POPULATION Given the population 2, 3, 10 3 = size of population 2 3  10  Σ 15

3 2 5 Σ 0

9 4 25 Σ 38

15 Σ 5 3 N mean of population

38 √114 3 3 standard deviation of the population σ

Median of the population 3

Range of population 38 Range   10 3 7  Proportion of odds in the population 3 N variance of population 1/3 ____________________________________________________________________________________________ Σ

SAMPLES Look at all samples of size 2 with replacement. There are nine samples. For each sample find the mean, proportion of odds, standard deviation, variance, median, and range. 2 = size of each sample Sample 2,2 2,3 2,10 3,2 3,3 3,10 10,2 10,3   10,10

Median Range ̂ 2 0 0 0 2 0 1/2 2.5 1 2.5 1/2 1/√2 32 6 8 6 0 4√2 1/2 2.5 1 2.5 1/2 1/√2 3 1 0 0 3 0 9/2 6.5 7 6.5 1/2 3/√2 32 6 8 6 0 4√2 49/2 6.5 7 6.5 1/2 7/√2 0 10 0 0 0 10 Σ 45 Σ 3 Σ 8√2 Σ 96 Σ 45 Σ 25

example of calculations for the standard deviation of the sample 2,3 2 3 /2 2.5 Σ 2 2.5 1/2 3 2.5 1/2 1/2 n 1 2 1 1 1/2 1/√2 √2/2   s example of calculations for the standard deviation of the sample 3,10 3 10 6.5 2 7 7 Σ 3 6.5 10 6.5 2 2 49/2 n 1 2 1 1 s

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49/2

7/√2

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7√2/2

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SAMPLING DISTRIBUTIONS A sampling distribution of a sample statistic is a probability distribution ∑ Recall the mean of a probability distribution of  is Σ ∑ Σ The variance of a probability distribution of is SAMPLING DISTRIBUTION OF SAMPLE MEANS The first two columns in the table below are the probability distribution of the sample means for all samples of size 2. The probability distribution is called the sampling distribution of the sample means.   .       .    

( / / / / / /

2  5 3 2.5 5 2.5 3  5 2 6  5 1 6.5 5 1.5 10 5 5 Σ 0 Σ

9 6.25 4 1 2.25 25 19/3

Mean of the probability distribution of the sample means 1 5 2 Σ P Σ 2· · 5 9 2 9 The mean of the population from above is 5 UNBIASED Variance of the probability distribution of the sample means 1 25 2 1 Σ Σ P 9· · 4· 9 4 9 9

19/3

Standard deviation of the probability distribution of the sample means 19 3

√57 3

√114/3

√114

√2

3√2

√

√228 6

 

√57 3

√

____________________________________________________________________________________________S

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SAMPLING DISTRIBUTION OF THE SAMPLE PROPORTIONS    

  /   /   /  

/     /  

̂

  1/3 1/6 2/3   Σ 0

̂ 1/9 1/36 4/9 Σ

Mean of the sampling distribution of the sample proportions 4 1 4 1 1 ̂ ̂ Σ ̂ 0· · 1· 9 2 9 9 3 From above the proportion of odds of the population is 1/3 UNBIASED Variance of the probability distribution of the proportions 1 4 1 4 4 1 Σ ̂ P ̂ Σ ̂ · · · 9 9 36 9 9 9

1 9

Standard deviation of the probability distribution of the proportion 1 9

1 3

1 2 3 3 2

1

1 9

 

1 3

1

____________________________________________________________________________________________

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SAMPLING DISTRIBUTION OF SAMPLE VARIANCES       /   38/3 38/3 /   /   1/2 38/3 … /   /   49/2 38/3 …   /   32 38/3 … / Σ 0 Σ Mean of the sampling distribution of the sample variances 3 1 2 38 Σ 0· · 9 2 9 3 From above the population variance is 38/3   is UNBIASED estimator ____________________________________________________________________________________________ SAMPLING DISTRIBUTION OF THE SAMPLE STANDARD DEVIATIONS    

    16√2/9   4√2 16√2/9   1/2 16√2/9   7/2 16√2/9   Σ 0

/ / / /

√   /√   /√   √ /  

… … … … …

Mean of the sampling distribution of the sample standard deviations 3 2 16√2 Σ 0· 4√2 · 9 9 9 √ From above the standard deviation of the population s is BIASED estimator ____________________________________________________________________________________________ SAMPLING DISTRIBUTION OF THE SAMPLE MEDIANS  

    .       .

  / / / / / /

 

         

3 2.5 2 1 1.5 5 Σ 0

9 6.25 4 1 1.25 25 …

Mean of the sampling distribution of the sample medians 1 5 2 45 Σ 2· · 5 9 2 9 9 From above the Median of the population 3 m is BIASED estimator ____________________________________________________________________________________________

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SAMPLING DISTRIBUTION OF THE SAMPLE RANGES   

 

       

/ / / / /  

         

16√2/9 4√2 16√2/9 1/2 16√2/9 7/2 16√2/9 Σ 0

… … … … …

Mean of the sampling distribution of the sample ranges 3 2 32 Σ 0· 1· 9 9 9 From above the Range of the population 8 range is BIASED estimator SUMMARY We found: 1. sampling distribution of sample means has expected value of population mean 2. sampling distribution of sample proportions has expected value of population proportion 3 sampling distribution of sample variances has expected value of population variance 4. sampling distribution of sample standard deviation does not have expected value of population standard deviation 5. sampling distribution of sample median does not have expected value of population standard deviation 6. sampling distribution of sample range does not have expected value of population range We say that , ̂ , are all unbiased estimators of population parameters , , We want unbiased and low variability. We want variance of sample   to be small (decrease as increases) We want variance of sample ̂   to be small (decrease as increases) We want variance of sample   to be small (decrease as increases)

respectively.

xxxx xxxx xxxx xx

xxxxx xxxxx xxxxx

.

xx x x xxx

Low bias high variability

Low bias low variability

High bias low variability

xx

High bias highvariability

Under certain conditions the sample median and the trimmed mean are also unbiased estimators of the population mean. If the population is normal the sample mean  has a smaller standard deviation than any other unbiased estimator of the population mean . This is what is mean by the best estimator.

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Finite Population Correction Factor If your sample size .05  then you can assume independence (with replacement) If your sample size .05 then you cannot assume independence. You are choosing without replacement. There is a correction factor for the standard deviation of the sampling distribution of the means. Given the POPULATION 2,3,10 . Find all samples of size 2 without replacement. 3  2  We previously calculated for the population 2,3,10 . 5 √114 3 SAMPLES Look at all samples of size 2 without replacement. There are six samples. For each sample find the mean. Sample 2,3 2,10 3,2 3,10 10,2 10,3 

5/2

5/2 6 5/2 13/2 6 13/2 Σ 30 5/2

1/3

6 1/3 13/2 1/3 Σ 5

1 3/2 Σ

25/4

1 9/4 0 Σ 19/6

Mean of the probability distribution of the sample means 5 1 Σ P Σ · 5 2 3 5 Variance of the probability distribution of the sample means 25 1 19 Σ P Σ · 4 3 6 Standard deviation of the probability distribution of the sample means 19 6

√19 √6

√114/3 √

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√114 6

1

√2

√

1

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·

3 3

2 1

√114 3√2

·

1 √2

√114 6

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