节: 三. 零 - 核心三 | 9.5 三级Z级

章节大纲

Do z -score probabilities always need to be calculated as the chance of a value either above or below a given score? How would you calculate the probability of a z -score between -0.08 and +1.92?
::z- score 概率总是需要被计算为值的概率, 或高于或低于给定分数的概率吗? 您如何计算 - 0.08 和 +1. 92 之间的 z- score 概率 ?

Z-Scores
::零分数

To calculate the probability of getting a value with a z -score between two other z -scores, you can either use a reference table to look up the value for both scores and subtract them to find the difference, or you can use technology. In this lesson, which is an extension of Z -scores and Z -scores II, we will practice both methods.
::要计算在另外两个z-score 之间以 z-score 获得值的概率, 您可以使用一个参考表格来查找两个分数的值, 并减去它们以找到差数, 或者您可以使用技术。在此课中, 这是 Z- score 和 Z- score II 的扩展, 我们将同时使用两种方法。

Historically, it has been very common to use a z -score probability table like the one below to look up the probability associated with a given z -score:
::从历史上看,使用z-core概率表,例如下面的z-z-score 概率表来查查与给定z-score 相关的概率,这是非常常见的:

Z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	Z
0.0	.5	0.504	0.508	0.512	0.516	0.5199	0.5239	0.5279	0.5319	0.5359	0.0
0.1	0.5398	0.5438	0.5478	0.5517	0.5557	0.5596	0.5636	0.5675	0.5714	0.5753	0.1
0.2	0.5793	0.5832	0.5871	0.591	0.5948	0.5987	0.6026	0.6064	0.6103	0.6141	0.2
0.3	0.6179	0.6217	0.6255	0.6293	0.6331	0.6368	0.6406	0.6443	0.648	0.6517	0.3
0.4	.6554	0.6591	0.6628	0.6664	0.67	0.6736	0.6772	0.6808	0.6844	0.6879	0.4
0.5	0.6915	0.695	0.6985	0.7019	0.7054	0.7088	0.7123	0.7157	0.719	0.7224	0.5
0.6	0.7257	0.7291	0.7324	0.7357	0.7389	0.7422	0.7454	0.7486	0.7517	0.7549	0.6
0.7	0.758	0.7611	0.7642	0.7673	0.7704	0.7734	0.7764	0.7794	0.7823	0.7852	0.7
0.8	0.7881	0.791	0.7939	0.7967	0.7995	0.8023	0.8051	0.8078	0.8106	0.8133	0.8
0.9	0.8159	0.8186	0.8212	0.8238	0.8264	0.8289	0.8315	0.834	0.8365	0.8389	0.9
1.0	0.8413	0.8438	0.8461	0.8485	0.8508	0.8531	0.8554	0.8577	0.8599	0.8621	1.0
1.1	0.8643	0.8665	0.8686	0.8708	0.8729	0.8749	0.877	0.879	0.881	0.883	1.1
1.2	0.8849	0.8869	0.8888	0.8907	0.8925	0.8944	0.8962	0.898	0.8997	0.9015	1.2
1.3	0.9032	0.9049	0.9066	0.9082	0.9099	0.9115	0.9131	0.9147	0.9162	0.9177	1.3
1.4	0.9192	0.9207	0.9222	0.9236	0.9251	0.9265	0.9279	0.9292	0.9306	0.9319	1.4
1.5	0.9332	0.9345	0.9357	0.937	0.9382	0.9394	0.9406	0.9418	0.9429	0.9441	1.5
1.6	0.9452	0.9463	0.9474	0.9484	0.9495	0.9505	0.9515	0.9525	0.9535	0.9545	1.6
1.7	0.9554	0.9564	0.9573	0.9582	0.9591	0.9599	0.9608	0.9616	0.9625	0.9633	1.7
1.8	0.9641	0.9649	0.9656	0.9664	0.9671	0.9678	0.9686	0.9693	0.9699	0.9706	1.8
1.9	0.9713	0.9719	0.9726	0.9732	0.9738	0.9744	0.975	0.9756	0.9761	0.9767	1.9
2.0	0.9772	0.9778	0.9783	0.9788	0.9793	0.9798	0.9803	0.9808	0.9812	0.9817	2.0
2.1	0.9821	0.9826	0.983	0.9834	0.9838	0.9842	0.9846	0.985	0.9854	0.9857	2.1
2.2	0.9861	0.9864	0.9868	0.9871	0.9875	0.9878	0.9881	0.9884	0.9887	0.989	2.2
2.3	0.9893	0.9896	0.9898	0.9901	0.9904	0.9906	0.9909	0.9911	0.9913	0.9916	2.3
2.4	0.9918	0.992	0.9922	0.9925	0.9927	0.9929	0.9931	0.9932	0.9934	0.9936	2.4
2.5	0.9938	0.994	0.9941	0.9943	0.9945	0.9946	0.9948	0.9949	0.9951	0.9952	2.5
2.6	0.9953	0.9955	0.9956	0.9957	0.9959	0.996	0.9961	0.9962	0.9963	0.9964	2.6
2.7	0.9965	0.9966	0.9967	0.9968	0.9969	0.997	0.9971	0.9972	0.9973	0.9974	2.7
2.8	0.9974	0.9975	0.9976	0.9977	0.9977	0.9978	0.9979	0.9979	0.998	0.9981	2.8
2.9	0.9981	0.9982	0.9982	0.9983	0.9984	0.9984	0.9985	0.9985	0.9986	0.9986	2.9
3.0	0.9987	0.9987	0.9987	0.9988	0.9988	0.9989	0.9989	0.9989	0.999	0.999	3.0
3.1	0.999	0.9991	0.9991	0.9991	0.9992	0.9992	0.9992	0.9992	0.9993	0.9993	3.1
3.2	0.9993	0.9993	0.9994	0.9994	0.9994	0.9994	0.9994	0.9995	0.9995	0.9995	3.2
3.3	0.9995	0.9995	0.9995	0.9996	0.9996	0.9996	0.9996	0.9996	0.9996	0.9997	3.3
3.4	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9997	0.9998	3.4
3.5	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	0.9998	3.5
3.6	0.9998	0.9998	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	3.6
3.7	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	3.7
3.8	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	0.9999	3.8
3.9	1	1	1	1	1	1	1	1	1	1	3.9
Z	0.000	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	Z

