---
description: 密度曲线的尾部
---

# 5-4. Areas of Tails of Distributions

> &#x54;_&#x68;e_ **left tail** _of a density curve_  $$y=f(x)$$ _of a continuous random variable_ _X_ _cut off by a value_ $$x^*$$ _of_ _X_ _is the region under the curve that is to the left of_ $$x^*$$ , _as shown by the shading in Figure  "Right and Left Tails of a Distribution" (a). The_ **right tail** _cut off by_ $$x^*$$ _is defined similarly, as indicated by the shading in Figure   "Right and Left Tails of a Distribution" (b)_.

![](https://saylordotorg.github.io/text_introductory-statistics/section_09/bbb58b35dd040bcf4d8b4d3f7c72b679.jpg)

## 1. Tails of the Standard Normal Distribution

At times it is important to be able to solve the kind of problem illustrated by the figure as follows. We have a certain specific area in mind, in this case the area 0.0125 of the shaded region in the figure, and we want to find the value $$z^*$$ of $$Z$$ that produces it. This is exactly the reverse of the kind of problems encountered so far. Instead of knowing a value $$z^*$$ of $$Z$$ and finding a corresponding area, we know the area and want to find $$z^*.$$ In the case at hand, in the terminology of the definition just above, we wish to find the value $$z^*$$ that cuts off a left tail of area 0.0125 in the standard normal distribution.

![Z Value that Produces a Known Area 0.0125](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1NAJz8AKVZJECIGng%2F-Lr1NqTJCTYX6kUn88Lr%2Fimage.png?alt=media\&token=0f32c549-40e5-449f-9468-91accb6c2963)

**EXAMPLE 17.** Find the value $$z^*$$ of $$Z$$ as determined by the above figure: the value $$z^*$$ that cuts off a left tail of area 0.0125 in the standard normal distribution. In symbols, find the number $$z^*$$ such that $$P(Z<z^*)=0.0125.$$&#x20;

**\[ Solution ]**

{% tabs %}
{% tab title="R Code" %}
```
qnorm(0.0125)
```
{% endtab %}

{% tab title="z*" %}
```
> qnorm(0.0125)
[1] -2.241403
```
{% endtab %}

{% tab title="Plot" %}
```
pnormGC(-2.241403, region="below", graph=TRUE)
```

![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1jKAEKpjsGqnaQFbc%2F-Lr1jgokh3VZCJf5qanh%2Fimage.png?alt=media\&token=c0f12aa6-d1e0-4331-8dec-e79ca91a0f3f)
{% endtab %}
{% endtabs %}



**EXAMPLE 18.** Find the value $$z^*$$of $$Z$$ as determined by  Figure : the value $$z^*$$ that cuts off a right tail of area 0.0250 in the standard normal distribution. In symbols, find the number $$z^*$$ such that $$P(Z>z^*)=0.0250.$$&#x20;

![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1QXHku1RuHu2G47ox%2F-Lr1Qvj0cBMyAURGe7rE%2Fimage.png?alt=media\&token=8f592b0d-8750-42b8-abda-d434cdcdeed7)

**\[ Solution ]**

* _The value of the standard normal random variable_ $$Z$$ _that cuts off **a right tail of area**_ _**c**_ _is denoted_ $$z_c$$ . _By symmetry, value of_ $$Z$$ _that cuts off a left tail of area_ _c_ _is_ $$−z_c.$$ _See the below "The Numbers "_.

![](https://saylordotorg.github.io/text_introductory-statistics/section_09/e4499588e283aa8c3339ac767a95ccef.jpg)

{% tabs %}
{% tab title="R Code" %}
```
qnorm(0.9750)
```
{% endtab %}

{% tab title="z*" %}
```
> qnorm(0.9750)
[1] 1.959964
```
{% endtab %}

{% tab title="Plot" %}
```
pnormGC(c(-1.959964,1.959964), region="outside", graph=TRUE)
```

![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1jKAEKpjsGqnaQFbc%2F-Lr1kDW-8qp3e_7gZJnq%2Fimage.png?alt=media\&token=f3284f5e-235b-4d4d-a80e-5e2a1cecfb49)
{% endtab %}
{% endtabs %}



**EXAMPLE 19.** Find $$z_{.01}$$ and $$−z_{.01}$$ , the values of $$Z$$ that cut off right and left tails of area 0.01 in the standard normal distribution.

