第4講正態(tài)分布及其應(yīng)用2004

1、exciting spirit, to , not up of urgency, and slow shall not of sense, and sat not of sense, moments has a lying sake thing, and sat with rules of procedure of the, and station with officer of momentum, keep a ran of drive, and rob of consciousness, and race of momentum, and spell of courage, to fire

2、 as of passion input work. the only way to ensure the healthy development of the cause, did not live up to the organizations trust and the porcelain people live up to expectations. (b) the practice of three suns, requires that we adhere to the problem oriented, improving the effectiveness of governm

3、ent. an idea that i have said many times, talking about issues is much more useful than the talk. because the results are in the past, talking about there, but the problem is eternal, not free, ubiquitous, if not resolved, will never go away on their own, and then only by solving a problem, work in

4、order to move forward, before moving forward progress of society. marxist philosophy tells us that catch to catch the nature of the problem, grab the contradiction to catch the main contradiction. this requires us not only to be good at resolving individual, specific issues, also to be good at looki

5、ng through the phenomenon of nature, found by accident law, so as to catch common, widespread problem. we some place and sector of comrade, work starting point is good of, wants to developed some policy, wants to help enterprise development, and for masses do points things, but is careless make a kn

6、ife cut pot, upper and lower general rough, led to file introduced hou hard implementation, some even help has pour busy, to enterprise increased has burden, damage has masses of interests; some place in do city development planning or industry development planning shi, to self for center, pursuit w

7、ishful thinking, not focused on and city general planning phase convergence, final conflict for planning land cannot be submitted for approval, the project cant fall, wasted time, energy and money. in addition, there are many comrades often complain about difficult, superiors know the problem instea

8、d of trying, often report to prepare funding conditions. in fact, if any conditions are prepared properly, who called that is not the same?, conversely, if you are really looking for the right reasons, there is a good solution, the government will provide support in terms of manpower, material and f

9、inancial resources. government agencies set is a science, departments does the difference between good and bad, the difference only work well. can get the attention and support, the key achievements and contributions to the department. we carry out government work, do anything to stick to problem-or

10、iented, be sure to have clear links to, take practical measures to achieve real results. this requires that we must reach out before the decision, . help, in order to help companies ride out the storm. last month, the city government carried out a steady growth and development of research, many comr

11、ades told me that, in the course of land, some farmers dont understand municipal government decisions, even plain, block construction. on this issue, we should not only see the appearance of land expropriation, but on the other hand, thought the masses why conflict? worried that without a job, there

12、 are still other worries? can then take specific measures to resolve it. for example, baic changhe automobile new bases in the land acquisition process and for surrounding farmers making a special people club sector statistics, finds out, and organizations develop the skills training, government 正態(tài)分

13、布及其應(yīng)用一、 正態(tài)分布的概念和特征根據(jù)頻數(shù)表資料繪制成直方圖，可以設(shè)想，如果將觀察人數(shù)逐漸增多，線段不斷分細(xì)，圖中直條將逐漸變窄，其頂端將逐漸接近一條光滑的曲線，這條曲線稱為頻數(shù)曲線或頻率曲線，略呈鐘型，兩頭低，中間高，左右對(duì)稱，近似于數(shù)學(xué)上的正態(tài)分布（normal distribution）。由于頻率的總和等于100%或1，故橫軸上曲線下的面積等于100%或1。 正態(tài)分布是一種橫重要的連續(xù)型分布，在生物統(tǒng)計(jì)學(xué)中，占有極其重要的地位。許多生物學(xué)現(xiàn)象所產(chǎn)生的數(shù)據(jù)，都服從正態(tài)分布。 1、正態(tài)分布的圖形 有了正態(tài)分布的密度函數(shù)f(x)，即正態(tài)分布的方程，就可給出圖形 -s m s 上式中右側(cè)m為均

14、數(shù)，s為標(biāo)準(zhǔn)差，x為自變量。當(dāng)x確定后，就可由此式求得其密度函數(shù)f(x),也就是相應(yīng)的縱坐標(biāo)的高度。所以，已知m和 s , 就能繪出正態(tài)曲線的圖形。2、正態(tài)分布的特征（1） 正態(tài)分布以m為中心，左右對(duì)稱。（2）正態(tài)分布有兩個(gè)參數(shù)，即m和s。m是位置參數(shù)，當(dāng)s恒定后，m 越大，則曲線沿橫軸越向右移動(dòng)；m 越小，則曲線沿橫軸越向左移動(dòng)。s是變異參數(shù)，當(dāng)m 恒定時(shí)，s越大，表示數(shù)據(jù)越分散，曲線越“胖”；s越小，表示數(shù)據(jù)越分散，曲線越“瘦”。（3）正態(tài)分布的偏斜度 g1=0，峭度 g2=0為了應(yīng)用方便，常將上式作如下變換， 也就是將原點(diǎn)移到 m 的位置，使橫軸尺度以s為單位，使m=0，s=1，則正態(tài)分

