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

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:

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

A reminder that

s

^{2}=[sum(xi-mean)^{2}]/N ... the variance formula for a population. s

^{2}= [sum(xi-mean) ^{2}]/(N-1)... the varianceformula for asample.

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 s

^{2}=0.66666667These values are parameters, since they arederived from a population.

Now, consider all possible samples of size 2, withreplacement. There would be 3^{2} = 9 such samples.

Sample | Sample Mean | Sample Var. = s | Sample St.Dev.=s |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| | | |

| 2.0 | 0.66666667 | 0.628539 |

The table shows that...

-the

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

mean of thedistribution of sample variances(s^{2}) isequal to the variance ( s^{2}) of thepopulation. This illustrates that a sample variance(s^{2}) is anunbiased estimatorof the population variance. (The distribution of samplevariances "centers" around the variance of the population.)

Note: A sample standard deviation is**not** 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.)