Sampling Distributions for Beginning Statistics
Practice problems for these concepts can be found at:
- Sampling Distributions Solved Problems for Beginning Statistics
- Sampling Distributions Supplementary Problems for Beginning Statistics
Systematic Random Sampling
Systematic random sampling consists of choosing a sample by randomly selecting the first element and then selecting every kth element thereafter. The systematic method of selecting a sample often saves time in selecting the sample units.
EXAMPLE 7.4 In order to obtain a systematic sample of 50 of the nation's 3143 counties, divide 3143 by 50 to obtain 62.86. Round 62.86 down to obtain 62. From a list of the 3143 counties, select one of the first 62 counties at random. Suppose county number 35 is selected. To obtain the other 49 counties, add 62 to 35 to obtain 97, add 2 × 62 to 35 to obtain 159, and continue in this fashion until the number 49 × 62 + 35 = 3073 is obtained. The counties numbered 35, 97, 159, . . . , 3073 would represent a systematic sample from the nations counties.
In cluster sampling, the population is divided into clusters and then a random sample of the clusters are selected. The selected clusters may be completely sampled or a random sample may be obtained from the selected clusters.
EXAMPLE 7.5 A large company has 30 plants located throughout the United States. In order to access a new total quality plan, the 30 plants are considered to be clusters and five of the plants are randomly selected. All of the quality control personnel at the five selected plants are asked to evaluate the total quality plan.
In stratified sampling, the population is divided into strata and then a random sample is selected from each strata. The strata may be determined by income levels, different stores in a supermarket chain, different age groups, different governmental law enforcement agencies, and so forth.
EXAMPLE 7.6 Super Value Discount has 10 stores. To assess job satisfaction, one percent of the employees at each of the 10 stores are administered a job satisfaction questionnaire. The 10 stores are the strata into which the population of all employees at Super Value Discount are divided. The results at the 10 stores are combined to evaluate the job satisfaction of the employees.
Sampling Distribution of the Sampling Mean
The mean of a population, μ, is a parameter that is often of interest but usually the value of μ is unknown. In order to obtain information about the population mean, a sample is taken and the sample mean,, is calculated. The value of the sample mean is determined by the sample actually selected. The sample mean can assume several different values, whereas the population mean is constant. The set of all possible values of the sample mean along with the probabilities of occurrence of the possible values is called the sampling distribution of the sampling mean. The following example will help illustrate the sampling distribution of the sample mean.
EXAMPLE 7.7 Suppose the five cities with the most African-American-owned businesses measured in thousands is given in Table 7.2.
If X represents the number of African-American-owned businesses in thousands for this population consisting of five cities, then the probability distribution for X is shown in Table 7.3.
The population mean is μ = Σ xP(x) = 30 × .2 + 33 × .2 + 36 × .2 + 39 × .2 + 42 × .2 = 36, and the variance is given by σ2 = Σx2 P(x) – μ2 = 900 × .2 + 1089 × .2 + 1296 × .2 + 1521 × .2 + 1764 × .2 – 1296 = 1314 – 1296 = 18. The population standard deviation is the square root of 18, or 4.24.
The number of samples of size 3 possible from this population is equal to the number of combinations possible when selecting three cities from five. The number of possible samples is = 10. Using the letters A, B, C, D, and E rather than the name of the cities, Table 7.4 gives all the possible samples of three cities, the sample values, and the means of the samples.
The sampling distribution of the mean is obtained from Table 7.4. For random sampling, each of the samples in Table 7.4 is equally likely to be selected. The probability of selecting a sample with mean 39 is .1 since only one of 10 samples has a mean of 39. The probability of selecting a sample with mean 36 is .2, since two of the samples have a mean equal to 36. Table 7.5 gives the sampling distribution of the sample mean.
When the sample mean is used to estimate the population mean, an error is usually made. This error is called the sampling error, and is defined to be the absolute difference between the sample mean and the population mean. The sampling error is defined by
- sampling error = | – μ | (7.1)
EXAMPLE 7.8 In Example 7.7, the population of the five cities with the most African-American-owned businesses is given. The mean of this population is 36. Table 7.5 gives the possible sample means for all samples of size 3 selected from this population. Table 7.6 gives the sampling errors and probabilities associated with all the different sample means.
From Table 7.6, it is seen that the probability of no sampling error in this scenario is .20. There is a 60% chance that the sampling error is 1 or less.
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