课程： 8.6 以正常分配方式工作-interactive

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以正常分配方式工作

Lesson Objectives
::经验教训目标

Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages.
::使用数据集的平均值和标准差,使其符合正常分布,并估计人口百分比。

Use calculators, spreadsheets, and tables to estimate areas under the normal curve .
::使用计算器、电子表格和表格来估计正常曲线下的区域。

Introduction: Normal Distributions in Business
::导言:商业正常分布

The standard normal distribution has a wide range of uses in nearly every field. By measuring deviance from the mean in units of standard deviations, we can apply this function to model any scenario with normally distributed data . Additionally, although the standard normal distribution is a continuous distribution, large discrete, normally distributed datasets can be approximated using the normal distribution.
::标准的正常分布几乎在每个字段都有广泛的用途。通过测量标准偏差单位平均值的偏差,我们可以应用此函数用正常分布的数据来模拟任何假设情景。此外,虽然标准的正常分布是连续分布,但大离散的通常分布的数据集可以用正常分布来比较。

Barry owns a farm stand. Anything that he does not sell in a week is considered spoiled, and he must either give it away or throw it away. He anticipates a mean weekly demand of 164 heads of lettuce with a standard deviation of 25 heads of lettuce. Assuming the weekly demand is normally distributed, use the interactive to answer the given questions.
::巴里拥有一个农场摊位。他一周内不出售的东西都被视为被宠坏了,他必须放弃或扔掉。他预计每星期平均需要164个生菜头,标准偏差为25个生菜头。假设每周需求正常分配,则使用互动方式回答问题。

Discussion Question: How many heads of lettuce should Barry produce?
::讨论问题:Barry应该生产多少个生菜头?

Activity 1: Normal Distribution in Medicine
::活动1:医药正常分配

Doctors use to examine the proportions of the population exhibiting various heights, weights, cholesterol levels, etc. Normal Distributions help to identify when a trait is abnormally high or abnormally low. Given the mean and standard deviation of a normally distributed dataset, we can determine the proportion of data that falls above or below any data value.
::正常分布有助于识别异常高或异常低的特征。鉴于通常分布的数据集的平均值和标准偏差,我们可以确定任何数据值以上或以下的数据比例。

Example
::示例示例示例示例

Cholesterol levels among males over the age of 20 are normally distributed with a mean of 188 mg/dL and a standard deviation of 42 mg/dL. High cholesterol can lead to heart attack and stroke. Patients are considered to have high cholesterol if they have a cholesterol level of over 240. Use a normal distribution to approximate the proportion of males over the age of 20 with high cholesterol.
::20岁以上男性的胆固醇水平通常平均分配为188毫克/升/升,标准偏差为42毫克/升。高胆固醇可能导致心脏病发作和中风,如果患者的胆固醇水平超过240,则被视为高胆固醇。使用正常分布方法,接近20岁以上男性的胆固醇比例。

To understand how a cholesterol level of 240 relates to the entire population, we will need to convert it to a z score.
::为了了解240的胆固醇水平与全体人口的关系,我们需要将其转换为z分。

\begin{aligned} \frac{240 - 188}{42} \approx 1.24 \end{aligned}

$\begin{align*}\frac{240 - 188}{42} ≈ 1.24\end{align*}$

Next, we must find the proportion of data with a z score greater than 1.24. Using a table, we see that the proportion of the population with a cholesterol z score of less than 1.24 is 0.8925. We can subtract this from 1 to find the proportion of the population with a cholesterol z score of greater than 1.24.
::其次,我们必须找到z分大于1.24的数据比例。用一张表格,我们看到胆固醇z分小于1.24的人口比例为0.8925。我们可以从1中减去这一比例,以找到胆固醇z分大于1.24的人口比例。

\begin{aligned} 1 - 0.8925 = 0.1075 \end{aligned}

$\begin{align*}1 - 0.8925 = 0.1075\end{align*}$

Answer: Approximately 10.75% males over the age of 20 have a cholesterol level over 240.
::答复:大约10.75%的20岁以上的男性胆固醇水平超过240。

Before we can use the standard normal distribution, we must convert any specific data values to a z score. Use the standard normal deviation to answer the questions below.
::在使用标准正态分布之前,我们必须将任何特定数据值转换为z分。使用标准正态偏差回答下面的问题。

Discussion Question: The normal distribution of cholesterol levels above applies to all adult males over 20. What potential danger could arise when comparing a 27-year-old male with a 70-year-old male using this normal distribution?
::讨论问题:以上胆固醇水平的正常分布适用于20岁以上的所有成年男性,如果将27岁的男性与70岁的男性使用这一正常分布进行对比,会产生什么潜在危险?

