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c:ms:2017:schedule:week03 [2017/03/22 14:29] – [Sampling Distribution, Standard Error] hkimscilc:ms:2017:schedule:week03 [2022/05/15 11:25] – hkimscil
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-====== Week 3 내용 ======+ㄹ====== Week 3 내용 ======
 ===== SPSS ===== ===== SPSS =====
 <del>Chapter 3</del>, Chapter 4 <del>Chapter 3</del>, Chapter 4
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 {{:hist.jpg}} {{:hist.jpg}}
  
-</WRAP> 
-<WRAP half column> 
 data file: {{:Ex3-1.sav}} 읽지 않은 지문에 대한 답을 한 학생들의 점수 (Katz, 1990). data file: {{:Ex3-1.sav}} 읽지 않은 지문에 대한 답을 한 학생들의 점수 (Katz, 1990).
  
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   * Data file: [[http://www.uvm.edu/~dhowell/fundamentals7/DataFiles/Tab5-1.dat|Web site]] or {{:Tab5-1.sav}} p.86-7   * Data file: [[http://www.uvm.edu/~dhowell/fundamentals7/DataFiles/Tab5-1.dat|Web site]] or {{:Tab5-1.sav}} p.86-7
   * [[:range]]   * [[:range]]
-  * [[:outlier]]+  * [[:outliers]]: It is beyond our scope. Please just refer to it. Won't be appearing in tests. 
   * 평균편차   * 평균편차
   * [[:Variance]] 변량    * [[:Variance]] 변량 
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 <code>Descriptives  <code>Descriptives
- SET Statistic Std. Error + SET Statistic Std. Error 
-ATTRACT 4 Mean 2.6445 .14651 +ATTRACT 4 Mean 2.6445 .14651 
- 95% Confidence Interval for Mean Lower Bound 2.3379  + 95% Confidence Lower Bound 2.3379  
- Upper Bound 2.9511  + Interval for Upper Bound 2.9511  
- 5% Trimmed Mean 2.6483  + Mean 
- Median 2.5950  + 5% Trimmed Mean 2.6483  
- Variance .429  + Median 2.5950  
- Std. Deviation .65520  + Variance .429  
- Minimum 1.20  + Std. Deviation .65520  
- Maximum 4.02  + Minimum 1.20  
- Range 2.82 + Maximum 4.02  
 + Range 2.82
  Interquartile Range .82   Interquartile Range .82
- Skewness -.001 .512 + Skewness -.001 .512 
- Kurtosis .438 .992 + Kurtosis .438 .992 
- 32 Mean 3.2615 .01541+ 32 Mean 3.2615 .01541
  95% Confidence Interval for Mean Lower Bound 3.2292   95% Confidence Interval for Mean Lower Bound 3.2292
- Upper Bound 3.2938 + Upper Bound 3.2938
  5% Trimmed Mean 3.2622   5% Trimmed Mean 3.2622
- Median 3.2650 + Median 3.2650
  Variance .005   Variance .005
  Std. Deviation .06892   Std. Deviation .06892
- Minimum 3.13  + Minimum 3.13  
- Maximum 3.38  + Maximum 3.38  
- Range .25 + Range .25
  Interquartile Range .11   Interquartile Range .11
  Skewness -.075 .512  Skewness -.075 .512
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   * [[:Standard Deviation]] 표준편차   * [[:Standard Deviation]] 표준편차
  
-  * $\displaystyle S_x^2 = \displaystyle \frac {\Sigma X^2 - \frac{(\Sigma X)^2}{N} } {N-1} $ +  * Variance calculation formula 
- +    * <wrap #variance_calculation_formula /> $ \displaystyle S_x^2 = \displaystyle \frac {\Sigma X^2 - \frac{(\Sigma X)^2}{N} } {N-1} $ 
-  * $\displaystyle \sigma_x^2 = \displaystyle \frac {\Sigma X^2 - \frac{(\Sigma X)^2}{N} } {N} = \displaystyle \frac {\Sigma X^2}{N} - \frac {(\Sigma X)^2}{N^2} = \displaystyle \frac {\Sigma X^2}{N} - \bigg(\frac {\Sigma X}{N}\bigg)^2 = \displaystyle \frac {\Sigma X^2}{N} - \mu^2   $ +    * $ \displaystyle \sigma_x^2 = \displaystyle \frac {\Sigma X^2 - \frac{(\Sigma X)^2}{N} } {N} = \displaystyle \frac {\Sigma X^2}{N} - \frac {(\Sigma X)^2}{N^2} = \displaystyle \frac {\Sigma X^2}{N} - \bigg(\frac {\Sigma X}{N}\bigg)^2 = \displaystyle \frac {\Sigma X^2}{N} - \mu^2 $
  
   * [[:Degrees of Freedom]] N-1   * [[:Degrees of Freedom]] N-1
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   * [[:Central Limit Theorem]]   * [[:Central Limit Theorem]]
   * [[:Standard Error]]   * [[:Standard Error]]
 +===== CLT에 관한 정리 =====
 우선, Expected value (기대값)와 Variance (분산)의 연산은 아래와 같이 계산될 수 있다. 우선, Expected value (기대값)와 Variance (분산)의 연산은 아래와 같이 계산될 수 있다.
  
-<WRAP box 450px> +X,Y 가 서로 독립적이라고 할 때:
-X,Y are Independent variables. +
 \begin{eqnarray} \begin{eqnarray}
 E[aX] = a E[X] \\ E[aX] = a E[X] \\
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 Var[X+Y] = Var[X] + Var[Y]   Var[X+Y] = Var[X] + Var[Y]  
 \end{eqnarray} \end{eqnarray}
- 
-</WRAP> 
  
 이때, 한 샘플의 평균값을 $X$ 라고 하면, 평균들의 합인 $S_k$ 는  이때, 한 샘플의 평균값을 $X$ 라고 하면, 평균들의 합인 $S_k$ 는 
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 와 같다. 와 같다.
  
-이렇게 얻은 샘플들(k 개의)의 평균인 $A_k$ 는, +이렇게 얻은 샘플들(k 개의)의 평균인 $ A_k $ 는, 
  
-$$A_k = \displaystyle \frac{(X_1 + X_2 + . . . + X_k)}{k} = \frac{S_{k}}{k} $$+$$ A_k = \displaystyle \frac{(X_1 + X_2 + . . . + X_k)}{k} = \frac{S_{k}}{k} $$
  
 라고 할 수 있다.  라고 할 수 있다. 
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 이때,  이때, 
  
-$$+$$ 
 \begin{align*} \begin{align*}
 E[S_k] & = E[X_1 + X_2 + . . . +X_k] \\ E[S_k] & = E[X_1 + X_2 + . . . +X_k] \\
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 $$ $$
    
-$$+$$ 
 \begin{align*} \begin{align*}
 Var[S_k] & = Var[X_1 + X_2 + . . . +X_k]  \\ Var[S_k] & = Var[X_1 + X_2 + . . . +X_k]  \\
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 이다. 이다.
  
-그렇다면, $A_k$ 에 관한 기대값과 분산값은: +그렇다면, $ A_k $ 에 관한 기대값과 분산값은: 
  
-$$+$$ 
 \begin{align*} \begin{align*}
 E[A_k] & = E[\frac{S_k}{k}] \\ E[A_k] & = E[\frac{S_k}{k}] \\
c/ms/2017/schedule/week03.txt · Last modified: 2022/05/15 11:25 by hkimscil
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