Chapter 5. Sampling distributions

5.1. Sampling and sampling distributions

Suppose that we want to select a sample of n objects from a population

of N objects.

A simple random sample is selected such that every object has an equal probability of being selected and the objects are selected independently-

the selection of one object does not change the probability of selecting any other objects.

It is important that a sample represents the population as a whole. If a marketing manager wants to assess reactions to a new food product, she would not sample only her friends and neighbors. People must be selected randomly and independently. Random selection is our insurance policy against allowing personal influence the selection.

We use sample information to make inferences about the parent population. The distribution of all values of interest in this population can be represented by a random variable. It would be too ambitious to attempt to describe the entire population distribution based on a small random sample of observations. However, we may well be able to make quite firm inferences about important characteristics of the population distribution, such as the population mean and variance.

Sampling distribution:

The probability distribution of is called its sampling distribution. It lists the various values that can assume and the probability for each value of .

In general, the probability distribution of a sample statistic is called its sampling distribution.

Let us consider sampling distribution in example.

Example:

Suppose that there are only five employees working for a small company. The following data give the annual salaries (in thousands of dollars) of these employees:

17; 24; 35; 35; 43

Let X denote the annual salary of an employee. We can write the frequency distribution of annual salaries as in table 5.1

X	f
17 24 35 43	1 1 2 1

Table5.1 Table 5.2

X	P (X)
17 24 35 43	1/5=0.2 1/5=0.2 2/5=0.4 1/5=0.2

Population frequency Population

distribution probability distribution

Dividing the frequencies of classes by the population size we obtain the relative frequencies, which can be used as probabilities of those classes. Table 5.2, which lists the probabilities of various X values, presents the probability distribution of the population.

Now, let us consider all possible samples of three salaries each, that can be selected, without replacement, from the population. The total number of possible samples, given by the combination

Total number of samples= .

Suppose we assigns letters A, B, C, D, and E to the salaries of five employees so that

A=17; B=24; C=35; D=35; E=43.

Then 10 possible samples of three salaries are

ABC, ABD, ABE, ACD, ACE,

ADE, BCD, BCE, BDE, CDE.

These 10 samples and their respective means are listed in Table 5.3.

Note that the values of means of samples in Table 5.3 are rounded to two decimal places.

Sample	Salaries in the sample
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE	17, 24, 35 17, 24, 35 17, 24, 43 17, 35, 35 17, 35, 43 17, 35, 43 24, 35, 35 24, 35, 43 24, 35, 43 35, 35, 43	25.33 25.33 28.00 29.00 31.67 31.67 31.33 34.00 34.00 37.67

Table 5.3

All possible samples and

their means when the sample size is 3.

By using the values of given in Table 5.3, we record the frequency distribution of in Table 5.4.

Table 5.4

	f
25.33 28.00 29.00 31.33 31.67 34.00 37.67	2 1 1 1 2 2 1

Frequency distribution of

when the sample size is 3.

By dividing the frequencies of various vales of by the sum of all frequencies, we obtain the relative frequencies of classes, which can be used as probabilities of classes. These probabilities are listed in Table 5.5.

This table gives the sampling distribution of .

Table5.5.

	P (X)
25.33 28.00 29.00 31.33 31.67 34.00 37.67	2/10=0.20 1/10=0.10 1/10=0.10 1/10=0.10 2/10=0.20 2/10=0.20 1/10=0.10

Sampling distribution of

when the sample size is 3.

If we draw just one sample of three salaries from the population of five salaries, we may draw any of 10 possible samples. Hence, the sample mean can assume any of the values listed in Table 5.5 with the corresponding probability. This probability function is graphed in Figure 5.1. For example, the probability that the mean of a randomly drawn sample of three salaries is

31.67 is 0.20. This can be written as