14.10 正常曲线

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  普通曲线

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  Introduction
  ::导言

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  When students ask their teachers to curve exams, what they often mean is they want everyone to simply get a higher grade. Curving a grade can also mean fitting to a bell curve, where lots of people get Cs, some people get Ds and Bs, and very few people get As and Fs. Even though this 2nd interpretation is not what most students mean, the normal curve is one of the most widely used and applied probability distributions. What other examples follow a normal distribution ?
  ::当学生要求教师参加曲线考试时,他们通常的意思是,他们希望每个人都获得更高的分数。 缩短一个分数还意味着适合钟曲线,因为许多人获得C,一些人获得D和B,而很少有人获得A和F。 尽管这种第二解释不是大多数学生所想的,但正常曲线是最广泛使用和应用的概率分布之一。还有什么其他例子可以跟随正常分布呢?

  

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  Standard Normal Distribution
  ::标准正常分配标准

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  The Standard Normal Distribution is graphed from the following function and is represented by the Greek letter phi, $\begin{array}{r}\varphi \end{array}$ .
  ::标准正常分配表按以下函数编排,由希腊字母“”表示。

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  $\begin{array}{r}\varphi (x)=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{1}{2}{x}^2}\end{array}$
  :x) = 12e - 12x2

  

  
  

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  This distribution represents a population with a mean of 0 and a standard deviation of 1. The numbers along the $\begin{array}{r}x\end{array}$ -axis represent standard deviations. For data that are normally distributed, the states that:
  ::此分布表示平均值为0,标准偏差为1。x轴上的数字代表标准偏差。对于通常分布的数据,规定:

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  • Approximately 68% of the data will be within 1 standard deviation of the mean.
    ::大约68%的数据将低于平均值的1个标准差。
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  • Approximately 95% of the data will be within 2 standard deviations of the mean.
    ::大约95%的数据将低于平均值的2个标准差。
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  • Approximately 99.7% of the data will be within 3 standard deviations of the mean.
    ::大约99.7%的数据将低于平均值的3个标准差。

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  Some other important points about the normal distribution:
  ::关于正常分配的其他一些要点:

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  • The total area between the normal curve and the  $\begin{array}{r}x\end{array}$ -axis is 1, and this area represents all possible probabilities.
    ::正常曲线与 X 轴之间的总面积为 1, 这个区域代表所有可能的概率 。
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  • If data are distributed normally, you can use the normal distribution to determine the percentage of the data between any two values by calculating the area under the curve between those two values. 
    ::如果数据是正常分布的,可以使用正常分布来确定任何两个数值之间数据的百分比,方法是计算这两个数值之间的曲线下的区域。
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  • Many histograms approximate a normal curve, but a true normal curve is infinitely smooth.
    ::许多直方图接近正常曲线,但真正的正常曲线绝对平滑。

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  On a TI calculator, the normalcdf  command calculates the normal cumulative distribution function, which is the area between any two values for data that is 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.  W hen using normalcdf, four numbers are needed to be inputted: normalcdf $\begin{array}{r}(x,y,\mu ,\sigma ).\end{array}$  The 1st number is the lower boundary of the interval , the 2nd number is the upper boundary of the interval, the 3rd number is the mean, and the 4th number is the standard deviation.
  ::在 TI 计算器上, uscdf 命令计算正常的累积分布函数, 也就是通常分布的数据的两个值之间的区域, 只要你知道数据的平均和标准偏差。 您的计算器将此函数嵌入其中, 它产生的准确答案与经验规则相对。 当使用 uscdf 时, 需要输入四个数字 : uscdf( x,y, li, Q) 。 第一个数字是间隔的下边界, 第二个数字是间隔的上边界, 第三个数字是平均值, 第四个数字是标准偏差 。

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  There  is a 2nd programmed feature, invNor m , in the distribution menu that performs the inverse of normalcdf. Instead of being given the standard deviation and asked to find the probability, you are given the probability and asked to find the standard deviation.
  ::分发菜单中有第2个编程特性, invNorm, 用于执行反正的正态 cdf 。 没有给您提供标准偏差, 并要求您查找概率, 而是给您提供概率, 并要求您找到标准偏差 。

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  The following video  introduces the standard normal distribution:
  ::以下录像介绍标准正常分发情况:

    

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  The following video  explains how to draw a normal distribution using Microsoft Excel:
  ::以下影片解释如何使用微软Excel进行正常分发:

    

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  Play, Learn, and Explore with The Normal Curve using the topic of food safety:  .
  ::以食品安全为题的“正常曲线”游戏、学习和探索:.

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  Examples
  ::实例

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  Example 1
  ::例1

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  The amount of rain each year in Connecticut follows a normal distribution. What is the probability of getting more than 1 standard deviation below the normal amount of rain?
  ::康涅狄格州每年的降雨量遵循正常的分布。 降雨量低于正常的降雨量超过1个标准差的概率是多少?

