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

Notice that the range of the confidence interval, $1,303.753$ and $1,560.912$, is around 250 Cost units, much narrower than the range shown by the vertical bar in Figure 3.5, which is around 1,000 Cost units. Why? Is this yet another approximation? No. The reason for the big difference is that the mean of all potentially observable values of the Cost variable for jobs with any fixed number of widgets, is very much different than a single potentially observable value of the Cost variable.
It makes sense, intuitively and by the Law of Large Numbers, that you can estimate a mean value more and more precisely, with a margin of error tending towards 0 , when you increase your sample size. However, you cannot estimate a single value with such precision, no matter how large is your sample size.

To understand the distinction between a single value of $Y$ and the mean value of the potentially observable $Y^{\prime}$ s, suppose you have a model for how stock returns behave:
(Return on Company A’s stock) $=\beta_0+\beta_1($ Return on the $S \& P 500$ index $)+\varepsilon$.
(Return on Company A’s stock) $=0.0032+0.67$ (Return on the $S \& P 500$ index) $+e$
Now, suppose we tell you that the return on the SEPP 500 index was $0.005(0.5 \%)$ yesterday. What can you tell us about the return on Company A’s stock? Was it exactly $0.0032+0.67(0.005)=0.00655(0.655 \%)$ ? We hope you can see that the answer is no, it was not $0.655 \%$. In all likelihood, it was not even very close to $0.655 \%$. It might even have been a negative number-it is quite common that individual stock prices move in a direction counter to the market. You cannot predict the actual value of company A’s stock return with precision using this model, because company A’s stock return is not a deterministic function of the $S \mathcal{E} P 500$ return. There are unique features of Company A that make it differ from the market, sometimes substantially, on any given day.

Having a large sample size here means that the estimate of the mean of all potentially observable returns of Company A’s stock, given the SEPP 500 market return is $0.005$, is very close to $0.0032+0.67(0.005)=0.00655$. However, the mean of all potentially observable returns is very different from a single potentially observable return: The individual return differs from the conditional mean (which is approximately $0.00655$ ), by the random error term $e$. This error term can be quite large, leading possibly to a negative Company A return when the $S \mathcal{E P} 500$ return is positive.

统计代写|回归分析作业代写Regression Analysis代考|Hypothesis Testing and p-Values

Some researchers will do nearly anything to get a publication. The incentives are great: Fame, tenure, promotion, annual salary, raises, prime class assignments, and clout in one’s department are a function of quality and quantity of publications.

Historically, statistical results were required to be “statistically significant” to be publishable. In terms of confidence intervals, this means that the interval for the effect (e.g., the $\beta$ ) in question must exclude 0 so that you can confidently state the direction of the effect (positive or negative) of the given $X$ variable on $Y$.

Researchers used the $p$-values that are reported routinely by regression software to determine “statistical significance.” But p-values are easily manipulated, and unscrupulous researchers can analyze data “creatively” to get nearly any $p$-value they would like to see. This has led to an unfortunate practice known as p-hacking, where researchers try analyses many different ways until they get a $p$-value that is statistically significant, and then try to publish the results. Because of their potential for misuse, there is a strong movement in the scientific community away from use of $p$-values, as well as the phrase “statistical significance,” in favor of other statistics and characterizations.

When interpreted correctly and not misused, the $p$-value does provide interesting and somewhat useful information. Thus, we insist that you understand $p$-values very well, so that you can use them correctly and effectively, and so that you will not become a “p-hacker.”

To interpret the $p$-value correctly, you must consider the question, “Is the estimate of the effect of $X$ on $Y$ explainable by chance alone?” But to answer that question, you must first understand what it means for an estimated effect to be explained by chance alone. The following example explains this concept.

统计代写|回归分析作业代写Regression Analysis代考|STA4210

回归分析代考

统计代写|回归分析作业代写回归分析代考|精确推论:预测区间


请注意,置信区间$1,303.753$和$1,560.912$的范围大约是250个成本单位,比图3.5中垂直条所示的大约1000个成本单位的范围窄得多。为什么?这是另一种近似吗?不。产生巨大差异的原因是,对于具有固定数量的小部件的作业,Cost变量的所有潜在可观察值的平均值与Cost变量的单个潜在可观察值的平均值相差很大。凭直觉和根据大数定律,当你增加样本容量时,你可以越来越精确地估计出一个平均值,误差范围趋于0,这是有道理的。然而,无论你的样本容量有多大,你都无法以如此精确的精度估计单个值


为了理解$Y$的单个值和潜在可观察到的$Y^{\prime}$ s的平均值之间的区别,假设你有一个股票回报行为的模型:
(a公司股票的回报)$=\beta_0+\beta_1($$S \& P 500$指数的回报$)+\varepsilon$ .
(a公司股票的回报)$=0.0032+0.67$ ($S \& P 500$指数的回报)$+e$假设我们告诉你,昨天SEPP 500指数的回报率是$0.005(0.5 \%)$。你能告诉我们A公司股票的回报率吗?确切地说是$0.0032+0.67(0.005)=0.00655(0.655 \%)$吗?我们希望你能看到答案是否定的,它不是$0.655 \%$。十有八九,它甚至不太接近$0.655 \%$。这甚至可能是一个负数——个股价格与市场走势相反是很常见的。使用该模型无法精确预测A公司股票收益的实际值,因为A公司股票收益不是$S \mathcal{E} P 500$收益的确定函数。A公司有其独特的特点,使其在任何一天都与市场不同,有时甚至有很大的不同


这里有一个大的样本量意味着a公司股票的所有潜在可观察回报的平均值的估计,假定SEPP 500市场回报是$0.005$,非常接近$0.0032+0.67(0.005)=0.00655$。然而,所有潜在可观察到的回报的平均值与单个潜在可观察到的回报的平均值非常不同:通过随机误差项$e$,个体回报与条件平均值(大约是$0.00655$)不同。这个误差项可能相当大,当$S \mathcal{E P} 500$回报为正时,可能导致a公司的回报为负

统计代写|回归分析作业代写回归分析代考|假设检验和p-值


一些研究人员为了得到一篇论文几乎会做任何事。这种激励机制是很强大的:名声、终身职位、晋升、年薪、加薪、主要的课堂任务以及在一个部门的影响力都是出版物的质量和数量的函数


历史上,统计结果必须具有“统计意义”才能发表。就置信区间而言,这意味着有关的效果(例如$\beta$)的区间必须排除0,以便您可以有把握地陈述$Y$上给定$X$变量的效果的方向(正或负)


研究人员使用由回归软件例行报告的$p$ -值来确定“统计显著性”。但是p值很容易被操纵,无道德的研究人员可以“创造性地”分析数据,得到几乎任何他们想看到的$p$值。这导致了一种被称为p-hacking的不幸做法,研究人员尝试用多种不同的方法进行分析,直到得到统计上显著的$p$ -值,然后试图发布结果。由于它们可能被滥用,在科学界有一种强烈的运动,不使用$p$ -values以及短语“统计显著性”,而倾向于其他统计和描述


如果正确解释且没有误用,$p$ -value确实提供了有趣且有些有用的信息。因此,我们坚持要求您非常好地理解$p$ -values,以便您能够正确有效地使用它们,从而不会成为一个“p-hacker”。


要正确解释$p$ -值,你必须考虑这个问题,“对$X$对$Y$的影响的估计仅凭偶然可以解释吗?”但要回答这个问题,你必须首先理解,一个估计的效应只能用偶然解释是什么意思。下面的例子解释了这个概念。

统计代写|回归分析作业代写Regression Analysis代考

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