ผลต่างระหว่างรุ่นของ "Probstat/week15 practice 1"

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แถว 1: แถว 1:
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: ''This is part of [[probstat]].''
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== Data generation ==
 
Generate 5 sets of samples from normal populations, each of size 10, on a spreadsheet software.  Make sure that the mean and variance of each set can be modified easily.
 
Generate 5 sets of samples from normal populations, each of size 10, on a spreadsheet software.  Make sure that the mean and variance of each set can be modified easily.
  
 
[[Image:Prob-anova-data.png]]
 
[[Image:Prob-anova-data.png]]
  
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== Variance estimations ==
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'''Estimator 1.''' The first estimator for <math>\sigma^2</math> is
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<center>
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<math>\frac{1}{m}\sum_{i=1}^m \sum_{j=1}^n \frac{(X_{ij} - \bar{X}_{i\centerdot})^2}{n-1}.</math>
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</center>
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This estimator is always correct.  (Even when <math>H_0</math> is not true.)
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'''Estimator 2.''' The second estimator is calculated from the sample means.  It will be correct only when <math>H_0</math> is true.
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<center>
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<math>n \sum_{i=1}^m \frac{(\bar{X}_{i\centerdot} - \bar{X}_{\centerdot\centerdot})^2}{m-1}</math>
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</center>
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== Hypothesis testing ==
 
Then compute  
 
Then compute  
  
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<math>SS_b = n \sum_{i=1}^m (\bar{X}_{i\centerdot} - \bar{X}_{\centerdot\centerdot})^2</math>
 
<math>SS_b = n \sum_{i=1}^m (\bar{X}_{i\centerdot} - \bar{X}_{\centerdot\centerdot})^2</math>
 
</center>
 
</center>
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Recall that
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<center>
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<math>\frac{SS_w}{\sigma^2} \sim \chi_{m(n-1)}^2</math>,
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</center>
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and
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<center>
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<math>\frac{SS_b}{\sigma^2} \sim \chi_{m-1}^2</math>.
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</center>
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Also, since <math>S_i^2</math> is independent of <math>\bar{X}_{i\centerdot}</math>, <math>SS_w</math> and <math>SS_b</math> are independent.  Thus, the statistic
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<center>
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<math>
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\frac{SS_b/(m-1)}{SS_w/m(n-1)} \sim F_{m-1,m(n-1)}.
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</math>
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</center>
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Therefore, we can use the table for the F-distribution to test for <math>H_0</math>.

รุ่นแก้ไขปัจจุบันเมื่อ 03:00, 4 ธันวาคม 2557

This is part of probstat.

Data generation

Generate 5 sets of samples from normal populations, each of size 10, on a spreadsheet software. Make sure that the mean and variance of each set can be modified easily.

Prob-anova-data.png

Variance estimations

Estimator 1. The first estimator for is

This estimator is always correct. (Even when is not true.)

Estimator 2. The second estimator is calculated from the sample means. It will be correct only when is true.

Hypothesis testing

Then compute

and

Recall that

,

and

.

Also, since is independent of , and are independent. Thus, the statistic

Therefore, we can use the table for the F-distribution to test for .