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  Solution:
  ::解决方案 :

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  You are looking for the area of the shaded portion of the normal distribution shown below. By the empirical rule, you know that approximately 34% of the data is in between -1 and 0 standard deviations from the mean. Also, 50% of the data is above 0 standard deviations from the mean. Therefore, approximately 84% of the data is unshaded. Therefore,  $\begin{array}{r}100\mathrm{\%}-84\mathrm{\%}=16\mathrm{\%}\end{array}$ of the data is shaded. The approximate probability is 16%.
  ::您正在寻找以下显示的正常分布的阴影部分区域。 根据经验规则, 您知道大约34%的数据在 - 1 和 0 标准偏差之间。 另外, 50% 的数据高于 0 标准偏差。 因此, 大约 84% 的数据没有阴影。 因此, 100 84 16% 的数据是阴影。 大约概率为 16% 。

  

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  Example 2
  ::例2

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  On your 1st college exam, you score an 82. After the exam, the professor tells the class that the mean was a 62, and the standard deviation was 10. Assuming the data are normally distributed, what percentage of the class did better than you?
  ::在你的第一次大学考试中, 你得了82分, 考试之后,教授告诉班级, 平均值是62分, 标准偏差是10分, 假设数据正常分配, 哪个比例的班级比你好?

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  Solution:
  ::解决方案 :

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  An 82 is 20 away from the mean, so is 2 standard deviations from the mean. Therefore, this question is asking for the percentage of students that are above +2 standard deviations above the mean.
  ::82比平均值低20,2比平均值差2个标准差。 因此,这个问题要求学生中超过+2标准差的百分比高于平均值。

  

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  Either you can use the fact that your score was exactly 2 standard deviations above the mean, or you can calculate the probability using the actual numbers. 
  ::或者你可以用实际数字来计算概率。

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  • normalcdf(2, 1E99, 0, 1) = 0.022750 or 2.275%
    ::=0.0022750 或 2.2.75%
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  • normalcdf(82, 1E99, 62, 10) = 0.022750 or 2.275%
    ::=0.0022750或2.2.75%

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  2.275% of the class did better than you on the exam. 
  ::2.275%的班级考试成绩都比你好

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  Example 3
  ::例3

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  The quality control technician of a widget-making factory observes that widgets that are 3 standard deviations too large or 3 standard deviations too small from the precise widget size are unusable. Assuming the data are normally distributed, w hat is the probability of producing a usable widget?
  ::部件制造厂的质量控制技术员认为,3个标准偏差太大或3个标准偏差小于精确部件大小的部件是无法使用的。 假设数据通常分布,那么生产可用部件的概率是多少?

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  Solution:
  ::解决方案 :

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  This question is essentially asking for the area between -3 standard deviations and +3 standard deviations. The empirical rule says this should be 99.7%. Use the normalcdf function to find the exact value.
  ::这个问题基本上是为 - 3 标准偏差和 + 3 标准偏差之间的区域问题。 经验规则说这应该是99.7%。 使用正统 cdf 函数来找到准确值 。

  

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  normalcdf(-3, 3, 0, 1) = 0.997300 or 99.73%
  ::= 0.997300 或 99.73%

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  The quality control technician would decide if this is a high enough success rate for producing a usable widget.
  ::质量控制技术员将决定这是否是生产可用部件的足够高的成功率。

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  Example 4
  ::例4

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  Recall the question from the Introduction: What other examples follow a normal distribution?
  ::回顾导言中的问题:正常分配之后还有其他哪些例子?

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  Solution:
  ::解决方案 :

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  Height, weight, and other measures of people, animals, or plants are normally distributed.
  ::人、动物或植物的重量、重量和其他测量尺度通常分布在高度、重量和其他方面。

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  Example 5
  ::例5

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  What is the probability that a person in Texas is exactly 6 feet tall?
  ::在得克萨斯州,一个人身高6英尺高的概率是多少?

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  Solution:
  ::解决方案 :

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  Since height is a continuous variable , meaning any number within a reasonable domain interval is possible, the probability of choosing any single number is 0. Many people may be close to 6 feet tall, but in reality they are 5.99 or 6.0001 feet tall. There must be someone in Texas who is the closest to being exactly 6 feet tall, but even that person when measured accurately enough will still be slightly off from 6 feet. This is why, instead of calculating the probability for a single outcome, you calculate the probability between a certain interval, like between 5.9 feet and 6.1 feet. For continuous variables, the probability of any specific outcome, like 6 feet, will always be 0.
  ::由于高度是一个连续的变量, 意指在合理的域间距内的任何数字都是可能的, 选择任何单一数字的概率是 0。 许多人可能接近6英尺高, 但实际上他们有5. 99或6. 0001英尺高。 得克萨斯州一定有人最接近6英尺高, 但即使准确测量得足够准确, 也从6英尺略差一点。 这就是为什么你没有计算单一结果的概率, 而是计算某一间距之间的概率, 如5. 9英尺到6.1英尺。 对于连续变量来说, 任何具体结果的概率, 如6英尺, 总是为0。

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  Example 6
  ::例6

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  Two percent of high-school football players are invited to play at a competitive college level. How many standard deviations above the average player would someone need to be to have this opportunity?
  ::2%的高中足球运动员被邀请在竞争性大学水平上参加比赛。 需要多少比普通球员标准差的人才能有这个机会?

