### 统计代写|工程统计作业代写Engineering Statistics代考|“Student’s” t-Distribution

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## 统计代写|工程统计作业代写Engineering Statistics代考|“Student’s” t-Distribution

W. S. Gossett, publishing his work under the pseudonym “Student,” developed the $t$-distribution. The statistic would become the basis for the $t$-test so widely used for the evaluation of engineering data.

The $t$-statistic is very similar to the standard normal $z$-statistic, but instead of using the true population mean and standard deviation, it uses the sample standard deviation.
$$T=\frac{X-\mu}{s}$$
Because it is based on sample data, not the entire population, the degrees of freedom $\nu$ is one less than the number of data used to calculate the sample average and $s$
$$v=n-1$$
Relative to the $z$-statistic, the $t$-statistic includes the uncertainty on both the sample average and sample standard deviation. Both the $z$ – and t-statistics are dimensionless regardless of the units on the variable $X$.
The random variable $t$ has the probability density function below:
$$\begin{gathered} f(t)=\frac{1}{\sqrt{v \pi}} \frac{\Gamma((v+1) / 2)}{\Gamma(v / 2)}\left(1+\frac{t^{2}}{v}\right)^{-(v+1) / 2} \text { for }-\infty<t<\infty \ C D F(t)=F(t)=\frac{1}{\sqrt{v \pi}} \frac{\Gamma((v+1) / 2)}{\Gamma\left(\frac{v}{2}\right)} \int_{-\infty}^{t}\left(1+\frac{x^{2}}{v}\right)^{-(v+1) / 2} d x \end{gathered}$$

Note that $\Gamma(v / 2)$ is the gamma function. The gamma function is related to the factorial and is not the gamma probability density distribution. Like the $z$-distribution, the distribution of $t$ is bilaterally symmetric about $t=0$. The $t$-distribution is illustrated in Figure $3.11$ for two values of $v$, the degrees of freedom. The resulting bell-shaped distribution resembles that of the standard normal. However, more of the area under the $t$-distribution is in the “tails” of the distribution. In the limit of large $n$ (effectively $\nu$ greater than about 150) the $t$ – and standard normal distributions differ in the tenths of a percent.

The use of the $t$-distribution will be described in subsequent chapters in the sections discussing confidence intervals and tests of hypotheses for the mean of experimental distributions.

The cumulative $t$-distribution, $F(t)$ from Equation (3.60) can be calculated by the Excel function T.DIST $(t, v, 1)$ where $t$ is calculated from the sample data. Alternately, if you wanted to know the $t$-value that represents a probability limit then use the Excel function T.INV $(C D F, v)$ to return a $t$-value that would represent that $C D F$ value. Alternately, calculate $\alpha$, the level of significance, the extreme right-hand area, as $\alpha=1-F(t)=1-C D F$, then use the Excel function T.INV $(1-\alpha, v)$.

That represented a one-sided evaluation, which considered the area under the $t$-distribution from $-\infty$ up to a particular $t$-value. But often, we desire to know either the positive or negative extreme values for $t$, the ” $t$ ” or “- ” deviations from the central “0” value. You may want to know the range of $t$-values that includes the central $95 \%$ (or some confidence fraction C) of all expected values from sampling the population.
$$P\left(t_{\text {negative limit }} \leq T \leq t_{\text {positive limit }}\right)=C$$
Here, the level of significance is again the extreme area. If the $95 \%$ interval is desired $(C=$ $0.95$ ) then $\alpha=1-0.95=1-C$. Splitting the two tail areas equally, to define the central limits, use $\alpha / 2$ to represent both the far right and far left areas in the tails. Then we seek the $t$-value calculated with $F(t)=1-\alpha / 2$. The Excel function T.INV $(1-\alpha / 2, v)$ will return the $t$-value representing the positive extreme expected value, and $-T$.INV $(1-\alpha / 2, v)$ will return the negative extreme. This is termed a two-sided (historically a two-tailed) test, because we are seeking the limits of the central area. Alternately, $T$.INV.2T $(1-\alpha, v)$ returns the same value.

