统计代写|AP统计辅导AP统计答疑|Normal 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|Density Curves

• Density curves are smooth curves that can be used to describe the overall pattern of a distribution. Although density curves can come in many different shapes, they all have something in common: The area under any density curve is always equal to one. This is an extremely important concept that we will utilize in this and other chapters. It is usually easier to work with a smooth density curve than a histogram, so we sometimes overlay the density curve onto the histogram to approximate the distribution. A specific type of density curve called a normal curve will be addressed in section 3.2. This “bell-shaped” curve is especially useful in many applications of statistics as you will see later on. We describe density curves in much the same way we describe distributions when using graphs such as histograms or stemplots.
• The relationship between the mean and the median is an important concept, especially when dealing with density curves. In a symmetrical density curve, the mean and median will be equal if the distribution is perfectly symmetrical or approximately equal if the distribution is approximately symmetrical. If a distribution is skewed left, then the mean will be “pulled” in the direction of the skewness and will be less than the median. If a distribution is skewed right, the mean is again “pulled” in the direction of the skewness and will be greater than the median. Figure $3.1$ displays distributions that are skewed left, skewed right, and symmetrical. Notice how the mean is “pulled” in the direction of the skewness.

统计代写|AP统计辅导AP统计答疑|Normal Distributions

One particular type of density curve that is especially useful in statistics is the normal curve, or normal distribution. Although all normal distributions have the same overall shape, they do differ somewhat depending on the mean and standard deviation of the distribution (Figure 3.3). If we increase or decrease the mean while keeping the standard deviation the same, we will simply shift the distribution to the right or to the left. The more we increase the standard deviation, the “wider” and “shorter” the density curve will be. If we decrease the standard deviation, the density curve will be “narrower” and “taller.” Remember that all density curves, including normal curves, have an area under the curve equal to one. So, no matter what value the mean and standard deviation take, the area under the normal curve is equal to one. This is very important, as you’ll soon see.

Example 1: Let’s assume that the number of miles that a particular tire will last roughly follows a normal distribution with $\mu=40,000$ miles and $\sigma=5000$ miles. Note that we can use shorthand notation $N(40,000$, 5000 ) to denote a normal distribution with mean equal to 40,000 and standard deviation equal to 5,000 . Since the distribution is not exactly normal but approximately normal, we can assume the distribution will

roughly follow the $68,95,99.7$ Rule. Using the $68,95,99.7$ Rule we can conclude the following (see Figure $3.5$ ):

About $68 \%$ of all tires should last between 35,000 and 45,000 miles $(\mu \pm \sigma)$

About $95 \%$ of all tires should last between 30,000 and 50,000 miles $(\mu \pm 2 \sigma)$

About $99.7 \%$ of all tires should last between 25,000 and 55,000 miles $(\mu \pm 3 \sigma)$

Using the $68,95,99.7$ Rule a little more creatively, we can also conclude:
About $34 \%$ of all tires should last between 40,000 and 45,000 miles.
About $34 \%$ of all tires should last between 35,000 and 40,000 miles.
About $21 / 2 \%$ of all tires should last more than 50,000 miles.
About $84 \%$ of all tires should last less than 45,000 miles.

统计代写|AP统计辅导AP统计答疑|Normal Calculations

• Example 2: Referring back to Example 1, let’s suppose that we want to determine the percentage of tires that will last more than 53,400 miles. Recall that we were given $N(40,000,5000)$. To get a more exact answer than we could obtain using the Empirical Rule, we can do the following:
Solution: Always make a sketch! (See Figure 3.6.)

Using substitution, we obtain $z=\frac{53,400-40,000}{5000}=2.68$
Notice that the formula for $z$ takes the difference of $x$ and $\mu$ and divides it by $\sigma$. Thus, a z-score is the number of standard deviations that $x$ lies above or below the mean. So, 53,400 is $2.68$ standard deviations above the mean. You should always get a positive value for $z$ if the value of $x$ is above the mean, and a negative value for $z$ if the value of $x$ is below the mean.
When we find the z-score, we are standardizing the values of the distribution. Since these values are values of a normal distribution, the distribution we obtain is called the standard normal distribution. This new distribution, the standard normal distribution, has a mean of zero and a standard deviation of one. We can then write $N(0,1)$
The advantage of standardizing any given normal distribution to the standard normal distribution is that we can now find the area under the curve for any given value of $x$ that is needed.

We can now use the z-score of $2.68$ that we obtained earlier. Using Table A, we can look up the area to the left of $z=2.68$. Notice that Table A has two sides – one for positive values for $z$ and the other for negative values for $z$. Using the side of the table with the positive values for $z$, follow the left-hand column down until you reach $2.6$. Then go across the top of the table until you reach $.08$. By crossreferencing $2.6$ and $.08$, we can obtain the area to the left of $z=2.68$, which is $0.9963$.

In other words, $99.63 \%$ of tires will last less than 53,400 miles. We want to know what percent of tires will last more than 53,400 miles, so we subtract $0.9963$ from 1. Remember that the total area under any density curve is equal to one.
We obtain $1-0.9963=0.0037$.

统计代写|AP统计辅导AP统计答疑|Density Curves

• 密度曲线是可用于描述分布的整体模式的平滑曲线。尽管密度曲线可以有许多不同的形状，但它们都有一个共同点：任何密度曲线下的面积总是等于一。这是一个非常重要的概念，我们将在本章和其他章节中使用。使用平滑的密度曲线通常比使用直方图更容易，因此我们有时将密度曲线叠加到直方图上以近似分布。3.2 节将介绍一种称为正态曲线的特定类型的密度曲线。这种“钟形”曲线在许多统计应用中特别有用，您将在后面看到。我们描述密度曲线的方式与我们在使用直方图或茎图等图形时描述分布的方式非常相似。
• 平均值和中位数之间的关系是一个重要的概念，尤其是在处理密度曲线时。在对称密度曲线中，如果分布完全对称，则平均值和中位数将相等，如果分布近似对称，则平均值和中位数将近似相等。如果分布向左偏，则均值将被“拉”向偏斜方向，并且将小于中位数。如果分布向右偏斜，则均值再次向偏斜方向“拉动”，并将大于中位数。数字3.1显示左偏、右偏和对称的分布。请注意均值是如何向偏度方向“拉动”的。

统计代写|AP统计辅导AP统计答疑|Normal Calculations

• 示例 2：回到示例 1，假设我们要确定能够持续超过 53,400 英里的轮胎的百分比。回想一下，我们得到了ñ(40,000,5000). 为了得到比我们使用经验法则获得的更准确的答案，我们可以执行以下操作：
解决方案：总是画草图！（见图 3.6。）

有限元方法代写

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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。