节: Z分点二 | 9.4 Z分二

章节大纲

Knowing the z -score of a given value is great, but what can you do with it? How does a z -score relate to probability? What is the probability of occurrence of a z -score less than +2.47?
::了解给定值的 z 分数是巨大的, 但是你能用它做什么呢? z 分数与概率有什么关系? z 分数小于 +2. 47 的概率是多少?

Z-Scores
::零分数

Since z -scores are a measure of the number of SD’s between a value and the mean, they can be used to calculate probability by comparing the location of the z -score to the area under a normal curve either to the left or right. The area can be calculated using calculus, but we will just use a table to look up the area.
::z-scores是衡量数值和平均值之间SD数字的尺度,因此,它们可以通过将z-score的位置与正常曲线下的区域或向左或向右比较,用来计算概率。这个区域可以用微积分来计算,但我们只需要用一张表格来查看这个区域。

I believe that the concept of comparing z -scores to probability is most easily understood with a graphic like the one we used in the lesson on the , so I included one below. Be sure to review the Examples to see how the scores work.
::我认为,将z-scores与概率比较的概念最容易理解,其图形就像我们在关于这个的教训中使用的图形,所以我下面包括一个。你一定要审查这些例子,看看分数是如何运作的。

Like the graphic we viewed in the Empirical Rule lesson, this one only provides probability percentages for integer values of z -scores (standard deviations). In order to find the values for z -scores that aren’t integers, you can use a table like the one below. To find the value associated with a given z -score, you find the first decimal of your z -score on the left or right side and then the 2 ^nd decimal of your z -score across the top or bottom of the table. Where they intersect you will find the decimal expression of the percentage of values that are less than your sample (see the first example).
::和我们在“经验规则”课中看到的图形一样,这里只提供z-crows整数值(标准偏差)的概率百分率。为了找到非整数的z-crows的值,您可以使用下面的表格。要找到与给定z-score相关的数值,您可以在左侧或右侧找到z-crows的第一个小数点,然后在表格上方或下方找到z-crows的第二小数点。在它们相互交错时,您会发现比样本少的数值百分比的小数表示(见第一个例子 ) 。

Z	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	Z
0.0	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	0.6554	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	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	Z

Z -score tables like the one above describe the probability that a given value, or any value less than it, will occur in a given set. This particular table assumes you are looking to find the probability associated with a positive z -score. You may have additional work to do if the z -score is negative.
::类似上表的Z- 位数表描述给定值或低于给定值的任何值在给定集中的概率。该特定表格假设您正在寻找找到正兹- 位数的概率。如果z- 位数为负值,您可能还要做更多的工作。

To find the percentage of values greater than a negative Z score, just look up the matching positive Z score value.
::要找到大于负Z分数的值百分比, 只要查查正Z分数的匹配值。

To find the percentage of values less than a negative z -score, subtract the chart value from 1.
::要找到低于负z-分数的值百分比,请从 1 中减去图表值。

To find the percentage of values greater than a positive z -score, subtract the chart value from 1.
::要找到大于正兹分数的值百分比,请从 1 中减去图表值。

Calculating Probability
::计算概率

1. What is the probability that a value with a z -score less than 2.47 will occur in a normal distribution ?
::1. 在正常分布中,z-score 小于2.47的数值会发生的可能性有多大?

Scroll up to the table above and find “2.4” on the left or right side. Now move across the table to “0.07” on the top or bottom, and record the value in the cell: 0.9932 . That tells us that 99.32% of values in the set are at or below a z-score of 2.47.
::滚动到上面的表格, 在左侧或右侧找到“ 244 ” 。现在移动到表格上方或下方的“ 0.07 ” , 记录单元格中的值: 0.9932。这告诉我们, 集中的99.32%的值位于或低于2.47 的 az-score 。

2. What is the probability that a value with a z -score greater than 1.53 will occur in a normal distribution?
::2. z-score大于1.53的数值在正常分布中发生的可能性有多大?

Scroll up to the table of z -score probabilities again and find the intersection between 1.5 on the left or right and 3 on the top or bottom, record the value in the cell: 0.937 .
::向上滚动到z-core概率表,发现在左或右1.5与上或下3之间的交叉点,记录单元格中的值:0.937。

That decimal lets us know that 93.7% of values in the set are below the z -score of 1.53. To find the percentage that is above that value, we subtract 0.937 from 1.0 (or 93.7% from 100%), to get 0.063 or 6.3% .
::这个小数点让我们知道这组中93.7%的值低于1.53的z点。要找到高于该值的百分比,我们从1.0中减去0.937(或从100%中减去93.7%),以获得0.063或6.3%。

3. What is the probability of a random selection being less than 3.65, given a normal distribution with $\begin{align*}\mu=5\end{align*}$ and $\begin{align*}\sigma=2.2\end{align*}$ ?
::3. 随机选择少于3.65的概率是多少?

