### 统计代写|AP统计辅导AP统计答疑|Sampling Distributions

AP统计学与大学的统计学课程在核心内容上是一致的，只是涉及的深度稍浅，AP统计学主要包含以下四部分内容。 第一部分 如何获取数据，获取数据的方式有哪些呢？ 获取数据的方式主要包括普查、抽样调查、观测研究和实验设计等。

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|AP统计辅导AP统计答疑|Sampling Distributions

• Understanding sampling distributions is an integral part of inferential statistics. Recall that in inferential statistics you are making conclusions or assumptions about an entire population based on sample data. In this chapter, we will explore sampling distributions for means and proportions. In the remaining chapters, we will call upon the topics of this and previous chapters in order to study inferential statistics.
• From this point on, it’s important that we understand the difference between a parameter and a statistic. A parameter is a number that describes some attribute of a population. For example, we might be interested in the mean, $\mu$, and standard deviation, $\sigma$, of a population. There are many situations for which the mean and standard deviation of a population are unknown. In some cases, it is the population proportion that is not known. That is where inferential statistics comes in. We can use a statistic to estimate the parameter. A statistic is a number that describes an attribute of a sample. So, for the unknown $\mu$ we can use the sample mean, $\bar{x}$, as an estimate of $\mu$. It’s important to note that if we were to take another sample, we would probably get a different value for $\bar{x}$. In other words, if we keep sampling, we will probably keep getting different values for $\bar{x}$ (although some may be the same). Although $\mu$ may be unknown, it is a fixed number, as a population can have only one mean. The notation for the standard deviation of a sample is $s$. (Just remember that $s$ is for “sample.”) We sometimes use $s$ to estimate $\sigma$, as we will see in later chapters.To summarize the notation, remember that the symbols $\mu$ and $\sigma$ (parameters) are used to denote the mean and standard deviation of a population, and $\bar{x}$ and $s$ (statistics) are used to denote the mean and standard deviation of a sample. You might find it helpful to remember that $s$ stands for “statistic” and “sample” while $p$ stands for “parameter” and “population.” You should also remember that Greek letters are typically used for population parameters. Be sure to use the correct notation! It can help convince the reader (grader) of your AP* Exam that you understand the difference between a sample and a population.
• Consider again a population with an unknown mean, $\mu$. Sometimes it is simply too difficult or costly to determine the true mean, $\mu$. When this is the case, we then take a random sample from the population and find the mean of the sample, $\bar{x}$. As mentioned earlier, we could repeat the sampling process many, many times. Each time we would recalculate the mean, and each time we might get a different value. This is called sampling variability. Remember, $\mu$ does not change. The population mean for a given population is a fixed value. The sample mean, $\bar{x}$, on the other hand, changes depending on which individuals from the population are chosen. Sometimes the value of $\bar{x}$ will be greater than the true population mean, $\mu$, and other times $\bar{x}$ will be smaller than $\mu$. This means that $\bar{x}$ is an unbiased estimator of $\mu$.
• The sampling distribution is the distribution of the values of the statistic if all possible samples of a given size are taken from the population. Don’t confuse samples with sampling distributions. When we talk about sampling distributions, we are not talking about one sample; we are talking about all possible samples of a particular size that we could obtain from a given population.

## 统计代写|AP统计辅导AP统计答疑|Sample Means and the Central Limit Theorem

The following activity will help you understand the difference between a population and a sample, sampling distributions, sampling variability, and the Central Limit Theorem. I learned of this activity a few years ago from AP Statistics consultant and teacher Chris True. I am not sure where this activity originated, but it will help you understand the concepts presented in this chapter. If you’ve done this activity in class, that’s great! Read through the next few pages anyway, as it will provide you with a good review of sampling and the Central Limit Theorem.

The activity begins with students collecting pennies that are currently in circulation. Students bring in enough pennies over the period of a few days such that I get a total of about 600 to 700 pennies between all of my AP Statistics classes. Students enter the dates of the pennies into the graphing calculator (and Fathom) as they place the pennies into a container. These 600 to 700 pennies become our population of pennies. Then students make a guess as to what they think the distribution of our population of pennies will look like. Many are quick to think that the distribution of the population of pennies is approximately normal. After some thought and discussion about the dates of the pennies in the population, students begin to understand that the population distribution is not approximately normal but skewed to the left. Once we have discussed what we think the population distribution should look like, we examine a histogram or dotplot of the population of penny dates. As you can see in Figure 6.2, the distribution is indeed skewed to the left.

