"It has been proved beyond a shadowof a doubt that smoking is one of the leading causes ofstatistics."

Fletcher Knebel


9.1 SAMPLING DISTRIBUTIONS (Pages 456 - 469)

OVERVIEW: One examines samples inorder to come to reasonable conclusions about the population fromwhich the sample is chosen. One must be statistically literate inorder to gleen meaningful information from a sample. This involves anawareness of what the sample results tell us, along with what theydon't tell us. A statistic calculated from a sample may suffer frombias or high variability, and hence not represent a good estimate ofa population parameter.

Parameter: An index that isrelated to a population.

Statistic: An index that isrelated to a sample.

Sampling distribution of a statistic: The distribution of values of a statistic taken from allpossible samples of a specific size.

A statistic is unbiasedif the mean of the sampling distribution isequal to the true value of the parameter being estimated.

A reminder that

s2 =[sum(xi-mean)2]/N ... the variance formula for a population.

s2 = [sum(xi-mean)2]/(N-1)... the varianceformula for a sample.

The following example demonstrates some of thestatistical concepts developed in this section.

Consider the three element population P={1,2,3}.

The mean of P is m = 2

The standard deviation of P is s = 0.81649658

The variance of P is s2 =0.66666667

These values are parameters, since they arederived from a population.

Now, consider all possible samples of size 2, withreplacement. There would be 32 = 9 such samples.

Sample

Sample Mean

Sample Var. = s2

Sample St.Dev.=s

1,1

1

0

0

1,2

1.5

.5

.707107

1,3

2

2

1.4142

2,1

1.5

.5

.707107

2,2

2

0

0

2,3

2.5

.5

.707107

3,1

2

2

1.4142

3,2

2.5

.5

.707107

3,3

3

0

0

MEANS

2.0

0.66666667

0.628539

The table shows that...

-the mean ofthe distribution of sample means is themean ( m)of the population. This illustrates that a sample mean is anunbiased estimator of the population mean. (The distribution of sample means"centers" around the mean of the population.)

-the mean of thedistribution of sample variances(s2) isequal to the variance ( s2) of thepopulation. This illustrates that a sample variance(s2) is anunbiased estimator of the population variance. (The distribution of samplevariances "centers" around the variance of the population.)

Note: A sample standard deviation isnot an unbiasedestimator of the population standard deviation. In this example, themean of the sample standard deviations (s) is 0.628539, and thestandard deviation of the population is s = 8.81649658. (Thedistribution of sample standard deviations does "not center" aroundthe standard deviation of the population.)

 

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