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Ranny asked in 教育與參考考試 · 2 decades ago

一題統計考題...20點

某自然課程的任課老師,對於全班100名同學的期中考與期末考成績,進行初步的統計分析。結果期中考成績(X1)的平均數為70.0分,標準差(不偏估計值)為10.0;期末考成績(X2)平均數為80.0分,標準差(不偏估計值)為8.0;而這兩次考試的共變數為50.0。

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問題一、學期總成績【(X1+X2)/2】為期中考(X1)與期末考(X2)的平均,則學期總成績的變異數(不偏估計值)=?

問題二、若將期中考成績進行直線轉換:X1\'=0.9X1+10;期末考成績也進行直線轉換:X2\'=0.8X2+20。則X1\'與X2\'兩者的相關係數(r)=?

Update:

為什麼這裡處處強調「不偏估計值」啊?

2 Answers

Rating
  • Bug
    Lv 4
    2 decades ago
    Favorite Answer

    回答這幾題之前,必須先學會兩個觀念

    Var(X1+X2)=Var(X1)+Var(X2)+ 2*Cov(X1,X2)

    Var(a*X + b) = a^2 * Var(X)

    可參考

    http://www.sinica.edu.tw/as/ssrc/ckuan/pdf/project...

    第50、69~73頁

    答一:

    Var[ (X1+X2)/2 ] = (1/2)^2 * Var (X1 + X2)

    =0.25 * [ Var(X1) + Var(X2) + 2 * Cov(X1,X2) ]

    =0.25 * [ 10^2 + 8^2 + 2 * 50 ]

    =66

    答二:

    Var(X1') = (0.9)^2 * Var(X1) = 81

    Var(X2') = (0.8)^2 * Var(X2) = 40.96

    Cov(X1', X2') = (0.9) * (0.8) * Cov(X1, X2) = 36

    相關係數 cor(X1' , X2') = Cov(X1', X2') / [ Var(X1')^0.5 * Var(X2')^0.5 ]

    =0.625

    為什麼這裡處處強調「不偏估計值」啊?

    答:因為變異數有不偏估計值(unbiased estimator)跟最大可能估計值(maximum likelihood estimator)兩種,差別在於base是(n-1)或是(n)

    Source(s): 自己、統計學
  • 2 decades ago

    很詳細

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