Since the proliferation of the Internet, however, you can also use a free online calculator.
::自互联网扩散以来,您也可免费使用在线计算器。

Calculating Probability
::计算概率

What is the probability associated with a z -score between 1.2 and 2.31?
::1.2至2.31之间z分数的概率是多少?

To evaluate the probability of a value occurring within a given range , you need to find the probability of both the upper and lower values in the range, and subtract to find the difference.
::要评估某个数值在给定范围内发生的概率,您需要找到该范围内上值和下值的概率,并减去以找到差数。

First find $\begin{aligned} z = 1.2 \end{aligned}$ on the z -score probability reference above: .8849 Remember that value represents the percentage of values below 1.2.
::上文z-score 概率参考第一组发现z=1.2 :. 8849 记住,该数值代表低于1.2的数值的百分比。

Next, find and record the value associated with $\begin{aligned} z = 2.31 \end{aligned}$ : .9896
::下一步,查找并记录z=2.31:.9896的数值。

Since approximately 88.49% of all values are below $\begin{aligned} z = 1.2 \end{aligned}$ and approximately 98.96% of all values are below $\begin{aligned} z = 2.31 \end{aligned}$ , there are $\begin{aligned} 98.96 % - 88.49 % = 10.47 % \end{aligned}$ of values between .
::由于所有值中约有88.49%低于z=1.2,而所有值中约有98.96%低于z=2.31, 两者之间的数值为98.9688.4910.47%。

Calculating the Probability of a Value Occurring in a Normal Distribution
::计算正常分配中发生值的概率

1. What is the probability that a value with a z -score between -1.32 and +1.49 will occur in a normal distribution ?
::1. -1.32和+1.49之间的z-分数值在正常分布中发生的可能性有多大?

Let’s use the online calculator on "Math Portal" for this one.
::让我们使用“马思门户”上的在线计算器。

When you open the page, you should see a window like this:
::当您打开页面时, 您应该看到这样的窗口 :

All you need to do is select the radio button to the left of the first type of probability, input “-1.32” into the first box, and 1.49 into the second. When you click “Compute”, you should get the result
::您需要做的就是选择第一种概率左左侧的无线电按钮,输入第一个框中的“-132” 输入“-1.32” ,输入第二个框中的1.49。当单击“计算”时,您应该得到结果

\begin{aligned} P (- 1.32 < Z < 1.49) = 0.8385 \end{aligned}

::P(1.321.49=0.8385)

Which tells us that there is approximately and 83.85% probability that a value with a z -score between 1.32 and 1.49 will occur in a normal distribution.
::这告诉我们,在1.32至1.49之间,一个z-分数值的概率大约为83.85%,在正常分布时,该值的概率为1.32至1.49之间。

Notice that the calculator also details the steps involved with finding the answer:
::注意计算器还详细说明找到答案的步骤:

Estimate the probability using a graph, so you have an idea of what your answer should be.
::使用图表来估计概率, 所以你知道答案应该是什么。

