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Statistics in Applied Science and Statistics in Applied Science and TechnologyTechnology

Chapter 7 - Sampling Distribution of Means

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Key Concepts in This ChapterKey Concepts in This Chapter

Distribution of a population Distribution of sample means Central limit theorem Standard error of the mean Z score of a sample mean Student’s t distribution t score and degree of freedom

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The Distribution of a Population and the The Distribution of a Population and the Distribution of its Sample Means Distribution of its Sample Means

A distribution of sample means is the set of values of sample means obtained from all possible samples of the same size (n) from a given population.

A distribution of a population includes a set of intervals and displays their frequency (numbers of cases or occurrences) in each intervals for that given population.

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Central Limit TheoremCentral Limit Theorem

The central limit theorem states that for a randomly selected sample of size n (n25, but the larger n is, the better the approximation) with a mean of and standard deviation :• The distribution of sample means is

approximately normal regardless of whether the population distribution is normal or not

x

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Central Limit Theorem (Cont’d)Central Limit Theorem (Cont’d)

• The mean of the distribution of sample means is equal to the mean of the population distribution - that is,

• The standard deviation of the distribution of sample means is equal to the standard deviation of the population () divided by the square root of the sample size (n), that is,

x

nx

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Standard Error of the MeanStandard Error of the Mean

The standard deviation of the sample means, referred to as the standard error of the mean, is denoted as SE( ), that is,

SE ( ) is a rough measure of the average amount by which sample mean deviate from population mean (amount of sampling error).

x

x

nxxSE )(

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In practice, the standard error of In practice, the standard error of the mean is calculated by:the mean is calculated by:

Where: S - sample standard deviation

- standard error of the mean estimated from a sample

n

ssx

xs

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Z score of a sample meanZ score of a sample mean

Z score of a sample mean establishes the relative position of in a distribution of sample means and can be calculated by:

x

n

xZ

/

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Student’s t distributionStudent’s t distribution

When sample standard deviation is used to calculate z score of a sample mean, we no longer have the standard normal distribution, instead we have so called Student’s t distribution

t distribution is similar to the standard normal distribution and approximate standard normal distribution when sample size exceeds 30.

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tt score and degree of freedom score and degree of freedom

The equation for t score is:

Degree of freedom (df) can be calculated by:

ns

xt

/

1ndf