the corresponding parameter of the population from which the sample was drawn.

A point estimate is a single numerical value used to estimate the corresponding population parameter. For example, the sample mean is a point estimate for the population mean, and the sample proportion is a point estimate for the population proportion. However, having access to the data of a sample and a knowledge of statistical theory, we can do more than just providing a point estimate. The sampling distribution of a statistic-if available-would provide information on biasedness/unbiasedness (several statistics, such as $\bar{x}, p$, and $s^{2}$, are unbiased) and variance.

Variance is important; a small variance for a sampling distribution indicates that most possible values for the statistic are close to each other, so that a particular value is more likely to be reproduced. In other words, the variance of a sampling distribution of a statistic can be used as a measure of precision or reproducibility of that statistic; the smaller this quantity, the better the statistic as an estimate of the corresponding parameter. The square root of this variance is called the standard error of the statistic; for example, we will have the standard error of the sample mean, or $\mathrm{SE}(\bar{x})$; the standard error of the sample proportion, $\mathrm{SE}(p)$; and so on. It is the same quantity, but we use the term standard deviation for measurements and the term standard error when we refer to the standard deviation of a statistic. In the next few sections we introduce a process whereby the point estimate and its standard error are combined to form an interval estimate or confidence interval. A confidence interval consists of two numerical values, defining an interval which, with a specified degree of confidence, we believe includes the parameter being estimated.

## 统计代写|生物统计学作业代写Biostatistics代考|ESTIMATION OF MEANS

The results of Example $4.1$ are not coincidences but are examples of the characteristics of sampling distributions in general. The key tool here is the central limit theorem, introduced in Section 3.2.1, which may be summarized as follows: Given any population with mean $\mu$ and variance $\sigma^{2}$, the sampling distribution of $\bar{x}$ will be approximately normal with mean $\mu$ and variance $\sigma^{2} / n$ when the sample size $n$ is large (of course, the larger the sample size, the better the
ESTIMATION OF MEANS 153
approximation; in practice, $n=25$ or more could be considered adequately large). This means that we have the two properties
\begin{aligned} \mu_{\bar{x}} &=\mu \ \sigma_{\bar{x}}^{2} &=\frac{\sigma^{2}}{n} \end{aligned}
as seen in Example 4.1.
The following example shows how good $\bar{x}$ is as an estimate for the population $\mu$ even if the sample size is as small as 25 . (Of course, it is used only as an illustration; in practice, $\mu$ and $\sigma^{2}$ are unknown.)

Example 4.2 Birth weights obtained from deliveries over a long period of time at a certain hospital show a mean $\mu$ of $112 \mathrm{oz}$ and a standard deviation $\sigma$ of $20.6 \mathrm{oz}$. Let us suppose that we want to compute the probability that the mean birth weight from a sample of 25 infants will fall between 107 and $117 \mathrm{oz}$ (i.e., the estimate is off the mark by no more than $5 \mathrm{oz}$ ). The central limit theorem is applied and it indicates that $\bar{x}$ follows a normal distribution with mean
$$\mu_{\bar{x}}=112$$
and variance
$$\sigma_{\bar{x}}^{2}=\frac{(20.6)^{2}}{25}$$
or standard error
$$\sigma_{\bar{x}}=4.12$$
It follows that
\begin{aligned} \operatorname{Pr}(107 \leq \bar{x} \leq 117) &=\operatorname{Pr}\left(\frac{107-112}{4.12} \leq z \leq \frac{117-112}{4.12}\right) \ &=\operatorname{Pr}(-1.21 \leq z \leq 1.21) \ &=(2)(0.3869) \ &=0.7738 \end{aligned}
In other words, if we use the mean of a sample of size $n=25$ to estimate the population mean, about $80 \%$ of the time we are correct within $5 \mathrm{oz}$; this figure would be $98.5 \%$ if the sample size were 100 .

## 统计代写|生物统计学作业代写Biostatistics代考|ESTIMATION OF MEANS

ESTIMATION OF MEANS 153

μX¯=μ σX¯2=σ2n

μX¯=112

σX¯2=(20.6)225

σX¯=4.12

## 广义线性模型代考

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。