**\[ Solution ]**

![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1RxHkW3Pp7kODdb0d%2F-Lr1RziUft3_8lzKBglh%2Fimage.png?alt=media\&token=9904d8c5-d240-4028-a121-be44eaa3809c)

{% tabs %}
{% tab title="R Code" %}
```
qnorm(0.99)
```
{% endtab %}

{% tab title="z*" %}
```
> qnorm(0.99)
[1] 2.326348
```
{% endtab %}

{% tab title="Plot" %}
```
pnormGC(2.326348, region="above", graph=TRUE)
```

![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1jKAEKpjsGqnaQFbc%2F-Lr1kWsIUsnv0UrMYJBy%2Fimage.png?alt=media\&token=495c0b3e-120f-44af-acaa-8d152bcd09bf)
{% endtab %}
{% endtabs %}

## **2.** Tails of General Normal Distributions

Suppose $$X$$ is a normally distributed random variable with mean $$μ$$ and standard deviation $$σ$$ . To find the value $$x^*$$ of $$X$$ that cuts off a left or right tail of area _c_ in the distribution of $$X$$ :

1. find the value $$z^*$$ of $$Z$$ that cuts off a left or right tail of area _c_ in the standard normal distribution;
2. $$z^*$$ is the _z_-score of $$x^*$$; compute $$x^*$$ using the destandardization formula   $$x^*=μ+z^* σ$$



**EXAMPLE 20.**  Find  $$x^*$$ such that $$P(X<x*)=0.9332$$ , where $$X$$ is a normal random variable with mean $$μ = 10$$ and standard deviation $$σ = 2.5$$ .

**\[ Solution ]**

![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1RxHkW3Pp7kODdb0d%2F-Lr1SwtkM5xMQgu7w7ls%2Fimage.png?alt=media\&token=2e13e3f6-715b-461b-92fd-a275d34296cf)

{% tabs %}
{% tab title="R Code" %}
```
qnorm(0.9332, mean=10, sd=2.5)
```
{% endtab %}

{% tab title="x*" %}
```
> qnorm(0.9332, mean=10, sd=2.5)
[1] 13.75014
```
{% endtab %}

{% tab title="Plot" %}
```
pnormGC(13.75014, region="below", mean=10, sd=2.5, graph=TRUE)
```

![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1jKAEKpjsGqnaQFbc%2F-Lr1ksueXvXm6L5K2RhB%2Fimage.png?alt=media\&token=a966b3eb-5e5e-4803-90df-e8a46df0a2a4)
{% endtab %}
{% endtabs %}



**EXAMPLE 21.**  Find $$x^*$$ such that $$P(X>x^*)=0.65$$ , where _X_ is a normal random variable with mean $$μ = 175$$ and standard deviation $$σ = 12$$ .

**\[ Solution ]**

![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1TGQvXggCK68as9yD%2F-Lr1TOwgli5QXxgWwOuo%2Fimage.png?alt=media\&token=5c3865c3-aa4a-4f67-afa1-ae6876c0f868)

{% tabs %}
{% tab title="R Code" %}
```
qnorm(1-0.65, mean=175, sd=12)
```
{% endtab %}

{% tab title="x*" %}
```
> qnorm(0.35, mean=175, sd=12)
[1] 170.3762
```
{% endtab %}

{% tab title="Plot" %}
```
pnormGC(170.3762, region="above", mean=175, sd=12, graph=TRUE)
```

![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1jKAEKpjsGqnaQFbc%2F-Lr1lG9b7uIentfog7aQ%2Fimage.png?alt=media\&token=5c58ae9e-44f1-4b79-a87b-98e73dd938fb)
{% endtab %}
{% endtabs %}



**EXAMPLE 22.**  Scores on a standardized college entrance examination (_CEE_) are normally distributed with mean 510 and standard deviation 60. A selective university decides to give serious consideration for admission to applicants whose _CEE_ scores are in the top 5% of all _CEE_ scores. Find the minimum score that meets this criterion for serious consideration for admission.