Find the probability of $\begin{aligned} z < 1.49 \end{aligned}$ , using a reference. (0.9319)
::使用引用查找 z<1.49 的概率。 (0. 9319)

Find the probability of $\begin{aligned} z < - 1.32 \end{aligned}$ , again, using a reference. (0.0934)
::使用引用再次查找 z1.32 的概率。 (0. 0934)

Subtract the values: $\begin{aligned} 0.9319 - 0.0934 = 0.8385 o r 83.85 % \end{aligned}$
::减去数值: 0.9319-0.0934=0.8385或83.85%

2. What is the probability that a random selection will be between 8.45 and 10.25, if it is from a normal distribution with $\begin{aligned} μ = 10 \end{aligned}$ and $\begin{aligned} σ = 2 \end{aligned}$ ?
::2. 随机选择在8.45至10.25之间,如果是从10和2的正常分布中随机选择的概率是多少?

This question requires us to first find the z -scores for the value 8.45 and 10.25, then calculate the percentage of value between them by using values from a z -score reference and finding the difference.
::这个问题要求我们首先找到值为8.45和10.25的z-分数,然后通过使用z-分数参考值来计算它们之间的值百分比,并找出差数。

1. Find the z -score for 8.45, using the z -score formula: $\begin{aligned} \frac{(x - μ)}{σ} \end{aligned}$
::1. 使用z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-z-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{aligned} \frac{8.45 - 10}{2} = \frac{- 1.55}{2} \approx - 0.78 \end{aligned}

2. Find the z -score for 10.25 the same way:
::2. 以同样方式查找10.25的z-score:

\begin{aligned} \frac{10.25 - 10}{2} = \frac{0.25}{2} \approx .13 \end{aligned}

3. Now find the percentages for each, using a reference (don’t forget we want the probability of values less than our negative score and less than our positive score, so we can find the values between):
::3. 现在找出每种数值的百分比,使用参考(不要忘记,我们想要的数值概率低于我们的负分,低于我们的正分,这样我们才能在以下两点之间找到数值):

\begin{aligned} P (Z < - 0.78) & = .2177 o r 21.77 % \\ P (Z < .13) & = .5517 o r 55.17 % \end{aligned}

::P(0.78)=2177或21.77%P(.13)=5517或55.17%

4. At this point, let’s sketch the graph to get an idea what we are looking for:
::4. 此时此刻,让我们勾画一下图表,以了解我们正在寻找的是什么:

5. Finally, subtract the values to find the difference:
::5. 最后,减去找到差数的数值:

$\begin{aligned} .5517 - .2177 = .3340 o r a b o u t 33.4 % \end{aligned}$

::5517-21777=3340或约33.4%

There is approximately a 33.4% probability that a value between 8.45 and 10.25 would result from a random selection of a normal distribution with mean 10 and standard deviation 2.
::随机选择平均值为10和标准差2的正常分布,可能会产生8.45至10.25之间的数值,概率约为33.4%。

Earlier Problem Revisited
::重审先前的问题

Do z-score probabilities always need to be calculated as the chance of a value either above or below a given score? How would you calculate the probability of a z-score between -0.08 and +1.92?
::z- score 概率总是需要被计算为值的概率, 或高于或低于给定分数的概率吗? 您如何计算 - 0.08 和 +1. 92 之间的 z- score 概率 ?

After this lesson, you should know without question that z -score probabilities do not need to assume only probabilities above or below a given value, the probability between values can also be calculated.
::在此课后,您应该毫无疑问地知道,z-core概率不需要仅仅假设某一数值以上或以下的概率,也可以计算数值之间的概率。

The probability of a z -score below -0.08 is 46.81%, and the probability of a z -score below 1.92 is 97.26%, so the probability between them is $\begin{aligned} 97.26 % - 46.81 % = 50.45 % \end{aligned}$ .
::z-score 低于 - 0.08 的概率为 46.81%, z-score 低于 1. 92% 的概率为 97.26%, 因此它们之间的概率为 97.26 46. 81 50.45%。

Examples
::实例

What is the probability of a z -score between -0.93 and 2.11?
::0.93到2.11之间 z -score的概率是多少?

What is $\begin{aligned} P (1.39 < Z < 2.03) \end{aligned}$ ?
::P(1.392.03)是什么?

What is $\begin{aligned} P (- 2.11 < Z < 2.11) \end{aligned}$ ?
::P(-2.11)2.11是什么?

Solutions:
::解决办法:

Example 1
::例1

What is the probability of a z-score between -0.93 and 2.11
::0.93和2.11之间z分数的概率是多少?

