Sampling Size

Principles
The information contained in a sample does not depend appreciably on the size of the lot from which the sample is taken (provided the lot is at least 10 times the sample)
The information in a sample increases when the size of the sample increases (larger samples are more faithful to the entire lot)
The information contained in a sample does not depend on the proportion that the sample represents of the lot (just on the sheer number)

Principles
The information contained in a sample does not depend appreciably on the size of the lot from which the sample is taken (provided the lot is at least 10 times the sample).

Principles
The information in a sample increases when the size of the sample increases. Larger samples are more faithful to the entire lot.

Principles
The information contained in a sample does not depend on the proportion that the sample represents of the lot (just on the sheer number). In any case, larger samples contain more information than smaller ones.

Key principle
Larger sample sizes yield better information
However, we want to minimize the amount of data collected and still achieve “good” results
Minimize time for data collection
Minimize sample degradation
Etc.

Sampling Sizes for Various Applications
Estimating Averages, single variable
Estimating Variances, single variable

Single Variable - Estimating Average
It can be shown that the length of a confidence interval for μ is given by:
Where t is the critical value of Student’s t at the level of significance desired with degree of freedom equal to DF for σ estimation
So, given a required value of L and an estimated standard deviation, we can determine how many measurements are necessary
Assumptions
We have an estimate of the standard deviation of a measurement
We want to have a certain level of confidence of our average value
The t value can be obtained in numerous data tables or by using excel TINV((1-p),DoF), where p is the % confidence desired

Single Variable - Estimating Variance
Simple, approximate method (assume the distribution is normal):
We have a series of samples all of the same size N, taken from a population with a normal variance
Each sample will have it’s own variance. The distribution of these variances is NOT normal . It is a chi-square distribution with a mean at the population variance.
We are interested in the RATIO of estimated variance to actual variance

Single Variable - Estimating Variance
Complicated, exact method (because the distribution is not normal):
Then, you have to lookup in a chi-square table under the proper percentage column to find a value such that its quotient by the corresponding n is not less than (1- ε ) 2 and not more than (1+ ε ) 2 .
This is done in Excel by the following:
χ low -1 =CHIINV((1-p)/2,n)
χ hi -1 =CHIINV(1-(1-p)/2,n)
We have a series of samples all of the same size N, taken from a population with a normal variance
Each sample will have it’s own variance. The distribution of these variances is NOT normal . It is a chi-square distribution with a mean at the population variance.
We are interested in the RATIO of estimated variance to actual variance

The problem of fitting a line
y=mx+b
We want to ask a few major questions about this line
How good is our estimate of the slope?
How good is our estimate of the offset? (offset = (m-1)<x> + b)
Over what range do these values hold?

Errors in both x and y
y=mx+b  Y = α + β X
where x = X+ ε , y = Y+ δ
We assume that ε & δ are normally distributed with < ε > and < δ > = 0 and σ ε and σ δ are known
Want to know how many sites we need for good confidence over specific ranges for both slope (m) and average offset (<Y>-<X> = (m-1)<X>+b)
We can run Monte Carlo simulations to solve this

Offset calculation
The offset can be calculated by <Y> - <X>
We know that the standard deviation of this value will be equal to σ offset =sqrt( σ ε 2 + σ δ 2 +2(<XY>-<X><Y>))

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Sampling Size

by Daniel Moore on Sep 24, 2009

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