15、布變換為標(biāo)準(zhǔn)正態(tài)分布。(standard normal distribution) ，u稱為標(biāo)準(zhǔn)正態(tài)離差（standard normal deviate）標(biāo)準(zhǔn)正態(tài)分布的密度函數(shù)為： 一般用n（m，s2）表示均方為m，方差為s2的正態(tài)分布。于是標(biāo)準(zhǔn)正態(tài)分布用n（0，1）表示。 -3 -2 -1 0 1 2 3 68.26% 95.45% 99.74%下列一些值很重要，應(yīng)予記憶：u= -1 到 u=1 面積=0.6827u= -1.96 到 u=1.96 面積=0.9500u= -2.58 到 u=2.58 面積=0.9900標(biāo)準(zhǔn)正態(tài)分布有以下特征：（1） 在u=0時(shí)，j（u）達(dá)到最大值。（2）

16、當(dāng)u無(wú)論向哪個(gè)方向遠(yuǎn)離0時(shí)，j（u）的值都減小。（3） 曲線關(guān)于y軸對(duì)稱，即j（u）=j（-u）。（4） 曲線和橫軸所夾的面積等于1。二、標(biāo)準(zhǔn)正態(tài)曲線下面積的計(jì)算 p(u1 u u2) =j(u2) - j(u1)舉例已知高梁品種“三尺三”的株高服從正態(tài)分布n（156.2,4.822）,求： （1）x161cm 的概率；（3）x在152162cm 間的概率。（1） 依題意：所以，“三尺三”的株高低于161cm的概率為0.84134。（2） 依題意得： （3）依題意得：例題、某地農(nóng)民3539歲男性膽固醇的測(cè)定（毫克%）膽固醇分組頻數(shù)f累計(jì)頻數(shù)累計(jì)頻率85105125145165185205225

17、2455202840392410535255393132156166171174 2.8714.3730.4653.4575.8689.6695.4098.28100.00計(jì)算 1、總膽固醇水平在110mg150mg之間所占面積。2、總膽固醇水平在130mg190mg之間所占面積。解：計(jì)算 三、小結(jié)1、正態(tài)分布是一種連續(xù)性的分布，不少醫(yī)學(xué)現(xiàn)象服從正態(tài)分布或近似正態(tài)分布（如同性別、同年齡兒童的身高，同性別健康成人的紅細(xì)胞數(shù)、血紅蛋白量、脈搏數(shù)等，以及實(shí)驗(yàn)中的隨機(jī)誤差等）；或經(jīng)變量變換轉(zhuǎn)換為正態(tài)分布（如某些病人的潛伏期以及醫(yī)院病人住院天數(shù)等），可按正態(tài)分布規(guī)律來(lái)處理，它也是許多統(tǒng)計(jì)方法的理論基礎(chǔ)。

18、2、正態(tài)分布的特征是：（1） 曲線在橫軸上方，均數(shù)處最高；（2） 以均數(shù)為中心，左右對(duì)稱；（3） 確定正態(tài)分布的兩個(gè)參數(shù)是均數(shù)m和標(biāo)準(zhǔn)差s。正態(tài)分布用n（m，s2）表示，為了應(yīng)用的方便，常對(duì)變量x作u = (x-m)/s 使 m=0，s=1，則正態(tài)分布轉(zhuǎn)換為標(biāo)準(zhǔn)正態(tài)分布用n（0，1）表示。3、運(yùn)用正態(tài)曲線下面積的分布規(guī)律，可計(jì)算醫(yī)學(xué)參考值范圍和質(zhì)量控制等。 總體均數(shù)的估計(jì)和假設(shè)檢驗(yàn)醫(yī)學(xué)統(tǒng)計(jì)學(xué)的最基本問(wèn)題是研究總體與樣本的關(guān)系。總體與樣本的關(guān)系，可從兩方面研究：（4） 由已知的總體，研究樣本的分布規(guī)律，即由總體到樣本的研究過(guò)程；（5） 由樣本如何去推斷未知的總體，屬于從樣本到總體的研究過(guò)程。1、

