【統計學】第六章

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1.什麼是統計量？爲何要引進統計量？統計量中爲何不含任何未知參數？

統計量是樣本中的一個函數。由樣本構造具體的統計量，其實是對樣本所含的整體信息按某種要求進行加工處理，把分散在樣本中的信息集中到統計量的取值上，不一樣的統計推斷問題要求構造不一樣的統計量。
統計量在統計學中具備極其重要的地位，它是統計推斷的基礎。統計量在統計學中的地位至關於隨機變量在機率論中的地位。

2.判斷下列樣本函數中哪些是統計量？哪些不是統計量？

T1​=10(X1​+X2​+...+X10​)​

是統計量

T2​=min(X1​,X2​,...,X10​)

不是統計量

T3​=X10​−μ

不是統計量

T4​=σ(X10​−μ)​

不是統計量

3.什麼是次序統計量

次序統計量稱爲參數估計和假設檢驗的一類重要的統計量。中位數、分位數、四分位數都是次序統計量。

4.什麼是充分統計量

在實際統計推斷中，是用統計量的值進行判斷，而不是由樣本觀測值進行推斷。這就意味着，通過加工後，有了統計量的值便可。在統計學中，假如一個統計量能把含在樣本中有關整體的信息一點都不損失地提取出來，那對保證後邊的統計推斷質量具備重要意義。統計量加工過程當中一點信息都不損失的統計量稱爲充分統計量。

5.什麼是自由度

自由度是統計學中經常使用的一個概念，它具備解釋爲獨立變量的個數，還能夠解釋爲二次型的秩。

6.簡述 x2分佈、t分佈、F分佈及正態分佈之間的關係

1.X2分佈當X趨近於無窮大時，分佈的極限分佈是正態分佈
2.t分佈的密度函數是一偶函數
3.F分佈中，兩個自由度的位置不可互換。若是隨機變量X服從t(n)分佈，則X2服從F(1,n)的F分佈。

7.什麼是抽樣分佈？

爲了判斷整體的某些特徵，須要從整體中按必定的抽樣技術抽取若干個個體，將這一抽取過程稱爲抽樣。
抽樣分佈在整體X的分佈類型已知時，若對任意天然數n，都能導出統計量T=T(X1​,X2​,...Xn​)的分佈的數學表達式，這種分佈稱爲精確的抽樣分佈。

8.簡述中心極限定理的意義

中心極限定理設從均值爲μ、方差爲σ2（有限）的任意一個整體中抽取樣本量爲n的樣本，當n充分大時，樣本均值X的抽樣分佈近似服從均值爲μ、方差爲nσ2​的正態分佈。

練習題

調節一個裝瓶機使其對每一個瓶子的灌裝量均值爲μ盎司，經過觀察這臺裝瓶機對每一個瓶子的灌裝量服從標準差 1.0盎司的正態分佈。隨機抽取由這臺機器灌裝的9個瓶子造成一個樣本，並測定每一個瓶子的灌裝量。試肯定樣本均值偏離整體均值不超過0.3盎司的機率。

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