Sampling Distribution
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Lesson 14
Sampling distribution
14.1 Sampling Distribution of the Sample Mean
For a variable and a given sample size, the distribution of the variable ─ that is, the distribution of all possible sample means─ is called the sampling distribution of the sample mean. In statistics, the following terms and phrases are synonymous.
Sampling distribution of the sample mean
Distribution of the variable .
Distribution of all possible sample means of a given sample size.
Example 14.1
Suppose that the population of interest consists of the five supermarkets in Dhaka city whom, for convenience, we will call A, B, C, D and E. Suppose that the variable of interest is sales in the month of December, in million taka. Table 3.1 lists the supermarkets and the sales.
Super market
A B C D E
Sale (Million Taka)
76 78 79 81 86
Table 14.1: Sales of five supermarkets in the month of December
Obtain the sampling distribution of the sample mean for samples of size 2.
Make some observations about sampling error when the mean sale of a random sample of two supermarkets is used to estimate the population mean sale.
Find the probability that, for a random sample of size 2, the sampling error made in estimating the population mean by the sample mean will be 1 unit (1million Taka) or less; that is, determine the probability that will be within 1 unit (1million Taka) of .
Solution:
The population mean sale,
The population standard deviation is
There are possible distinct samples of size 2, but if we draw the samples with replacement there are possible samples. Table 14.2 represents the sampling distribution of the sample means and Figure 14.1 shows the stem and leaf display of the distribution.
Sample
Sales
76, 76
76, 78
76, 79
76, 81
76, 86
78, 76
78,78
78, 79
78, 81
78, 86
79, 76
79, 78
79, 79
79, 81
79, 86
81, 76
81, 78
81, 79
81, 81
81, 86
86, 76
86, 78
86, 79
86, 81
86, 86
Table 14.2: Possible samples and sample means for samples of size 2.
Figure 14.1: Stem and leaf plot for
sample means for samples of size 2.
The mean sale of the two supermarkets selected is not likely to equal the population mean of 80 Million Taka. Only two of the 25 samples has a mean of 80 Million Taka. Thus, in this case, the chances are only or 8% that will equal ; evidently, some sampling error is likely.
The sample means can be organized in a frequency distribution as:
Class interval
Frequency
Relative frequency
76-78
78-80
80-82
82-84
84-86
Table 14.3: Frequency distribution of sample means for samples of size 2.
The relative frequency provides the probability distribution as:
Sales