Activity 2: Measuring Intelligence
::活动2:衡量情报

In addition to using the standard normal distribution to find the proportion of data that falls into a range for any normally distributed dataset, we can also use it to find the data value at a certain percentile .
::除了使用标准的正常分布来找到任何正常分布数据集中属于范围的数据比例外,我们还可以使用它来找到某一百分位值的数据值。

Example
::示例示例示例示例

In the late 1800’s Sir Francis Galton was among the first to propose a link between the size of a person’s brain and their intelligence. That is the same Sir Francis Galton who invented the Galton Board, which we saw when studying binomial distributions. The sizes of adult brains are normally distributed with an approximate mean of 1260 cubic centimeters and a standard deviation of 127 cubic centimeters. What range of brain sizes represent the top 1%?
::在1800年代末期,弗朗西斯·加尔顿爵士是最早提出将一个人的大脑大小与其智力联系起来的建议的人之一。这就是在研究二进制分布时我们看到的发明加尔顿董事会的弗朗西斯·加尔顿爵士。成人大脑的大小通常以大约1,260立方厘米的平均值和127立方厘米的标准偏差来分布。大脑大小的哪个范围代表顶端的1 % ?

Our strategy is similar to the one that we used in the previous activity, but we will be working backward. We must first find the z score for the given proportion and then convert it to the corresponding data value.
::我们的战略与我们在前一活动中使用的战略相似,但我们将会向后工作。我们必须首先找到给定比例的z分, 然后将其转换为相应的数据值。

When looking for a proportion in the standard normal distribution table, we must keep in mind that the table displays the proportion of data falling to the left of the z score. Thus to find the top 1%, we must look for the z score, for which 99% of the data falls to the left. In this case, that value is 2.33. Since 1% of the data falls to the left of –2.33, we could also have taken the opposite value of –2.33.
::当在标准正常分布表中查找比例时,我们必须记住,表格显示数据落到z分左侧的比例。因此,要找到上1 % , 我们必须寻找Z分, 其中99%的数据落到左侧。在这种情况下,该值为2.33。由于1%的数据落到-2.33左侧,我们也可以选择正对值-2.33。

To convert 2.33 to a data value, we must use the z score formula:
::要将2.33转换为数据值, 我们必须使用 z 得分公式 :

\begin{aligned} z & = \frac{x - 𝜇}{𝜎} \\ 2.33 & = \frac{x - 1260}{127} \\ \times 127 & \times 127 \\ 295.91 & = x - 1260 \\ + 1260 & + 1260 \\ 1555.91 & = x \end{aligned}

$\begin{align*}z &= \frac{x – 𝜇}{𝜎}\\ \\ 2.33 &= \frac{x \:– 1260}{127}\\\\ \times 127 \:&\:\:\:\:\: \times 127 \\ \hline 295.91 &= x \:– 1260\\ +1260\:\:\:& \:\:\:\:\:\:+1260\\ \hline 1555.91 &= x\end{align*}$
::zx=xx=1260127x127x12727295.91x-1260+1260+12601555.91=x

Answer: An adult brain with a volume greater than or equal to 1,555.91 cubic centimeters will fall in the top 1% of brain volumes.
::答:体积大于或等于1 555.91立方厘米的成人大脑将下降至大脑体积的顶部1%。

Use the interactive below to practice finding a data value given a proportion of the data.
::使用下面的交互式数据来根据数据的一部分来寻找数据值。

Activity 3: Measuring Intelligence Part 2
::活动3:衡量情报第2部分

While using a table is a useful tool for finding proportions of data within a normal distribution, our accuracy is limited to the ten-thousandth place. For more accuracy, we can use technology. We can use a calculator, a spreadsheet, or any program with a function designed to find a cumulative value of the normal distribution function at any data value.
::使用表格是发现正常分布中数据比例的有用工具, 但是我们的准确性仅限于十千位。为了更精确, 我们可以使用技术。我们可以使用计算器、电子表格或任何程序, 或任何程序, 其功能旨在以任何数据值查找正常分布函数的累积值。

