# 统计代写|回归分析作业代写Regression Analysis代考|Application of the Theory: The Graduate Student GPA Data Analysis, Revisited

#### Doug I. Jones

Lorem ipsum dolor sit amet, cons the all tetur adiscing elit

couryes™为您提供可以保分的包课服务

couryes-lab™ 为您的留学生涯保驾护航 在代写回归分析Regression Analysis方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写回归分析Regression Analysis代写方面经验极为丰富，各种代写回归分析Regression Analysis相关的作业也就用不着说。

## 统计代写|回归分析作业代写Regression Analysis代考|Application of the Theory: The Graduate Student GPA Data Analysis, Revisited

Here is how the concepts presented in this chapter apply to this concrete situation.

All estimates and standard errors are matrix functions of the observed data set, as described and calculated above.

The fitted function is (Predicted GPA) $=2.7506999+0.0013572 \times$ GMAT +0.1793805 $\times \mathrm{PhD}$. This function defines the plane that minimizes the sum of squared vertical deviations from individual GPA values to the plane. Figure 7.4 is obtained using the same code that produced Figure 7.1.

The processes at work that gave these data on $n=494$ students’ GPAs could have given rise to a completely different set of $n=494$ GPAs, even with exact same $\mathrm{PhD}$ and GMAT data values as in the current data set. These other data are potentially observable only and do not refer to specific, existing other students. These other possible data simply reflect other possibilities that might have occurred at that particular point in time and place. Unbiasedness of the parameter estimates means that, while the estimates will be different for every other data set, on average they will be neither systematically above nor below the targets $\beta_0, \beta_1$ and $\beta_2$ that govern the production of the GPA data. In other words, unbiasedness implies that your estimates, $2.7506999,0.0013572$, and 0.1793805 , are randomly sampled values from distributions whose means are precisely $\beta_0, \beta_1$ and $\beta_2$, respectively.

The same conclusion regarding unbiasedness holds when you imagine the other data sets all having different $\mathrm{PhD}$ and GMAT data (the random- $X$ viewpoint). While this way of looking at the other data sets makes it easier to view them as simply belonging to a different set of $n=494$ students, it is still best not to think about it that way, because there never existed another set of 494 students coming from the same processes that produced these students. Rather, again, you should view these other possible data sets as potentially observable, just as in the fixed- $X$ viewpoint.

Again assuming the data-generating processes just described are well modelled via the classical model, then the standard deviations of all the other parameter estimates you would get from all these other data sets, assuming the same $\mathrm{PhD}$ and GMAT data for all data sets (the conditional- $x$ framework), would be approximately $0.1191639363,0.0002155794$, and 0.0503072073 . Thus, since data values from a distribution are typically within \pm two standard deviations of the mean, and because the means of the distributions of the estimated $\beta$ ‘s are in fact the true $\beta$ ‘s (by unbiasedness), you can expect, for example, that the true $\beta_2$ (measuring the true mean difference between GPA’s of PhD and Masters student who share a common GMAT) will be within the range $0.1793805 \pm 2(0.0503072073)$. In other words, you can claim confidently that $0.07876609<\beta_2<0.2799949$ (grade points). Under the assumptions of the classical model, an exact $95 \%$ confidence version of this interval uses the $T$ distribution to get the multiplier rather than using 2.0; the more precise interval is $0.0805365726<\beta_2<0.27822450$.

$95 \%$ of data sets in the conditional- $x$ samples will have the true $\beta_2$ inside of similarly constructed intervals; the same conclusion holds in the unconditional case because of the Law of Total Expectation: Over all possible random- $X$ samples, the average coverage level is the average of the conditional coverage levels $95 \%, 95 \%, \ldots$, etc. Because the average of a constant is just that constant, the interpretation of “95\%” holds in both the fixed- $X$ and random- $X$ frameworks.

## 统计代写|回归分析作业代写Regression Analysis代考|The $R$-Squared Statistic

Recall that the true $R^2$ statistic was introduced in Chapter 6 as
$$\Omega^2=1-\mathrm{E}{v(X)} / \operatorname{Var}(Y),$$
where $v(x)$ is the conditional variance of $Y$ given $X=x$, written as $v(x)=\operatorname{Var}(Y \mid X=x)$.
The number $\Omega^2$ is a measure of how well the ” $X$ ” variable(s) predict(s) your ” $Y$ ” variable. You can understand this concept in terms of separation of the distributions $p(y \mid X=x)$, for the two cases (i) $X=\mathrm{a}$ “low” value, and (ii) $X=\mathrm{a}$ “high” value. When these distributions are well-separated, then $X$ is a good predictor of $Y$.
For example, suppose the true model is
$$Y=6+0.2 X+\varepsilon,$$
where $X \sim \mathrm{N}\left(20,5^2\right)$ and $\operatorname{Var}(\varepsilon)=\sigma^2$. Then $\operatorname{Var}(Y)=0.2^2 \times 5^2+\sigma^2=1+\sigma^2$, and $v(x)=$ $\operatorname{Var}(Y \mid X=x)=\sigma^2$, implying that $\Omega^2=1-\sigma^2 /\left(1+\sigma^2\right)=1 /\left(1+\sigma^2\right)$ is the true $R^2$.

Three cases to consider are (i) $\sigma^2=9.0$, implying a low $\Omega^2=0.1$, (ii) $\sigma^2=1.0$, implying a medium value $\Omega^2=0.5$, and (iii) $\sigma^2=1 / 9$, implying a high $\Omega^2=0.9$. In all cases, let’s say a “low” value of $X$ is 15.0, one standard deviation below the mean, and a “high” value of $X$ is 25.0, one standard deviation above the mean.

Now, when $X=15$, the distribution $p(y \mid X=15)$ is the $\mathrm{N}\left(9.0, \sigma^2\right)$ distribution; and when $X=25$, the distribution $p(y \mid X=25)$ is the $\mathrm{N}\left(11.0, \sigma^2\right)$ distribution. Figure 8.1 displays these distributions for the three cases above, where the true $R^2$ is either $0.1,0.5$, or 0.9 (which happens in this study when $\sigma^2$ is either 9.0,1.0, or 1/9). Notice that there is greater separation of the distributions $p(y \mid x)$ when the true $R^2$ is higher.

# 回归分析代写

## 统计代写|回归分析作业代写Regression Analysis代考|Application of the Theory: The Graduate Student GPA Data Analysis, Revisited

$95 \%$ 条件- $x$样本中的数据集将在相似构造的区间内具有真实的$\beta_2$;由于总期望定律，在无条件情况下也得出同样的结论:在所有可能的随机- $X$样本中，平均覆盖水平是条件覆盖水平的平均值$95 \%, 95 \%, \ldots$，等等。因为一个常数的平均值就是那个常数，所以“95％”的解释在固定- $X$和随机- $X$框架中都成立。

## 统计代写|回归分析作业代写Regression Analysis代考|The $R$-Squared Statistic

$$\Omega^2=1-\mathrm{E}{v(X)} / \operatorname{Var}(Y),$$

$$Y=6+0.2 X+\varepsilon,$$

Days
Hours
Minutes
Seconds

# 15% OFF

## On All Tickets

Don’t hesitate and buy tickets today – All tickets are at a special price until 15.08.2021. Hope to see you there :)