平均值的信賴區間估計 (母體變異數已知)

問題一:已知樣本序列 x=(3 2 1 1 2 4 1 1 6 2 1 5 5 5 5 2 2 4 2 3) 的母體變異數為 2.92,請計算以下數值。(必須寫出計算過程)

1. x 的樣本平均值 xˉ 為何?

2. x 的樣本變異數 S2 為何?

3. x 母體平均值 μ 的 95% 信賴區間範圍為何?

解:
依序列由小至大排好 (1 1 1 1 1 2 2 2 2 2 2 3 3 4 4 5 5 5 5 6 )
序列總個數為 20
1.平均值為2.85
( 1 + 1 + 1 + 1 + 1 + 2 + 2 + 2 + 2 + 2 + 2 + 3 + 3 + 4 + 4 + 5 + 5 + 5 + 5 + 6 ) / 20 = 2.85

2.
{ ( 1 - 2.85 ) ^ 2 + ( 1 - 2.85 ) ^ 2 + ( 1 - 2.85 ) ^ 2 + ( 1 - 2.85 ) ^ 2 + ( 1 - 2.85 ) ^ 2 + ( 2 - 2.85 ) ^ 2 + ( 2 - 2.85 ) ^ 2 +( 2 - 2.85 ) ^2 + ( 2 - 2.85 ) ^ 2 + ( 2 - 2.85 ) ^ 2 + ( 2 - 2.85 ) ^ 2 + ( 3 - 2.85 ) ^ 2 + ( 3 - 2.85 ) ^ 2 + ( 4 - 2.85 ) ^ 2 + ( 4 - 2.85 ) ^ 2 + ( 5 - 2.85 ) ^ 2 + ( 5 - 2.85 ) ^ 2 + ( 5 - 2.85 ) ^ 2 + ( 5 - 2.85 ) ^ 2 + ( 6 - 2.85 )^ 2 } / ( 20 - 1 ) = 2.765789474 (答案由excel計算)

3.


問題二:已知樣本序列 x 如下所示,且 x 的母體變異數為 5.4,請計算以下的數值。

1. x 的樣本平均值 xˉ 為何?

2. x 的樣本變異數 S2 為何?

3. x 母體平均值 μ 的 95% 信賴區間範圍為何?

> x
[1] 8.934661 3.234151 5.063758 5.322139 3.611042 6.176047 7.596522 5.774294
[9] 2.017229 2.921471 7.550340 6.090995 7.824173 3.450554 1.985519 8.763690
[17] 6.828349 7.164586 2.160137 5.247192 5.045215 2.140142 6.033718 5.604483

解:
序列X由小至大排好
1.985519 2.017229 2.140142 2.160137 2.921471 3.234151
3.450554 3.611042 5.045215 5.063758 5.247192 5.322139
5.604483 5.774294 6.033718 6.090995 6.176047 6.828349
7.164586  7.550340 7.596522 7.824173 8.763690 8.934661
序列各數為 24
1.平均值為 5.272516958
( 1.985519 + 2.017229 + 2.140142 + 2.160137 + 2.921471 + 3.234151 + 3.450554 + 3.611042 + 5.045215 + 5.063758 + 5.247192 + 5.322139 + 5.604483 + 5.774294 + 6.033718 + 6.090995 + 6.176047 + 6.828349 + 7.164586 + 7.550340 + 7.596522 + 7.824173 + 8.763690 + 8.934661 ) / 24 = 5.272516958 (答案由excel計算)

2.{ ( 1.985519 - 5.272516958 ) ^ 2 + ( 2.017229 - 5.272516958 ) ^ 2 + ( 2.140142 - 5.272516958 ) ^ 2 + ( 2.160137 - 5.272516958 ) ^ 2 + ( 2.921471 - 5.272516958 ) ^ 2 + ( 3.234151 - 5.272516958 ) ^ 2 + ( 3.450554 - 5.272516958 ) ^ 2 + ( 3.611042 - 5.272516958 ) ^ 2 + ( 5.045215 - 5.272516958 ) ^ 2 + ( 5.063758 - 5.272516958 ) ^ 2 + ( 5.247192 - 5.272516958 ) ^ 2 + ( 5.322139 - 5.272516958 ) ^ 2 + ( 5.604483 - 5.272516958 ) ^ 2 + ( 5.774294 - 5.272516958 ) ^ 2 + ( 6.033718 - 5.272516958 ) ^ 2 + ( 6.090995 - 5.272516958 ) ^ 2 + ( 6.176047 - 5.272516958 ) ^ 2 + ( 6.828349 - 5.272516958 ) ^ 2 + ( 7.164586 - 5.272516958 ) ^ 2 + ( 7.550340 - 5.272516958 ) ^ 2 + ( 7.596522 - 5.272516958 ) ^ 2 + ( 7.824173 - 5.272516958 ) ^ 2 + ( 8.763690 - 5.272516958 ) ^ 2 + ( 8.934661 - 5.272516958 ) ^ 2 } / ( 24- 1 ) = 4.690822712 (答案由excel計算)

3.

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