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# 统计代写| 假设检验作业代写Hypothesis testing代考|Power and Sample Size Analysis

##### 空白假设的早期选择

Paul Meehl认为，无效假设的选择在认识论上的重要性基本上没有得到承认。当无效假设是由理论预测的，一个更精确的实验将是对基础理论的更严格的检验。当无效假设默认为 “无差异 “或 “无影响 “时，一个更精确的实验是对促使进行实验的理论的一个较不严厉的检验。

1778年：皮埃尔-拉普拉斯比较了欧洲多个城市的男孩和女孩的出生率。他说 “很自然地得出结论，这些可能性几乎处于相同的比例”。因此，拉普拉斯的无效假设是，鉴于 “传统智慧”，男孩和女孩的出生率应该是相等的 。

1900: 卡尔-皮尔逊开发了卡方检验，以确定 “给定形式的频率曲线是否能有效地描述从特定人群中抽取的样本”。因此，无效假设是，一个群体是由理论预测的某种分布来描述的。他以韦尔登掷骰子数据中5和6的数量为例 。

1904: 卡尔-皮尔逊提出了 “或然性 “的概念，以确定结果是否独立于某个特定的分类因素。这里的无效假设是默认两件事情是不相关的（例如，疤痕的形成和天花的死亡率）。[16] 这种情况下的无效假设不再是理论或传统智慧的预测，而是导致费雪和其他人否定使用 “反概率 “的冷漠原则。

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• 马尔科夫过程 Markov process
• 随机最优控制stochastic optimal control
• 粒子滤波 Particle Filter
• 采样理论 sampling theory

## 统计代写| 假设检验作业代写Hypothesis testing代考|Power and Sample Size Analysis

Statistical power is the opposite of Type II errors, both mathematically $(1-\beta)$ and conceptually. Power is the ability of the test to detect an effect that exists in the population. In other words, the test correctly rejects a false null hypothesis.

For example, if your study has $80 \%$ power, it has an $80 \%$ chance of detecting an effect that exists. Let this point be a reminder that when you work with samples, nothing is guaranteed! When an effect exists in the population, your study might not detect it because you are working with a sample. Samples contain sample error, which can occasionally cause a random sample to misrepresent the population.
$80 \%$ power is a standard benchmark for studies. However, you’ll need to consider standards for your field or industry.

As you learned in the previous sections, while various factors affect power, researchers have the greatest control over sample size.

Determining a good sample size for a study is always an important issue. After all, using the wrong sample size can doom your study from the start. Fortunately, power analysis can find the answer for you. Power analysis combines statistical analysis, subject-area knowledge, and your requirements to help you derive the optimal sample size.

As you’ll see in this section, both under-powered and over-powered studies are problematic. Let’s learn how to find the right sample size for your study!

Before data collection and hypothesis testing begin, you must do a lot of preplanning. This planning includes identifying the data you will gather, how you will collect it, and how you will measure it, among many other details. A crucial part of the planning is determining how much data you need to collect. I’ll show you how to estimate the sample size for your study.

Before we get to estimating sample size requirements, let’s review the factors that influence statistical significance. This process will help you see the value of formally going through a power and sample size analysis rather than guessing.

## 统计代写|假设检验作业代写HYPOTHESIS TESTING代考|Low Power Tests Exaggerate Effect Sizes

I’m going to end this chapter with an advanced topic about statistical power. The previous sections of this chapter have shown you that a study with low power is unlikely to detect an effect when it exists. However, there is an additional danger to consider with low powered studies.

Clearly, a high-powered study is a good thing just for being able to identify these effects. Low power reduces your chances of discovering real findings. However, many analysts don’t realize that low power also tends to exaggerate the effect size when they detect effects.
In this section, I show how this unexpected relationship between power and exaggerated effect sizes exists. I’ll also tie it to other issues, such as the bias of effects published in journals and other matters about statistical power. I think this topic will be eye-opening and thought provoking! As always, I’ll use many graphs rather than equations.

Hypothetical Study Scenario
To illustrate how this effect size inflation works, I’ll simulate a study and conduct it many times at three power levels.

Imagine that we’re studying a fictitious medication that promises to increase your intelligence (IQ). Our experiment has two groups-a control group that doesn’t take the pill and the treatment group that does. Then, each group takes the same IQ test and we compare the results. The effect size is the difference between group means.

Because we’re simulating these studies, we can control the effect size and other properties of the population. I’ll set the effect size at 10 IQ points and define the two populations as follows:

• Control group: Normal distribution with a mean of 100 and a standard deviation of 15 .
• Treatment group: Normal distribution with a mean of 110 and a standard deviation of 15 .

I calculated the sample sizes I need to produce statistical power of $0.3$, $0.55$, and 0.8. The first two values represent low power studies, while the third value is a standard target value. The output below shows the power analysis results.

## 统计代写| 假设检验作业代写HYPOTHESIS TESTING代考|POWER AND SAMPLE SIZE ANALYSIS

80%功率是研究的标准基准。但是，您需要考虑您所在领域或行业的标准。

## 统计代写|假设检验作业代写HYPOTHESIS TESTING代考|LOW POWER TESTS EXAGGERATE EFFECT SIZES

• 对照组：正态分布，平均值为 100，标准差为 15。
• 治疗组：正态分布，平均值为 110，标准差为 15。

## 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。