# 5-4. Areas of Tails of Distributions > T*he* **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 $$P(Z\ qnorm(0.0125) [1] -2.241403 ``` {% endtab %} {% tab title="Plot" %} ``` pnormGC(-2.241403, region="below", graph=TRUE) ``` ![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1jKAEKpjsGqnaQFbc%2F-Lr1jgokh3VZCJf5qanh%2Fimage.png?alt=media\&token=c0f12aa6-d1e0-4331-8dec-e79ca91a0f3f) {% endtab %} {% endtabs %} **EXAMPLE 18.** Find the value $$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.$$ ![](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 $$P(X\ qnorm(0.9332, mean=10, sd=2.5) [1] 13.75014 ``` {% endtab %} {% tab title="Plot" %} ``` pnormGC(13.75014, region="below", mean=10, sd=2.5, graph=TRUE) ``` ![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1jKAEKpjsGqnaQFbc%2F-Lr1ksueXvXm6L5K2RhB%2Fimage.png?alt=media\&token=a966b3eb-5e5e-4803-90df-e8a46df0a2a4) {% endtab %} {% endtabs %} **EXAMPLE 21.** Find $$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** $$X$$\~ $$norm(\mu, \sigma)$$ 1\) find $$z^*$$ or $$x^*$$ `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.** $$P(Z\ qnorm(0.0125) [1] -2.241403 ``` {% endtab %} {% tab title="R Code" %} ``` pnormGC(-2.241403, graph=TRUE) ``` {% endtab %} {% tab title="Plot" %} ![](https://2234305379-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-Lqmt1Wp54aKlZ5eVcaS%2F-Lr1_bcYu4r-Upihg3ej%2F-Lr1hUHmQIft-2F9d7BW%2Fimage.png?alt=media\&token=231f9c8b-6ba6-483b-82de-366f405a0b75) {% endtab %} {% endtabs %} **EXAMPLE 25.** Find $$x^*$$ such that $$P(X\ x <- 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