Course: 14.14 摘要:概率和统计

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摘要:概率和统计
This chapter addressed concepts in probability as well as statistics.
::本章讨论概率概念和统计概念。

The probability lessons addressed counting permutations and combinations, as well as the Fundamental Counting Principle. Then the chapter moved on to the Binomial Theorem. Finally, you learned how to calculate the predicted cost of events using the expected value formula.
::概率教训涉及计算差数和组合,以及基本计数原则。然后该章转到Binomial定理。最后,你学会了如何使用预期值公式计算事件的预测成本。

You worked with univariate data and learned how to display it graphically and summarize it numerically. Additionally, you learned how to calculate mean, median, mode, and variance—and also learned when to use each. Moreover, you explored bivariate data and used the regression capabilities of your calculator to create mathematical models for real-world phenomena, while also exploring the difference between correlation and causation.
::此外,你还学会了如何计算平均值、中位数、模式和差异 — — 并学习了使用每种数据的时间。此外,你还探索了双轨数据,并利用计算器的回归能力为现实世界现象创建数学模型,同时也探索了相关性和因果关系之间的差别。

Chapter Summary
::章次摘要

A combination is the number of ways of choosing $\begin{aligned} k \end{aligned}$ objects from a total of $\begin{aligned} n \end{aligned}$ objects. (Order does not matter.)
::组合是指从总计 n 对象中选择 k 对象的方法数。 (命令无关紧要。)

The number of ways to find the combination or choose $\begin{aligned} k \end{aligned}$ objects from a group of $\begin{aligned} n \end{aligned}$ objects is:
$\begin{aligned} _{n} C_{k} = (\binom{n}{k}) = \frac{n!}{k! (n - k)!} \end{aligned}$

::从一组 n 对象中查找组合或选择 k 对象的方法数是:nCk=(nk)=n!k! (n-k)!

A permutation is the number of ways of choosing and arranging $\begin{aligned} k \end{aligned}$ objects from a total of $\begin{aligned} n \end{aligned}$ objects. (Order does matter.)
::变换是指从总计 n 对象中选择和排列 k 对象的方法数量。 (命令确实重要。)

The formula to calculate how to choose and arrange $\begin{aligned} k \end{aligned}$ objects from a group of $\begin{aligned} n \end{aligned}$ objects (or calculate a permutation):
$\begin{aligned} _{n} P_{k} = k! (\binom{n}{k}) = k! \cdot \frac{n!}{k! (n - k)!} = \frac{n!}{(n - k)!} \end{aligned}$
.
::计算如何从一组 n 对象中选择和排列 k 对象的公式( 或计算一个变异性): nPk=k! (nk)=k! k! k! (n- k)! (n- k)!=n! (n- k)!

The Fundamental Counting Principle states that if one event has m possible outcomes and a 2nd independent event has n possible outcomes, then there are $\begin{aligned} m \times n \end{aligned}$ total possible outcomes for the two events together.

::基本计数原则指出,如果一项活动可能取得结果,而第2项独立活动可能取得结果,那么这两项活动可能取得的总结果是 mxn 。

The probability of an event is the number of outcomes you are looking for (called successes) divided by the total number of outcomes.
::事件的概率是您正在寻找的结果数量(所谓的成功)除以结果的总数。

The notation $\begin{aligned} P (E) \end{aligned}$ is read "the probability of event $\begin{aligned} E \end{aligned}$ ."
$\begin{aligned} P (E) = \frac{# s u c c e s s e s}{# p o s s i b l e o u t c o m e s} \end{aligned}$

::标注P(E)为“事件E的概率”。 P(E) 成功# 可能的结果

The complement of an event is the event not happening.
::事件的补充是事件没有发生。

Independent events are events where the occurrence of the 1st event does not impact the probability of the 2nd event.
::独立事件是指第一次事件的发生不影响第二次事件的概率的事件。

The Binomial Theorem (or binomial expansion ) describes the algebraic expansion of powers of a binomial.
::二元论(或二元论扩展)描述了二元论力量的代数扩张。

The coefficients in the binomial expansion appear as the entries of Pascal's Triangle. In Pascal's Triangle, each entry is the sum of the two above it.
::二进制扩展中的系数显示为帕斯卡尔三角的条目。在帕斯卡尔三角中,每个条目是上面两个条目的总和。

The coefficients can also be generated using combinations. The numbers in the combination are associated with a particular row and column in Pascal's Triangle. For example, 5 combinations of 3 would be associated with the 5th row and 3rd column of the triangle.
::也可以使用组合生成系数。组合中的数值与Pascal三角形中特定的行和列相关联。例如, 5 组合3 与三角形第五行和第三列相关联。

The Binomial Theorem is:

$\begin{aligned} (a + b)^{n} = \sum_{i = 0}^{n} (\binom{n}{i}) a^{i} b^{n - i} \end{aligned}$

:a+b)ni=0n(ni)aibn-i

::Binomial定理是a+b)ni=0n(ni)aibn-i

A weighted average is an average that multiplies each component by a factor representing its frequency or probability.
::加权平均数是每个组成部分乘以代表其频率或概率的一个系数的平均数。

The expected value is the return or cost you can expect on average, given many trials.
::预计价值是,鉴于许多审判,平均回报率或预期成本。

The payoff of a game is the expected value of the game minus the cost.
::游戏的回报是游戏的预期值减去成本。

The mean is the arithmetic average of the data.
::平均值是数据的算术平均数。

The median is the number in the middle of a dataset. When the data has an odd number of counts, the median is the middle number after the data have been ordered. When the data has an even number of counts, the median is the average of the two most central numbers.
::中位数是数据集中间的数字。当数据有奇数计数时,中位数是数据排序后的数字中位数。当数据有偶数计数时,中位数是两个中位数的平均值。

