# 统计代写|回归分析作业代写Regression Analysis代考|ST 503

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## 统计代写|回归分析作业代写Regression Analysis代考|Exact Inferences: Confidence Intervals

To interpret the estimate and its standard error, you should have a mental conversation with yourself, saying something like this:
How to think about the estimate and its standard error
Hmmm, the estimated slope is shown in the output as 1.6199, and the standard error is shown in the output as $0.1326$. So the actual slope is most likely in the range $1.6199 \pm 2(0.1316)$, or roughly between $1.6 \pm 0.26$. AHA! The true slope is most likely a positive number! So the $X$ variable has a positive relation to $Y$ !

We used $2.0$ rather than $1.96$ as a multiplier of the standard error because the result is only approximate anyway, so why not? We might as well simplify things by using another approximation, $2.0$ instead of 1.96. It just makes life easier. And it works well in practice, so we generally recommend that you follow the advice given by the above mental conversation.

But there are precise, mathematically exact results that you can use in the case where the data are produced by the classical model. The theory is mathematically deep, but you probably have seen it before, to one degree or another. It involves “Student’s $T$ distribution,” which is ubiquitous in statistics. In a nutshell, the issue revolves around how to deal with the estimate $\hat{\sigma}$ of $\sigma$ in the standard error formula. After all, as shown above, the first interval formula involving $1.96$ and $\sigma$ is exact; the only reason for calling the second interval formula “approximate” is because of the substitution of $\hat{\sigma}$ for $\sigma$. The effect of using $\hat{\sigma}$ rather than $\sigma$ can be precisely, exactly, quantified. A mathematical theorem states that if the classical regression model produces the real data, then the additional variability incurred when you use $\hat{\sigma}$ rather than $\sigma$ is precisely accounted for by using the $T$ (Student’s T) distribution rather than the $\mathrm{Z}$ (standard normal) distribution.

## 统计代写|回归分析作业代写Regression Analysis代考|Practical Interpretation of the Confidence Interval

We now discuss the practical interpretation of the confidence interval for the slope parameter. As with everything in regression, these interpretations involve conditional distributions.

If the linearity assumption is true, then the parameter $\beta_{1}$ is the difference between the means of the conditional distributions of $Y$ for cases where the $X$ variable differs by one unit. Specifically:
$$\mathrm{E}(Y \mid x+1)-\mathrm{E}(Y \mid x)=\left{\beta_{0}+\beta_{1}(x+1)\right}-\left(\beta_{0}+\beta_{1} x\right)=\beta_{0}+\beta_{1} x+\beta_{1}-\beta_{0}-\beta_{1} x=\beta_{1}$$
Thus, the mean of the distribution of potentially observable $Y$ when $X=x+1$ is precisely $\beta_{1}$ higher than the mean of the distribution of potentially observable $Y$ when $X=x$. In particular, the mean of the distribution of Cost when Widgets $=1,001$ is exactly $\beta_{1}$ higher than the mean of the distribution of Cost when Widgets $=1,000$. And it does not matter which two values $(x+1, x)$ that you compare: The mean of the distribution of Cost when Widgets $=1,601$ is exactly $\beta_{1}$ higher than the mean of the distribution of Cost when Widgets $=1,600$.

Here and throughout the book, we will refer to $\beta_{1}$ as a measure of the effect of $X$ on $Y$. In general, the word effect has the following meaning:
The meaning of the phrase ” $X$ has an effect on $Y^{\prime \prime}$
When the conditional distribution $p\left(y \mid X=x_{1}\right)$ differs from $p\left(y \mid X=x_{2}\right)$, for some specific values $x_{1}$ and $x_{2}$ of the variable $X$, then $X$ has an effect on $Y$.

# 回归分析代写

## 统计代写|回归分析作业代写Regression Analysis代考|Exact Inferences: Confidence Intervals

Hmmm，估计斜率在输出中显示为 1.6199，标准误差在输出中显示为0.1326. 所以实际的斜率最有可能在这个范围内1.6199±2(0.1316), 或大致介于1.6±0.26. 啊哈！真正的斜率很可能是一个正数！所以X变量与是 !

## 统计代写|回归分析作业代写Regression Analysis代考|Practical Interpretation of the Confidence Interval

\mathrm{E}(Y \mid x+1)-\mathrm{E}(Y \mid x)=\left{\beta_{0}+\beta_{1}(x+1)\right}-\左(\beta_{0}+\beta_{1} x\right)=\beta_{0}+\beta_{1} x+\beta_{1}-\beta_{0}-\beta_{1} x=\测试版_{1}\mathrm{E}(Y \mid x+1)-\mathrm{E}(Y \mid x)=\left{\beta_{0}+\beta_{1}(x+1)\right}-\左(\beta_{0}+\beta_{1} x\right)=\beta_{0}+\beta_{1} x+\beta_{1}-\beta_{0}-\beta_{1} x=\测试版_{1}

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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。

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