**\[ Solution ]**



![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1TGQvXggCK68as9yD%2F-Lr1TVYrLSwAmsXlXVpt%2Fimage.png?alt=media\&token=7960112d-f0fa-4dad-a9e7-604e812eb53e)

{% tabs %}
{% tab title="R Code" %}
```
qnorm(1-0.05, mean=510, sd=60)
```
{% endtab %}

{% tab title="x*" %}
```
> qnorm(1-0.05, mean=510, sd=60)
[1] 608.6912
```
{% endtab %}

{% tab title="Plot" %}
```
pnormGC(608.6912, region="above", mean=510, sd=60, graph=TRUE)
```

![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1jKAEKpjsGqnaQFbc%2F-Lr1lWkR1li_IiIksbbP%2Fimage.png?alt=media\&token=03eab8b7-cdb9-4acc-8274-4d5a5fa44d6c)
{% endtab %}
{% endtabs %}



**EXAMPLE 23.**  All boys at a military school must run a fixed course as fast as they can as part of a physical examination. Finishing times are normally distributed with mean 29 minutes and standard deviation 2 minutes. The middle 75% of all finishing times are classified as “average.” Find the range of times that are average finishing times by this definition.

**\[ Solution ]**

![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1TGQvXggCK68as9yD%2F-Lr1TfpSAhqajqjKL547%2Fimage.png?alt=media\&token=bc8a7645-9df6-46f3-82c8-5ae55bddd0e1)

{% tabs %}
{% tab title="R Code" %}
```
a <- qnorm(0.125, mean=29, sd=2) ; a
b <- qnorm(1-0.125, mean=29, sd=2) ; b
```
{% endtab %}

{% tab title="x*" %}
```
> a <- qnorm(0.125, mean=29, sd=2) ; a
[1] 26.6993
> b <- qnorm(1-0.125, mean=29, sd=2) ; b
[1] 31.3007
```
{% endtab %}

{% tab title="Plot" %}
```
pnormGC(c(26.6993, 31.3007), region="between", mean=29, sd=2, graph=TRUE)
abline(h=0, col="gray")
```

![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1jKAEKpjsGqnaQFbc%2F-Lr1m1rOI5-ZFtgz3EYo%2Fimage.png?alt=media\&token=eabad35e-84bb-453c-97ad-68cc49f66bef)
{% endtab %}
{% endtabs %}

## **3. Using qnorm() and pnormGC in R**&#x20;

$$X$$\~ $$norm(\mu, \sigma)$$&#x20;

1\) find $$z^*$$ or $$x^*$$&#x20;

`qnorm(p, mean = , sd = )`

2\) plotting (See 5-3. [pnormGC()](https://kmis.gitbook.io/statistics/chapter-5.-continuous-random-variables/5-3.-probability-computations-for-general-normal-random-variables#3-using-pnormgc))

`pnormGC(p, region= , mean=, sd =, graph = )`



**EXAMPLE 24.**    $$P(Z<z^*)=0.0125.$$&#x20;

{% tabs %}
{% tab title="R Code" %}
```
qnorm(0.0125)
```
{% endtab %}

{% tab title="z*" %}
```
> qnorm(0.0125)
[1] -2.241403
```
{% endtab %}

{% tab title="R Code" %}
```
pnormGC(-2.241403, graph=TRUE)
```
{% endtab %}

{% tab title="Plot" %}
![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1_bcYu4r-Upihg3ej%2F-Lr1hUHmQIft-2F9d7BW%2Fimage.png?alt=media\&token=231f9c8b-6ba6-483b-82de-366f405a0b75)
{% endtab %}
{% endtabs %}



**EXAMPLE 25.** Find  $$x^*$$ such that $$P(X<x*)=0.9332$$ , where $$X$$ is a normal random variable with mean $$μ = 10$$ and standard deviation $$σ = 2.5$$ .

{% tabs %}
{% tab title="R Code" %}
```
x <- qnorm(0.9332, mean=10, sd=2.5); x <- round(x, 2); x
pnormGC(x, region = "below", mean=10, sd=2.5, graph=TRUE)
```
{% endtab %}

{% tab title="x*" %}
```
> x <- qnorm(0.9332, mean=10, sd=2.5); x <- round(x, 2); x
[1] 13.75
```
{% endtab %}

{% tab title="Plot" %}
![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1hgx3Nb6EKdd497lV%2F-Lr1ibFcprsj38ibY0C0%2Fimage.png?alt=media\&token=d11512f7-7b57-4ee6-9a5f-34ea93c99f5c)
{% endtab %}
{% endtabs %}



**密度曲线的尾部** tail of a density curve \
**一般正态分布** General Normal Distributions