## 统计代写|AP统计辅导AP统计答疑|Sample Proportions and the Central Limit Theorem

• Now that we’ve discussed sampling distributions, sample means, and the Central Limit Theorem, it’s time to turn our attention to sample proportions. Before we begin our discussion, it’s important to note that when referring to a sample proportion, we always use $\hat{p}$. When referring to a population proportion, we always use $p$. Note that some texts use $\pi$ instead of $p$. In this case, $\pi$ is just a Greek letter being used to denote the population proportion, not $3.1415 \ldots$
• The Central Limit Theorem also applies to proportions as long as the following conditions apply:
1. The sampled values must be independent of one another. Sometimes this is referred to as the $10 \%$ condition. That is, the sample size must be only $10 \%$ of the population size or less. If the sample size is larger than $10 \%$ of the population, it is unlikely that the individuals in the sample would be independent.
2. The sample must be large enough. A general rule of thumb is that $n p \geq 10$ and $n(1-p) \geq 10$. As always, the sample must be random.
• If these two conditions are met, the sampling distribution of $\hat{p}$ should be approximately normal. The mean of the sampling distribution of $\hat{p}$ is exactly equal to $p$. The standard deviation of the sampling distribution is equal to:
$$\sqrt{\frac{p(1-p)}{n}}$$
• Note that because the average of all possible $\hat{p}$ values is equal to $p$, the sample proportion, $\hat{p}$, is an unbiased estimator of the population proportion, $p$.

## 统计代写|AP统计辅导AP统计答疑|Sampling Distributions

• 了解抽样分布是推理统计的一个组成部分。回想一下，在推论统计中，您是根据样本数据对整个人口做出结论或假设。在本章中，我们将探讨均值和比例的抽样分布。在剩下的章节中，我们将调用本章和前几章的主题来研究推论统计。
• 从这一点开始，重要的是我们要了解参数和统计数据之间的区别。参数是描述总体某些属性的数字。例如，我们可能对均值感兴趣，μ, 和标准差,σ, 一个人口。在许多情况下，总体的均值和标准差是未知的。在某些情况下，人口比例是未知的。这就是推理统计的用武之地。我们可以使用统计来估计参数。统计量是描述样本属性的数字。所以，对于未知μ我们可以使用样本均值，X¯，作为估计μ. 重要的是要注意，如果我们要取另一个样本，我们可能会得到不同的值X¯. 换句话说，如果我们继续采样，我们可能会不断得到不同的值X¯（虽然有些可能是一样的）。虽然μ可能未知，它是一个固定数字，因为总体只能有一个均值。样本标准差的符号是s. （请记住s代表“样本”。）我们有时使用s估计σ，我们将在后面的章节中看到。总结符号，记住符号μ和σ（参数）用于表示总体的平均值和标准差，以及X¯和s（统计）用于表示样本的均值和标准差。您可能会发现记住这一点很有帮助s代表“统计”和“样本”，而p代表“参数”和“人口”。您还应该记住，希腊字母通常用于人口参数。一定要使用正确的符号！它可以帮助说服 AP* 考试的读者（评分者）您了解样本和总体之间的区别。
• 再次考虑一个均值未知的总体，μ. 有时确定真正的平均值太困难或太昂贵，μ. 在这种情况下，我们从总体中随机抽取样本并找到样本的均值，X¯. 如前所述，我们可以多次重复采样过程。每次我们都会重新计算平均值，每次我们可能会得到不同的值。这称为抽样变异性。记住，μ不改变。给定总体的总体均值是一个固定值。样本均值，X¯另一方面，变化取决于从人口中选择了哪些个体。有时价值X¯将大于真实的总体均值，μ, 其他时候X¯会小于μ. 这意味着X¯是一个无偏估计量μ.
• 如果给定大小的所有可能样本均取自总体，则抽样分布是统计值的分布。不要将样本与抽样分布混淆。当我们谈论抽样分布时，我们不是在谈论一个样本。我们谈论的是我们可以从给定人群中获得的所有可能的特定大小的样本。

## 统计代写|AP统计辅导AP统计答疑|Sample Proportions and the Central Limit Theorem

• 现在我们已经讨论了抽样分布、样本均值和中心极限定理，是时候将注意力转向样本比例了。在我们开始讨论之前，重要的是要注意，在提到样本比例时，我们总是使用p^. 当提到人口比例时，我们总是使用p. 请注意，有些文本使用圆周率代替p. 在这种情况下，圆周率只是一个用来表示人口比例的希腊字母，而不是3.1415…
• 只要满足以下条件，中心极限定理也适用于比例：
1. 采样值必须相互独立。有时这被称为10%健康）状况。也就是说，样本量必须只有10%人口规模或更少。如果样本量大于10%在总体中，样本中的个体不太可能是独立的。
2. 样本必须足够大。一般的经验法则是np≥10和n(1−p)≥10. 与往常一样，样本必须是随机的。
• 如果满足这两个条件，则抽样分布p^应该是大致正常的。抽样分布的均值p^正好等于p. 抽样分布的标准差等于：
p(1−p)n
• 请注意，因为所有可能的平均值p^值等于p, 样本比例,p^, 是人口比例的无偏估计量，p.

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## MATLAB代写

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