The mode is the most often occurring number in the data. If two or more numbers occur equally frequently, then the data is said to be bimodal or multimodal .
::模式是数据中最经常出现的数字,如果两个或两个以上的数字同样频繁出现,则数据据说是双式或多式数据。

With descriptive statistics , your goal is to describe the data that you find in a sample or is given in a problem.
::在描述性统计数据时,您的目标是描述您在样本中发现或遇到问题时提供的数据。

With inference statistics , your goal is use the data in a sample to draw conclusions about a larger population.
::通过推断统计,你的目标是使用抽样中的数据来得出关于人口增加的结论。

The rank of an observation is the number of observations that are less than or equal to the value of that observation.
::观察的等级是少于或等于观察价值的观察次数。

Data are divided into four parts by the 1 st quartile $\begin{aligned} (Q 1) \end{aligned}$ , 2nd quartile $\begin{aligned} (Q 2) \end{aligned}$ , and 3 rd quartile $\begin{aligned} (Q 3) \end{aligned}$ . The 2 nd quartile is also known as the median.
::数据分为四个部分,即第1四分位数(Q1)、第2四位数(Q2)和第3四分位数(Q3),第二个四分位数也称为中位数。

Variance is a measure of how spread out the data are.
::差异是衡量数据分布的尺度。

The square root of the variance is the standard deviation .
::差异的平方根是标准偏差。

Both the variance and the standard deviation can be calculated from a sample or from the whole population . The formulas are slightly different in each case, so it is important to know whether your data is just a sample or is from the whole population.
::差异和标准偏差都可以从抽样中或从全部人群中计算出来。公式在每种情况下略有不同, 所以重要的是要知道您的数据是样本还是来自全部人群。

The absolute deviation is the sum total of how different each number is from the mean.
::绝对偏差是每个数字与平均值之差的总和。

The mean absolute deviation is an alternate measure of how spread out the data are. While this method might seem more intuitive, in statistics it has been found to be too limited and is not commonly used.
::绝对偏差是衡量数据分散程度的另一种方法。虽然这种方法看起来更直观,但在统计中发现它太有限,不常用。

Mean and variance for the population: $\begin{aligned} x_{1}, x_{2}, x_{3}, \dots, x_{n} \end{aligned}$

$\begin{aligned} μ & = \frac{1}{n} \cdot \sum_{i = 1}^{n} x_{i} \\ σ^{2} & = \frac{1}{n} \cdot \sum_{i = 1}^{n} (μ - x_{i})^{2} \end{aligned}$

::平均和人口差异: x1,x2,x3,...,xn 1ni=1nxx12=1n_xx12=1n_i=1n(xxi)2

Mean and variance for a sample from a population: $\begin{aligned} x_{1}, x_{2}, x_{3}, \dots, x_{m} \end{aligned}$

$\begin{aligned} \bar{x} & = \frac{1}{m} \cdot \sum_{i = 1}^{m} x_{i} \\ s^{2} & = \frac{1}{m - 1} \cdot \sum_{i = 1}^{m} (\bar{x} - x_{i})^{2} \end{aligned}$

::x 1mi=1mxis2=1m-1i=1m(x x)2

::从组群中取样的平均值和差异: x1, x2, x3,..., xm x {%1mi=1mxis2=1m-1_1i=1m(x *xxxxxxxxxxx)2

A standard normal distribution is a normal distribution with mean of 0 and a standard deviation of 1.
::标准正常分配是指平均值为0和标准差为1的正常分配。

The empirical rule states that for data that are normally distributed, approximately 68% of the data will fall within 1 standard deviation of the mean, approximately 95% of the data will fall within 2 standard deviations of the mean, and approximately 99.7% of the data will fall within 3 standard deviations of the mean. It is a good way to quickly approximate probabilities.
::经验规则规定,对于通常分发的数据,大约68%的数据将低于平均值的1个标准差,大约95%的数据将低于平均值的2个标准差,大约99.7%的数据将低于平均值的3个标准差。这是快速接近概率的好方法。

Normalcdf is the normal cumulative distribution function and calculates the area between any two values for data that are normally distributed, as long as you know the mean and standard deviation for the data. Your calculator has this function built in, and it produces an exact answer as opposed to the empirical rule.
::usrcdf 是正常的累积分布函数, 计算正常分布数据的两个值之间的区域, 只要您知道数据的平均偏差和标准偏差。您的计算器将此函数嵌入其中, 它产生的准确答案不同于经验规则。

A scatterplot creates an $\begin{aligned} (x, y) \end{aligned}$ point from each data pair.
::散射图为每对数据创建一个(x,y)点。

Bivariate data are two sets of data that are paired.
::双轨数据是两组对齐数据。

The correlation coefficient , $\begin{aligned} r \end{aligned}$ , is a number in the interval [-1, 1]. It indicates the strength of the correlation between two variables.
::相关系数(r)是间隔[-1,1]中的一个数字,表示两个变量之间相互关系的强度。

Regression is a method of attempting to fit a model to observed data in order to predict new values.
::倒退是试图将模型与观察到的数据相匹配以预测新值的一种方法。

The best fit for a scatter plot may be a linear, quadratic, or exponential model.
::最适合散射图的可能是线性模型、二次模型或指数模型。

Review
::回顾

Try the following cumulative review problems to practice the concepts studied in this chapter:
::尝试下列累积审查问题,以实践本章所研究的概念:

Section outline

General

摘要:概率和统计

Chapter Summary ::章次摘要

Review ::回顾

Chapter Summary
::章次摘要

Review
::回顾