Relationship between sample data and population values
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Relationship Between Sample Data and Population Values

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Relationship between sample data and population values

Relationship Between Sample Data and Population Values

You will encounter many situations in business where a sample will be taken from a population, and you will be required to analyze the sample data. Regardless of how careful you are in using proper sampling methods, the sample likely will not be a perfect reflection of the population.


Sampling distribution

Sampling Distribution

A Sampling Distribution is the probability distribution for a statistic. Its description includes:

  • all possible values that can occur for the statistic; and

  • the probability of each value or each interval of values for a given sample.


Example

Example


Population parameters

Population Parameters

  • Population Mean (µX):

    µX = 170,000 / 5 = $34,000

  • Population Standard Deviation (X):

    X = [SQRT(898*106) / 5] = $13,401.49


Draw a random sample of three

Draw a Random Sample of Three

  • How many random samples of three can you draw from this population?

    5C3 = 10 samples of three can be drawn form this population. Each sample has a 1 / 5C3 , or 1 / 10 chance of being selected.

  • List the sample space and find sample means.


Ten possible samples

Ten Possible Samples


The sampling distribution of sample means x

The Sampling Distribution of Sample Means ( X )

  • The mean of the samples means:

    µX = ( X1 + X2 + …. + Xn ) / NCn

    µX = 340,000 / 10 = $34,000

  • The Standard Deviation the samples means, better known as the Standard Error of the Mean:

    X = SQRT[( Xi - µX )2 / NCn]


Standard error of the mean

Standard Error of the Mean

  • The standard error of the mean indicates the spread in the distribution of all possible sample means.

  • Xis also equal to the population standard deviation divided by the SQRT of the sample size

    X = X / SQRT(n)


A finite population correction factor fpc

A Finite Population Correction Factor (fpc)

  • For n > 0.05N, the finite population correction factor adjusts the standard error to most accurately describe the amount of variation.

  • The fpc is SQRT[( N - n ) / ( N - 1 )]


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