## 统计代写|工程统计作业代写Engineering Statistics代考|Chi-Squared Distribution

Let $Y_{\nu} Y_{2}, Y_{3}, \ldots, Y_{n}$ be independent random variables each distributed with mean 0 and variance 1 . The random variable chi-squared:
$$\chi^{2}=\sum_{i=1}^{n} Y_{i}^{2}$$
has the chi-squared probability density function with $v=n-1$ degrees of freedom
$$f\left(\chi^{2}\right)=\frac{1}{2^{v / 2} \Gamma(v / 2)}\left[e^{-\chi^{2} / 2}\right]\left[\chi^{2}\right]^{(v / 2)-1} \text { for } 0 \leq \chi^{2} \leq \infty$$
and cumulative distribution
$$F\left(\chi^{2}\right)=\frac{1}{2^{v / 2} \Gamma(v / 2)} \int_{0}^{\chi^{2}} e^{-Y / 2}(Y)^{(v / 2)-1} d Y$$
If $Y$ in Equation (3.62) is defined as $(X-\bar{X}) / \sigma$ then
$$\chi^{2}=\sum_{i=1}^{n} Y_{i}^{2}=\sum_{i=1}^{n} \frac{\left(X_{i}-\bar{X}\right)^{2}}{\sigma^{2}}=\frac{(n-1) s^{2}}{\sigma^{2}}$$
Figure $3.12$ illustrates the probability density and cumulative chi-squared distributions, respectively. Values of the cumulative chi-squared $\left(\chi^{2}\right)$ distribution can be obtained from the Excel function $F\left(\chi^{2}\right)=\operatorname{CHISQ.DIST}\left(\chi^{2}, v, 1\right)$, and the $p d f$ by using $f\left(\chi^{2}\right)=$ CHISQ.DIST $\left(\chi^{2}, v, 0\right) .$

The inverse of the calculation, the value of $\chi^{2}$ given $F\left(x^{2}\right)$ and $v$ can be obtained by the Excel function $\chi^{2}=\mathrm{CHISQ} \cdot \mathrm{INV}(F, v)$.

Note: Some tables or procedures use $\chi^{2} / v$. Since Equation (3.62) indicates that $\chi^{2}$ increases linearly with $n$, and since degrees of freedom is often $v=n-1$, the scaling makes

sense. Mostly, this book will not scale $\chi^{2}$ by the degrees of freedom. But be aware that the use of either $\chi^{2} / v$ or $\chi^{2}$ is common.
The mean and variance of the chi-squared distribution are $v$ and $2 v$, respectively.
$$\begin{gathered} \mu=v \ \sigma=2 v \end{gathered}$$
So, if degrees of freedom is 10 , an average-like value of the $\chi^{2}$ statistic would be about 10 . $x^{2}=1$ would be an unexpectedly low value, and $\chi^{2}=20$ would be unexpectedly high.
This distribution has several applications, one of which is in calculating and evaluating probability intervals for single variances from normally distributed populations as shown in Chapters 5 and 6. The chi-squared distribution is also used as a nonparametric method of determining whether or not, based on sample data, a population has a particular distribution, as described in Chapter 7 . The chi-squared distribution goes from 0 to infinity, or $P\left(0 \leq \chi^{2} \leq \infty\right)=1 .$
The interval
$$P\left(\chi_{v, \alpha / 2}^{2} \leq \chi^{2} \leq \chi_{v, 1-\alpha / 2}^{2}\right)=1-\alpha$$
defines the values for the $\chi^{2}$-distribution such that equal areas are in each tail. The $x^{2}$-distribution is not symmetric about the mean as are the $Z$ – and $t$-distributions.

## 统计代写|工程统计作业代写Engineering Statistics代考|F-Distribution

The F-distribution (named in honor of Sir Ronald Fisher, who developed it) is the distribution of the random variable $F$, defined as
$$F=\frac{U / v_{1}}{V / v_{2}}=\frac{\chi_{1}^{2} / v_{1}}{\chi_{2}^{2} / v_{2}}$$
Using Equation (3.65) $\chi^{2}=\frac{(n-1) s^{2}}{\sigma^{2}}=\frac{v s^{2}}{\sigma^{2}}$
$$F=\frac{s_{1}^{2} / \sigma_{1}^{2}}{s_{2}^{2} / \sigma_{2}^{2}}$$
where $U$ and $V$ are independent variables distributed following the chi-squared distribution with $v_{1}$ and $v_{2}$ degrees of freedom, respectively. The symbol $F$ in Equation (3.69) does not represent any cumulative distribution but is a statistic, specifically, the ratio of two $\chi^{2}$ statistics, each scaled by their degrees of freedom. The probability density function of $F$ is
$$f(F)=\frac{\Gamma\left(\left(v_{1}+v_{2}\right) / 2\right)}{\Gamma\left(v_{1} / 2\right) \Gamma\left(v_{2} / 2\right)}\left(\frac{v_{1}}{v_{2}}\right)^{v_{1} / 2} \frac{F^{\left(v_{1}-2\right) / 2}}{\left(1+\left(v_{1} / v_{2}\right) F\right)^{\left(v_{1}+v_{2}\right) / 2}}$$
and the cumulative distribution of $F$ is
$$C D F(F)=\int_{0}^{F} f(F) d F$$

