Recall that a major objective of statistics is to make inferences about a population from an analysis of information contained in sample data.

Typically, we are interested in learning about some numerical feature of the population, such as

• the proportion possessing a stated characteristic;

• the mean and the standard deviation.

A numerical feature of a population is called a parameter.

The true value of a parameter is unknown. An appropriate sample-based quantity is our source about the value of a parameter.

A statistic is a numerical valued function of the sample observations.

Sample mean is an example of a statistic.

Chapter 7: Variation in repeated samples – Sampling distributions

The sampling distribution of a statisticThree important points about a statistic:

• the numerical value of a statistic cannot be expected to give us the exact value of the parameter;

• the observed value of a statistic depends on the particular sample that happens to be selected;

• there will be some variability in the values of a statistic over different occasions of sampling.

Because any statistic varies from sample to sample, it is a random variable and has its own probability distribution.

The probability distribution of a statistic is called its sampling distribution.

Often we simply say the distribution of a statistic.

Statistical inference about the population mean is of prime practical importance. Inferences about this parameter are based on the sample mean and its sampling distribution.

Distribution of the sample mean

Figure 7.4 (p. 275)

Distributions of for n = 3 and n = 10 in sampling from an asymmetric population.X

An example illustrating the central limit theorem

Consider a population with mean 82 and standard deviation 12.

If a random sample of size 64 is selected, what is the probability that the sample mean will lie between 80.8 and 83.2?

Solution: We have μ = 82 and σ = 12. Since n = 64 is large, the central limit theorem tells us that the distribution of the sample mean is approximately normal with

Converting to the standard normal variable:

Thus,

5.164

12)(,82)(

nXsdXE

5.1

82

X

n

XZ

Example on probability calculations for the sample mean

5762.2119.7881.]8.8.[

]5.1/)822.83(5.1/)828.80[(

]2.838.80[

ZP

ZP

XP