Example
::示例示例示例示例

In modern times, measures of intelligence have evolved beyond brain size. One commonly used measure of intelligence is the IQ test. The first IQ test was developed by Alfred Binet and Theodore Simon to rate the memory, logical thinking, and problem-solving ability of the test-taker on a scale from 0 to 200. IQ scores of the world population are normally distributed with a mean of 100 and a standard deviation of 15. What proportion of test-takers score 110 or below?
::现代时代,智力衡量尺度的发展超出了大脑的大小,一种常用的智力衡量尺度是智商测试。第一次智商测试是由阿尔弗雷德·比内特和西奥多·西蒙开发的,目的是从0到200的尺度评分测试师的记忆、逻辑思维和解决问题的能力。智商在世界人口中的得分通常平均为100分,标准偏差为15分。接受测试者达到或低于110分的比例是多少?

In the interactive below, we will use the cumulative normal distribution function to find the proportion of IQ test takers with an IQ of at most 110. We will use the following function:
::在下文互动部分,我们将使用累积正常分配函数来找到智商测试接受者的比例,其智商最多为110。我们将使用以下函数:

\begin{aligned} Normal(Insert the Mean, Insert the Standard Deviation, Insert the Data Value) \end{aligned}

$\begin{align*}\text{Normal(Insert the Mean, Insert the Standard Deviation, Insert the Data Value)}\end{align*}$
::正常( 插入平均值, 插入标准偏差, 插入数据值)

We are given the information below:
::我们得到的信息如下:

Mean = 100
::平均值=100

Standard Deviation = 15
::标准偏离=15

Data Value = 110
::数据值=110

Enter $\begin{align*}\text{Normal(100, 15, 110)}\end{align*}$ into the interactive below.
::将正常( 100、 15、 110) 输入到下面的交互式中。

Answer: The proportion of test-takers that score at most a 110 on an IQ test, rounded to the nearest millionth, is 0.747507.
::答复:在智商测试中最多得110分(四舍五入至百万分之一)的测试接受者比例为0.77507。

We can use the following function to return the data value when given the proportion of data falling to the left of it.
::我们可以使用以下函数返回数据值,如果考虑到数据在左边下降的比例。

\begin{aligned} InverseNormal(Insert the Mean, Insert the Standard Deviation, Insert the Proportion to the Left) \end{aligned}

$\begin{align*}\text{InverseNormal(Insert the Mean, Insert the Standard Deviation, Insert the Proportion to the Left)}\end{align*}$
::反向( 插入平均值, 插入标准偏差, 插入向左的比例)

Use the functions and interactive above to answer the following questions.
::使用上面的函数和互动来回答下列问题。

Summary

The standard normal distribution has a wide range of uses in nearly every field.
::标准的正常分布几乎在每个领域都有广泛的用途。

The normal distribution function is

$\begin{aligned} N (x) = \frac{1}{σ \sqrt{2 π}} e^{- \frac{(x - μ)^{2}}{2 σ^{2}}} \end{aligned}$
$\begin{align*}N(x) = \frac{1}{\sigma \sqrt{2\pi}} e ^{-\frac{(x - \mu)^{2}}{2\sigma ^{2}}}\end{align*}$
::正常的分布函数为 N( x) = 1-19632e- (x) 222- 132 。

The formula for z score is $\begin{align*}z = \frac{x - \mu}{\sigma}\end{align*}$
::兹得分的公式是z=x

章节大纲

常规

以正常分配方式工作

Introduction: Normal Distributions in Business ::导言:商业正常分布

Activity 1: Normal Distribution in Medicine ::活动1:医药正常分配

Activity 2: Measuring Intelligence ::活动2:衡量情报

Activity 3: Measuring Intelligence Part 2 ::活动3:衡量情报第2部分

Wrap-Up: Review Questions ::总结:审查问题

Introduction: Normal Distributions in Business
::导言:商业正常分布

Activity 1: Normal Distribution in Medicine
::活动1:医药正常分配

Activity 2: Measuring Intelligence
::活动2:衡量情报

Activity 3: Measuring Intelligence Part 2
::活动3:衡量情报第2部分

Wrap-Up: Review Questions
::总结:审查问题