This question requires us to first find the z -score for the value 3.65, then calculate the percentage of values below that z -score from a reference.
::这个问题要求我们首先找到3.65值的z-分数,然后从参考中计算出低于z-分数的数值百分比。

1. Find the z -score for 3.65, using the z -score formula: $\begin{align*}\frac{(x-\mu)}{\sigma}\end{align*}$
::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-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-x-xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

\begin{aligned} \frac{3.65 - 5}{2.2} = \frac{- 1.35}{2.2} \approx - 0.61 \end{aligned}

$\begin{align*}\frac{3.65-5}{2.2}=\frac{-1.35}{2.2}\approx -0.61\end{align*}$

2. Now we can scroll up to our z -score reference above and find the intersection of 0.6 and 0.01, which should be .7291 .
::2. 现在我们可以滚动到上文提到的z-score参考点,找到0.6和0.01的交叉点,即0.7291。

3. Since this is a negative z -score, and we want the percentage of values below it, we subtract that decimal from 1.0 (reference the three steps highlighted by bullet points below the chart if you didn’t recall this), to get $\begin{align*}1-.7291=.2709\end{align*}$
::3. 由于这是一个负z-分数,我们需要低于该分数的百分比,因此我们从1.0中减去小数点(用图表下方的圆点标明三个步骤,如果你不记得这一点的话),得出1-7291=2709。

There is approximately a 27.09% probability that a value less than 3.65 would occur from a random selection of a normal distribution with mean 5 and standard deviation 2. 2.
::大约27.09%的概率是,从随机选择的正常分布中,平均5和标准偏差2.2的值低于3.65的值。

Earlier Problem Revisited

::重审先前的问题

Knowing the z-score of a given value is great, but what can you do with it? How does a z-score relate to probability? What is the probability of occurrence of a z-score less than 2.47?
::了解给定值的 z 分数是巨大的, 但是你能用它做什么呢? z 分数与概率有什么关系? z 分数小于 2. 47 的发生概率是多少?

A z-score lets you calculate the probability that a randomly selected value will be greater or less than a particular value in a set.
::z-score 允许您计算随机选择值大于或小于一组中特定值的概率。

To find the probability of a z-score below +2.47, using a reference such as the table in the lesson above:
::使用上文课目中的表格等参考值,发现z-score低于+2.47的概率:

Find 2.4 on the left or right side
::在左侧或右侧找到2.4

Move across to 0.07 on the top or bottom.
::移动到上方或下方的 0.07 。

The cell you arrive at says: 0.9932, which means that apx 99.32% of the values in a normal distribution will occur below a z -score of 2.47.
::到达的细胞表示:0.9932, 也就是说正常分布中99.32%的Apx值将低于2.47的z分。

Examples
::实例

Example 1
::例1

What is the probability of occurrence of a value with a z-score greater than 1.24?
::z-score 大于 1. 24 的值发生概率是多少?

Since this is a positive z -score, we can use the value for $\begin{align*}z=1.24\end{align*}$ directly from the table, and just express it as a percentage: 0.8925 or 89.25%
::由于这是一个正z-分数,我们可以直接从表格中使用z=1.24的值,并以百分比表示:0.8925或89.25%。

Example 2
::例2

What is the probability of $\begin{align*}z<-.23\end{align*}$ ?
::23的概率是多少?

This is a negative z -score, and we want the percentage of values greater than it, so we need to subtract the value for $\begin{align*}z=+0.23\end{align*}$ from 1: $\begin{align*}1 - 0.591 = .409\end{align*}$ or 40.9%
::这是一个负 Z- z- score, 我们希望数值的百分比大于它, 所以我们需要从1: 1 - 0. 591= 409 或 40.9% 中减去 z 0. 23 的值。

Example 3
::例3

What is $\begin{align*}P(Z<2.13)\end{align*}$
::什么是P(2.13)

This is a positive z -score, and we need the percentage of values below it, so we can use the percentage associated with $\begin{align*}z=+2.13\end{align*}$ directly from the table: 0.9834 or 98.34%
::这是一个正z-分数,我们需要低于该值的百分比, 所以我们可以直接使用与z2.13有关的百分比, 即0.9834, 即98.34%。

Review
::回顾

Find the probabilities, use the table from the lesson above.
::查找概率,使用上面的课目中的表格。

What is the probability of a z -score less than +2.02?
::z-score 小于+2.02的概率是多少?

What is the probability of a z -score greater than +2.02?
::z-score 大于+2.02的概率是多少?

What is the probability of a z -score less than -1.97?
::z-score小于-1.97的概率是多少?

What is the probability of a z -score greater than -1.97?
::z-score 大于 -1.97的概率是多少?

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

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

What is $\begin{align*}P(Z<1.71)\end{align*}$ ?
::什么是P(1.71)?

What is $\begin{align*}P(Z>2.22)\end{align*}$ ?
::什么是P(2.22)?

What is $\begin{align*}P(Z<-1.19)\end{align*}$ ?
::什么是P(1.19)?

What is $\begin{align*}P(Z>-2.71)\end{align*}$ ?
::什么是P(2.71)?

What is $\begin{align*}P(Z<3.71)\end{align*}$ ?
::什么是P(3.71)?

What is the probability of the random occurrence of a value greater than 56 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出现超过56个数值的概率是多少?

What is the probability of a value of 329 or greater, assuming a normally distributed set with mean 290 and standard deviation 32?
::假设通常分配的数值为290和标准差32,329或以上数值的概率有多大?

What is the probability of getting a value below 1.2 from therandom output of a normally distributed set with $\begin{align*}\mu=2.6\end{align*}$ and $\begin{align*}\sigma=.9\end{align*}$ ?
::从通常分配的 2.6 和 .9 组合的随机输出中获得低于1.2 的值概率是多少?

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)