# 数学代写|数值分析代写numerical analysis代考|Hypothesis Testing Using p-values

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## 数学代写|数值分析代写numerical analysis代考|Hypothesis Testing Using p-values

The idea here is to compute a statistic $S$ from the data set $D$, and see how consistent the value of this statistic is with a hypothesis to be tested called the null hypothesis $H_0$. If the computed value of the statistic $S$ for data set $D$ is $s$, then we test if (7.3.7) $\quad \operatorname{Pr}\left[S \geq s \mid H_0\right]<p$
where $0<p<1$ is a pre-specified $p$-value. Typically, $p$ is taken to be 0.05 or 0.01 , representing testing at the $5 \%$ and $1 \%$ levels, respectively. If $\operatorname{Pr}\left[S \geq s \mid H_0\right]<p$ then we reject $H_0$, the null hypothesis. The idea is that under the null hypothesis, the chance of seeing the value of the statistic $S$ as extreme as the observed value $s$ is less than $p$.

For example, suppose a number of values $x_1, x_2, \ldots, x_N$ are measured. Assume that the null hypothesis $H_0$ is that each $x_i \sim \operatorname{Normal}\left(0, \sigma^2\right)$ independently with $\sigma^2$ given. If the statistic $S$ is the mean of the values, $S=(1 / N) \sum_{i=1}^N x_i$, under the null hypothesis, $S \sim \operatorname{Normal}\left(0, \sigma^2 / N\right)$. If $\operatorname{Pr}\left[S \geq s \mid H_0\right]=\int_s^{\infty}\left(2 \pi \sigma^2 / N\right)^{-1 / 2}$ $\exp \left(-N z^2 /\left(2 \sigma^2\right)\right) d z<p$ then we reject the null hypothesis.

While this is a standard method for identifying when “something interesting” is happening, there are a number of ways in which this approach can fail. This is particularly true where there are multiple tests or tests of multiple hypotheses. Situations like this often arise with, for example, high-throughput testing. For example, genetic markers can be tested for connection with, say, cancer likelihood. The null hypothesis for a given genetic marker would be that the genetic marker has no effect on the cancer likelihood. If we have an estimate $p_0$ of the “background” probability of a certain cancer occurring, the null hypothesis would be that a person with the specified genetic marker has probability $p_0$ of having cancer. Under the null hypothesis, the number of people in a random sample of $N$ people with the genetic marker that also have cancer is a random variable with the $\operatorname{Binomial}\left(N, p_0\right)$ distribution. The statistic $S$ used would be the number of people in the sample with cancer. Then
$$\operatorname{Pr}[S \geq k]=\sum_{j=k}^N\left(\begin{array}{c} N \ j \end{array}\right) p_0^j\left(1-p_0\right)^{N-j} .$$
If $\operatorname{Pr}[S \geq k]<p$ for a threshold $p$-value, then the null hypothesis would be rejected: the genetic marker seems to be positively correlated with cancer.

## 数学代写|数值分析代写numerical analysis代考|Random Algorithms

Many practical algorithms use random choices. Here we will look at three algorithms that use random choices: quicksort, primality testing, and estimating $\pi$. The reasons for the random choices differ according to the algorithm. In quicksort, the random choices means that the average case performance occurs with high probability. In primality testing, a test is applied to the random choices made. If the test for any specific choice is negative, then there is no need for any other test. For estimating $\pi$, many random choices are made, and averaging is used to obtain a more accurate estimate.

The $\pi$ estimation algorithm is called a Monte Carlo algorithm. Monte Carlo algorithms are inherently random; at no point does a Monte Carlo algorithm have a definite answer. For estimating $\pi$, we have only approximate value of $\pi$ at any stage of the algorithm. On the other hand, the primality testing algorithm can stop with a definite result once a single test is negative. This kind of algorithm is called a Las Vegas algorithm. The results of the quicksort algorithm, however, do not depend on the random choices: the result is the input list, but sorted. Random choices do not affect the final result, just the speed of obtaining it; it is neither a Monte Carlo nor a Las Vegas algorithm.

One example is the quicksort algorithm $[59$, Ch. 8] that first selects an element (called the pivot) of a list, and then splits the rest of the list into two sublists consisting of elements less than the pivot, and elements above the pivot. Each sublist is then recursively sorted. If we choose a fixed element as the pivot element, then for some input lists of length $n$ the method takes $\mathcal{O}\left(n^2\right)$ comparisons to sort the list, while for most input lists the method takes $\mathcal{O}(n \log n)$ comparisons. If instead, we choose the pivot entry randomly and uniformly from the list to sort, the method takes $\mathcal{O}(n \log n)$ comparisons with a very high probability for large $n$.

# 数值分析代考

## 数学代写|数值分析代写numerical analysis代考|Hypothesis Testing Using p-values

$\operatorname{Pr}\left[S \geq s \mid H_0\right]<p$ 然后我们拒绝 $H_0$ ，原假设。这个 想法是，在原假设下，看到统计值的机会 $S$ 与观察值一 样极端 $s$ 小于 $p$.

$\operatorname{Pr}\left[S \geq s \mid H_0\right]=\int_s^{\infty}\left(2 \pi \sigma^2 / N\right)^{-1 / 2}$ $\exp \left(-N z^2 /\left(2 \sigma^2\right)\right) d z<p$ 那么我们拒绝原假设。

$\operatorname{Binomial}\left(N, p_0\right)$ 分配。统计数据 $S$ 使用的是样本中患 有癌症的人数。然后
$$\operatorname{Pr}[S \geq k]=\sum_{j=k}^N(N j) p_0^j\left(1-p_0\right)^{N-j}$$

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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。

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