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POPULATION Given the population 2, 3, 10

3 = size of population 2 3 9 3  2 4 10  5 25

Σ 15 Σ 0 Σ 38

ΣN

153

5 mean of population

ΣN

383

variance of population

σ383

√1143

standard deviation of the population Proportion of odds in the population

1/3

Median of the population

3 Range of population Range   10 3 7 

____________________________________________________________________________________________ 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

example of calculations for the standard deviation of the sample 2,3

2 3 /2 2.5 Σn 1

2 2.5 3 2.52 1

1/2 1/21

1/2

  s 1/2 1/√2 √2/2 example of calculations for the standard deviation of the sample 3,10

3 102

6.5

Σn 1

3 6.5 10 6.52 1

72

72

149/2

s 49/2 7/√2 7√2/2

Sample Median Range 2,2 2 0 0 0 2 0 2,3 2.5 1/2 1/√2 1/2 2.5 1 2,10 6 0 4√2 32 6 8 3,2 2.5 1/2 1/√2 1/2 2.5 1 3,3 3 1 0 0 3 0 3,10 6.5 1/2 3/√2 9/2 6.5 7 10,2 6 0 4√2 32 6 8 10,3   6.5 1/2 7/√2 49/2 6.5 7 10,10 10 0 0 0 10 0

Σ 45 Σ 3 Σ 8√2 Σ 96 Σ 45 Σ 25

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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 The variance of a probability distribution of is ∑ Σ 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.

Mean of the probability distribution of the sample means

Σ P Σ 2 ·19

52·29

5 The mean of the population from above is 5

UNBIASED Variance of the probability distribution of the sample means

Σ P Σ 9 ·19

254·29

4 ·19

19/3

Standard deviation of the probability distribution of the sample means

193

√573

√√114/3√2

√1143√2

√2286

 √573

√

____________________________________________________________________________________________S

(   / 2  5 3 9 .   / 2.5 5 2.5 6.25   / 3  5 2 4   / 6  5 1 1 .   / 6.5 5 1.5 2.25   / 10 5 5 25

Σ 0 Σ 19/3

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SAMPLING DISTRIBUTION OF THE SAMPLE PROPORTIONS

Mean of the sampling distribution of the sample proportions

Σ 0 ·49

12·49

1 ·19

13

From above the proportion of odds of the population is 1/3

UNBIASED Variance of the probability distribution of the proportions

Σ P Σ 19·49

136

·49

49·19

19

Standard deviation of the probability distribution of the proportion

19

13

113232

19

 13

1

____________________________________________________________________________________________

        /   1/3 1/9 /   /   1/6 1/36   /   2/3 4/9

/     Σ 0 Σ

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SAMPLING DISTRIBUTION OF SAMPLE VARIANCES

Mean of the sampling distribution of the sample variances

Σ 0 ·39

12·29

383

From above the population variance is 38/3

  is UNBIASED estimator ____________________________________________________________________________________________ SAMPLING DISTRIBUTION OF THE SAMPLE STANDARD DEVIATIONS

Mean of the sampling distribution of the sample standard deviations

Σ 0 ·39

4√2 ·29

16√29

From above the standard deviation of the population √ s is BIASED estimator

____________________________________________________________________________________________ SAMPLING DISTRIBUTION OF THE SAMPLE MEDIANS

Mean of the sampling distribution of the sample medians

Σ 2 ·19

52·29

459

5 From above the Median of the population 3

m is BIASED estimator ____________________________________________________________________________________________

      /   38/3 38/3 /   /   1/2 38/3 … /   /   49/2 38/3 …   /   32 38/3 …

/ Σ 0 Σ

      /   16√2/9 …

√   /   4√2 16√2/9 …

/√   /   1/2 16√2/9 …

/√   /   7/2 16√2/9 …

√ /     Σ 0 …

        /   3 9 .   /   2.5 6.25   /   2 4   /   1 1 . / 1.5 1.25

/ 5 25     Σ 0 …

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SAMPLING DISTRIBUTION OF THE SAMPLE RANGES

Mean of the sampling distribution of the sample ranges

Σ 0 ·39

1 ·29

329

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 , , respectively. 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) 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.

      /   16√2/9 …   /   4√2 16√2/9 …   /   1/2 16√2/9 …   /   7/2 16√2/9 …

/     Σ 0 …

xxxxxxxxxxxxxxx

Low bias high variability

xx .

xxxx xxxx xxxx

xx x x xxx xx

Low bias low variability High bias low variability High bias highvariability

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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 √1143

SAMPLES Look at all samples of size 2 without replacement. There are six samples. For each sample find the mean.

Mean of the probability distribution of the sample means

Σ P Σ52·13

5 5

Variance of the probability distribution of the sample means

Σ P Σ254·13

196

Standard deviation of the probability distribution of the sample means

196

√19√6

√1146

√ 1√114/3√2

·3 23 1

√1143√2

·1√2

√1146

Sample 2,3 5/2 2,10 6 3,2 5/2 3,10 13/2 10,2 6 10,3  13/2

Σ 30

5/2 1/3 5/2 25/4 6 1/3 1 1 13/2 1/3 3/2 9/4 Σ 5 Σ 0 Σ 19/6

√ 1