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  Solution:
  ::解决方案 :

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  This situation is the inverse of the previous questions. Thus, you use the invNorm command on your calculator. You are looking for how many standard deviations above the mean include 98% of the data.
  ::这是前几个问题的反向。 因此, 您在计算器上使用 InvNorm 命令。 您正在寻找超过平均值的多少标准偏差, 包括98%的数据 。

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  invNorm(0.98) = 2.0537
  :0.98)=2.0537

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  A person would have to be greater than about 2 standard deviations above the mean to be in the top 2 percent.
  ::一个人必须超过比平均值高出2%的大约2个标准差。

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  Example 7
  ::例7

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  On average, a pumpkin at your local farm weighs 10 pounds with a standard deviation of 6 pounds. You go and find a pumpkin weighing 26 pounds. Of all the pumpkins at the farm, what percent weigh less than this enormous pumpkin?
  ::平均来说,你当地农场的南瓜体重为10磅,标准偏差为6磅。你去找一个26磅的南瓜。在农场的所有南瓜中,什么比例比这个巨大的南瓜还小?

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  Solution:
  ::解决方案 :

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  Normalcdf(-1E99, 26, 10, 6) = 0.9961 or 99.61%
  ::普通(-1E99、26、10、6)=0.9961或99.61%

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  The vast majority of the pumpkins weigh less than the 26-pound pumpkin you found.
  ::绝大多数南瓜的重量 低于你找到的26磅南瓜

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  Summary
  ::摘要

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  • 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。
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  • 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。

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  Review
  ::回顾

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  Consider the standard normal distribution for the following questions:
  ::审议下列问题的标准正常分配情况:

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  1. What is the mean?
  ::1. 是什么意思?

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  2. What is the standard deviation?
  ::2. 标准偏差是什么?

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  3. What is the percentage of the data below 1?
  ::3. 1以下数据的百分比是多少?

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  4. What is the percentage of the data below -1?
  ::4. 以下数据的百分比是多少?

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  5. What is the percentage of the data above 2?
  ::5. 以上2项数据的百分比是多少?

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  6. What is the percentage of the data between -2 and 2?
  ::6. -2和2之间数据的百分比是多少?

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  7. What is the percentage of the data between -0.5 and 1.7?
  ::7. -0.5和1.7之间数据的百分比是多少?

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  8. What is the probability of a value of 2?
  ::8. 值为2的概率是多少?

   

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  Assume that the mean weight of one-year-old girls in the United States is normally distributed, with a mean of about 9.5 kilograms and a standard deviation of approximately 1.1 kilograms.
  ::假设美国一岁女孩的平均重量通常分布在大约9.5公斤左右,标准偏差约为1.1公斤。

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  9. What percent of one-year-old girls weigh between 8 and 12 kilograms?
  ::9. 1岁女孩的体重在8至12公斤之间,占多大比例?

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  10. What percent of girls weigh above 12 kilograms?
  ::10 %的女童体重超过12公斤?

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  11. Girls in the bottom 5% by weight need their weight monitored every two months. How many standard deviations below the mean would a girl need to be to have her weight monitored? 
  ::11. 按重量计,体重在5%以下的女孩需要每两个月监测一次体重,比平均数低多少标准偏差需要监测其体重?

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  Suppose that adult women's heights are normally distributed with a mean of 65 inches and a standard deviation of 2 inches.
  ::假设成年妇女的身高通常平均分布为65英寸,标准偏差为2英寸。

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  12. What percent of adult women have heights between 60 inches and 65 inches?
  ::12. 成年妇女中高在60至65英寸之间的身高是多少?

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  13. Use the empirical rule to describe the range of heights for women within 1 standard deviation of the mean.
  ::13. 利用经验规则,在平均值的1个标准差内,说明妇女的高度范围。

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  14. What is the probability that a randomly selected adult woman is more than 64 inches tall?
  ::14. 随机挑选的成年妇女身高超过64英寸的概率有多大?

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  15. What percent of adult women are either less than 60 inches or greater than 72 inches tall?
  ::15. 成年妇女中有多大比例或不到60英寸,或超过72英寸?

   

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  Review (Answers )
  ::回顾(答复)

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  Please see the Appendix.
  ::请参看附录。