19、 從一個(gè)正態(tài)總體中抽取的樣本統(tǒng)計(jì)量分布 生物學(xué)中遇到最多的是正態(tài)總體。對(duì)于正態(tài)總體，可以用數(shù)學(xué)推演的方法，得出嚴(yán)格的樣本統(tǒng)計(jì)量的抽樣分布，有了嚴(yán)格的樣本分布規(guī)律，就可以由樣本來(lái)推斷總體了。標(biāo)準(zhǔn)差已知時(shí)的樣本平均數(shù)的分布u分布平均數(shù)為m，標(biāo)準(zhǔn)差為s的正態(tài)總體中，獨(dú)立隨機(jī)的抽取含量為n的樣本，則：樣本平均數(shù)是一服從正態(tài)分布的隨機(jī)變量，記為。將平均數(shù)標(biāo)準(zhǔn)化，則：其中的分母稱為平均數(shù)的標(biāo)準(zhǔn)誤差（standard error of mean）。如果變量是正態(tài)的或近似正態(tài)的，則標(biāo)準(zhǔn)化的變量服從或近似地服從n（0，1）分布。1、標(biāo)準(zhǔn)差已知時(shí)的樣本平均數(shù)的分布t分布 (t-distribution)若上式中的

20、s是未知的，可用樣本標(biāo)準(zhǔn)差代替總體標(biāo)準(zhǔn)差，此時(shí)標(biāo)準(zhǔn)化變量并不服從正態(tài)分布，而服從具有n-1自由度的t分布。 t分布也是一種對(duì)稱分布，它只有一個(gè)參數(shù)，即自由度（degree of freedom）t 分布與標(biāo)準(zhǔn)正態(tài)分布相比有以下特征：（1） 二者都是單峰分布，以0為中心，左右兩側(cè)對(duì)稱。（2） t分布的峰部較矮而尾部翹得較高，說(shuō)明遠(yuǎn)側(cè)t 值的個(gè)數(shù)相對(duì)較多，即尾部面積（概率p）較大。自由度越小這種情況越明顯。n逐漸增大時(shí)，t分布逐漸逼近標(biāo)準(zhǔn)正態(tài)分布；當(dāng) n= 時(shí)，t分布就完全成為標(biāo)準(zhǔn)正態(tài)分布了。2、統(tǒng)計(jì)推斷 統(tǒng)計(jì)推斷是討論如何通過(guò)樣本去推斷總體。由樣本推斷總體是以各種樣本統(tǒng)計(jì)量的抽樣分布為基礎(chǔ)的。

21、對(duì)總體做統(tǒng)計(jì)推斷（statistical inference ）, 可以通過(guò)兩條途徑進(jìn)行。1. 總體參數(shù)估計(jì)（estimation of population parameter）通過(guò)樣本統(tǒng)計(jì)量估計(jì)總體參數(shù)；2. 統(tǒng)計(jì)假設(shè)檢驗(yàn)（statistical test of hypothesis ）首先對(duì)所估計(jì)的總體提出一個(gè)假設(shè)，如: 假設(shè)這個(gè)總體的平均數(shù)m 等于某個(gè)值m 0 ，然后，通過(guò)樣本均數(shù)據(jù)去推斷這個(gè)假設(shè)是否可以接受，如果可以接受，樣本很可能抽自這個(gè)總體；否則很可能不是抽自這個(gè)總體。這兩種不同的統(tǒng)計(jì)分析方法在實(shí)際工作中可互相參照使用。1、 總體均數(shù)的估計(jì)區(qū)間估計(jì)（internal estima

22、tion）即按一定的概率估計(jì)總體均數(shù)在哪個(gè)范圍?？筛鶕?jù)資料的條件選用不同的方法。（1）s未知時(shí)。一般用t分布的原理作區(qū)間估計(jì)。不 根據(jù) 于是得可信度為1-a 時(shí)，計(jì)算總體均數(shù)可信區(qū)間的通式為：習(xí)慣上，常取1-a =0.95， 即95%可信區(qū)間；或取1-a =0.99， 即99%可信區(qū)間。例1、 對(duì)某人群隨機(jī)抽取20人，用某批號(hào)的結(jié)核菌素作皮試，平均侵潤(rùn)直徑為10.9mm，標(biāo)準(zhǔn)差為3.86mm。問(wèn)這批結(jié)核菌素在該人群中使用時(shí)，皮試的平均侵潤(rùn)直徑的95%可信區(qū)間是多少？解：本例n=20、自由度n=n-1=20-1=19、 a=0.05（雙側(cè)）查附表，得t0.05,19=2.093所以該人群皮試的平