The family of F-distributions is a two-parameter family in $v_{1}$ and $v_{2}$. The shape of the F-distribution is skewed (more of the area under the curve to the left side of the nominal value, a longer tail to the right), as illustrated in Figure 3.13. The range of all members is from 0 to $\infty$. This distribution is used to evaluate equality of variances. The $F$-distribution is termed “robust” by statisticians, meaning that the results of such statistical comparisons are likely to be valid even if the underlying populations are not normally distributed. The uses of the F-distribution are explained in Chapters 5, 6, and $12 .$

Values of the $p d f(F)$ can be returned by the Excel function $p d f(F)=F$.DIST $\left(\chi_{1}^{2} / \chi_{2}^{2}, v_{1}, v_{2}, 0\right)$, and of the cumulative $F$-distribution by $\operatorname{CDF}(F)=F$.DIST $\left(\chi_{1}^{2} / \chi_{2}^{2}, v_{1}, v_{2}, 1\right)$. The inverse of the distribution returns the chi-squared ratio for a given $C D F$ value $\frac{\chi_{1}^{2}}{\chi_{2}^{2}}=F$ INV $\left(C D F, v_{1}, v_{2}\right)$.
If the chi-squared ratio is $3.58058$ and the numerator and denominator degrees of freedom are 6 and 8 , then the CDF value is $0.95$. If, however, you choose to call #1 as #2, then the chi-squared ratio would be $0.279284$, and the degrees of freedom would be 8 then 6 . With these reversed values the $C D F$ value is $0.05$ the complement to the first.

## 统计代写|工程统计作业代写Engineering Statistics代考|“Student’s” t-Distribution

WS Gossett 以笔名“学生”出版了他的作品，开发了吨-分配。该统计数据将成为吨-test 如此广泛地用于评估工程数据。

F(吨)=1在圆周率Γ((在+1)/2)Γ(在/2)(1+吨2在)−(在+1)/2 为了 −∞<吨<∞ CDF(吨)=F(吨)=1在圆周率Γ((在+1)/2)Γ(在2)∫−∞吨(1+X2在)−(在+1)/2dX

## 统计代写|工程统计作业代写Engineering Statistics代考|Chi-Squared Distribution

χ2=∑一世=1n是一世2

F(χ2)=12在/2Γ(在/2)[和−χ2/2][χ2](在/2)−1 为了 0≤χ2≤∞

F(χ2)=12在/2Γ(在/2)∫0χ2和−是/2(是)(在/2)−1d是

χ2=∑一世=1n是一世2=∑一世=1n(X一世−X¯)2σ2=(n−1)s2σ2

μ=在 σ=2在

## 统计代写|工程统计作业代写Engineering Statistics代考|F-Distribution

F 分布（以开发它的 Ronald Fisher 爵士的名字命名）是随机变量的分布F， 定义为
F=在/在1在/在2=χ12/在1χ22/在2

F=s12/σ12s22/σ22

F(F)=Γ((在1+在2)/2)Γ(在1/2)Γ(在2/2)(在1在2)在1/2F(在1−2)/2(1+(在1/在2)F)(在1+在2)/2

CDF(F)=∫0FF(F)dF

F 分布族是一个二参数族在1和在2. 如图 3.13 所示，F 分布的形状是倾斜的（曲线下面积在标称值左侧的更多，在右侧的尾部较长）。所有成员的范围是从0到∞. 此分布用于评估方差的相等性。这F-分布被统计学家称为“稳健”，这意味着即使基础人口不是正态分布的，这种统计比较的结果也可能是有效的。第 5、6 章和第 5 章解释了 F 分布的使用。12.

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## MATLAB代写

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