Using the z -score probability table above, we can see that the probability of a value below -0.93 is .1762, and the probability of a value below 2.11 is .9826. Therefore, the probability of a value between them is $\begin{aligned} .9826 - .1762 = .8064 o r 80.64 % \end{aligned}$
::使用上面的 z-score 概率表,我们可以看到,值低于 -0.93 的概率是 1762,值低于 2.11 的概率是 9826。因此,它们之间的数值概率是 .9826 - 1762=.8064 或 80.64%。

Example 2
::例2

What is $\begin{aligned} P (1.39 < Z < 2.03) \end{aligned}$ ?
::P(1.392.03)是什么?

Using the z -score probability table, we see that the probability of a value below $\begin{aligned} z = 1.39 \end{aligned}$ is .9177, and a value below $\begin{aligned} z = 2.03 \end{aligned}$ is .9788. That means that the probability of a value between them is 9177%3D.0611%20%5C%20or%20%5C%206.11%5C%25"> $\begin{aligned} .9788 - .9177 = .0611 o r 6.11 % \end{aligned}$
::使用 z-score 概率表,我们看到,z=1.39以下值的概率为 9177,而z=2.03以下值为 .9788。这意味着它们之间值的概率为.9789177=.0611或6.11%。

Example 3
::例3

What is $\begin{aligned} P (- 2.11 < Z < 2.11) \end{aligned}$ ?
::P(-2.11)2.11是什么?

Using the online calculator on "Math Portal", we select the top calculation with the associated radio button to the left of it, enter “-2.11” in the first box, and “2.11” in the second box. Click “Compute” to get “ .9652 ”, and convert to a percentage. The probability of a z -score between -2.11 and +2.11 is about 96.52% .
::使用“ Matth Portal” 上的在线计算器,我们选择左侧相关无线电按钮的顶部计算,输入第一个框中的“ 2.11” 和第二个框中的“ 2111 ”。单击“ 计算” 以获得 “. 9652 ” , 并转换成一个百分比。 - 2.11 和+2.11 之间z分的概率约为96.52%。

Review
::回顾

Find the probabilities, use the table from the lesson or an online resource.
::查找概率, 使用课程中的表格或在线资源。

What is the probability of a z -score between +1.99 and +2.02?
::+1.99 和 +2.02 之间的z分数概率是多少?

What is the probability of a z -score between -1.99 and +2.02?
::-1.99和+2.02之间z分数的概率是多少?

What is the probability of a z -score between -1.20 and -1.97?
::-1.20和1.97之间 z-score 的概率是多少?

What is the probability of a z -score between +2.33 and-0.97?
::+2.33和-0.97之间z-分数的概率是多少?

What is the probability of a z -score greater than +0.09?
::z-score 大于 + 0.09 的概率是多少?

What is the probability of a z -score greater than -0.02?
::z-score 大于 - 0.02 的概率是多少?

What is $\begin{aligned} P (1.42 < Z < 2.01) \end{aligned}$ ?
::P(1.422.01)是什么?

What is $\begin{aligned} P (1.77 < Z < 2.22) \end{aligned}$ ?
::P(1.77)2.22是什么?

What is $\begin{aligned} P (- 2.33 < Z < - 1.19) \end{aligned}$ ?
::P(2.33)1.19是什么?

What is $\begin{aligned} P (- 3.01 < Z < - 0.71) \end{aligned}$ ?
::什么是P(3.01)0.71?

What is $\begin{aligned} P (2.66 < Z < 3.71) \end{aligned}$ ?
::P(2.66)3.71是什么?

What is the probability of the random occurrence of a value between 56 and 61 from a normally distributed population with mean 62 and standard deviation 4.5?
::从平均为62和标准偏差4.5的正常分布人口中随机产生56至61个数值的概率是多少?

What is the probability of a value between 301 and 329, assuming a normally distributed set with mean 290 and standard deviation 32?
::假定正常分配的数值为290和标准偏差32,301和329之间的数值的概率是多少?

What is the probability of getting a value between 1.2 and 2.3 from the random output of a normally distributed set with $\begin{aligned} μ = 2.6 \end{aligned}$ and $\begin{aligned} σ = .9 \end{aligned}$ ?
::从通常分配的2.6和.9的随机输出中获取1.2和2.3之间的值的概率是多少?

Review (Answers)
::回顾(答复)

Click to see the answer key or go to the Table of Contents and click on the Answer Key under the 'Other Versions' option.
::单击可查看答题键, 或转到目录中, 单击“ 其他版本” 选项下的答题键。

章节大纲

Z-Scores ::零分数

Calculating Probability ::计算概率

Earlier Problem Revisited ::重审先前的问题

Examples ::实例

Example 1 ::例1

Example 2 ::例2

Example 3 ::例3

Review ::回顾

Review (Answers) ::回顾(答复)