23、均侵潤(rùn)直徑的95%可信區(qū)間為9.1mm12.7mm 。（2）s已知或樣本例數(shù)n足夠大時(shí)，按正態(tài)分布原理作區(qū)間估計(jì)。例2、 由某地成年男子中抽得144人的樣本，求得紅細(xì)胞數(shù)的均數(shù)為5.381012/ l, 標(biāo)準(zhǔn)差為0.441012/ l, 試估計(jì)該地成年男子紅細(xì)胞均數(shù)的95%可信區(qū)間。該地成年男子紅細(xì)胞均數(shù)的95%可信區(qū)間為(5.31, 5.45) 1012/ l。 可信區(qū)間的涵義 可以理解為從總體中作隨機(jī)抽樣，每個(gè)樣本可以算出一個(gè)可信區(qū)間，如95%可信區(qū)間，意味著100次抽樣，算得100個(gè)可信區(qū)間，平均有95個(gè)可信區(qū)間包括總體均數(shù)（估計(jì)正確），只有5個(gè)可信區(qū)間不包括總體均數(shù)（估計(jì)錯(cuò)誤）。5%是

24、小概率事件，實(shí)際發(fā)生的可能性小，因此，在實(shí)際應(yīng)用中就認(rèn)為總體均數(shù)在算得的可信區(qū)間內(nèi), 這種估計(jì)方法會(huì)冒5%犯錯(cuò)的風(fēng)險(xiǎn)。 可信區(qū)間的兩個(gè)要素 一是準(zhǔn)確度，反映在可信度1-a的大小，即區(qū)間包含總體均數(shù)的概率的大小，當(dāng)然愈接近1愈好；二是精度，反映在區(qū)間的長(zhǎng)度，當(dāng)然長(zhǎng)度愈小愈好。在樣本例數(shù)確定的情況下，二者是矛盾的。一般情況下，在可信度確定的情況下，增加樣本例數(shù)，可減少區(qū)間長(zhǎng)度，提高精度。coordination with baic, fight for some of its recruitment quota, used to solve the employment problems of l

25、andless farmers around. due to properly solve the landless farmers worries, our work has been fully understood, guarantee baic changhe automobile demolition and subsequent construction of new base project smoothly. of course, for those who obey the law and regulations, irrational, strong-hold engine

26、ering, sales of building materials of the wrongful act, or to law and regulation, severely and quickly dealt with. there is also a problem, our superiors issued by various departments of various types of files, not mechanically copy, but based on actual jingdezhen, and meet the markets opinion, carr

27、y through the spirit of purpose that is also exploring fresh experience. (c) the practice of three suns, requires that we adhere to the face of contradictions and difficulties, to take in the pioneer. the event is difficult to see as dare to play are the basic qualities of leading cadres, best test

28、leaders spirit of play before the conflict difficult. xi put forward by general secretary of the cadres in five criteria had to take. for a long time, there is a mentality in our cadres at all levels: afraid of problems. of course, the inability to say no. strict is an advantage. but when overdone,

29、it is easy to go to the other extreme cant play. some leaders fear problems, come into conflict, walk around, do not do not solve the problem; once out of the problem, not to solve, but to do everything possible to avoid the question, problem or even big problems said to small problems, problem as s

30、mall no problem. in fact, such leaders will eventually lead to big trouble, this unit will eventually be the bigger problem. there is such a phenomenon, some comrades have already agreed not to implement, not to wait for minutes, a copy of form; oral work assigned by the leadership not to do it, to

31、wait for written instructions. named about program, but cant play, afraid of responsibility. because they dare not play, many gays believe in growing flowers and planting thorn less vulgar philosophy, without the principle of harmony, safe seats, mediocre, dawdle, found the problem is not raised, mo

32、re reluctant to face up to and resolve, until small problems become big problems. practice has proved that numerous small problems become difficult to pack in such a perfunctory questions, numerous projects were delayed waiting for, numerous development opportunities are lost in such a delay. as leading cadres must have a k