Sampling and Sampling Distributions: Part 2 Sample size and the sampling distribution of Sampling distribution of Sampling methods

Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of Suppose we select a simple random sample of 100 Suppose we select a simple random sample of 100 applicants instead of the 30 originally considered. applicants instead of the 30 originally considered. E ( ) =  regardless of the sample size. In our E ( ) =  regardless of the sample size. In our example, E ( ) remains at 990. example, E ( ) remains at 990. Whenever the sample size is increased, the standard Whenever the sample size is increased, the standard error of the mean is decreased. With the increase error of the mean is decreased. With the increase in the sample size to n = 100, the standard error of the in the sample size to n = 100, the standard error of the mean is decreased to: mean is decreased to:

Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of With n = 30, With n = 100,

Recall that when n = 30, P (980 < < 1000) = Recall that when n = 30, P (980 < < 1000) = Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of We follow the same steps to solve for P (980 < < 1000) We follow the same steps to solve for P (980 < < 1000) when n = 100 as we showed earlier when n = 30. when n = 100 as we showed earlier when n = 30. Now, with n = 100, P (980 < < 1000) = Now, with n = 100, P (980 < < 1000) = Because the sampling distribution with n = 100 has a Because the sampling distribution with n = 100 has a smaller standard error, the values of have less smaller standard error, the values of have less variability and tend to be closer to the population variability and tend to be closer to the population mean than the values of with n = 30. mean than the values of with n = 30.

Relationship Between the Sample Size and the Sampling Distribution of and the Sampling Distribution of Area =.7888 SamplingDistributionof

A simple random sample of n elements is selected from the population. Population with proportion p = ? n Making Inferences about a Population Proportion The sample data provide a value for the sample proportion. The value of is used to make inferences about the value of p. Sampling Distribution of

where: p = the population proportion The sampling distribution of is the probability distribution of all possible values of the sample proportion. Expected Value of

is referred to as the standard error of the is referred to as the standard error of the proportion. Sampling Distribution of Finite Population Infinite Population Standard Deviation of

The sampling distribution of can be approximated The sampling distribution of can be approximated by a normal probability distribution whenever the by a normal probability distribution whenever the sample size is large. sample size is large. The sample size is considered large whenever these The sample size is considered large whenever these conditions are satisfied: conditions are satisfied: np > 5 n (1 – p ) > 5 and Sampling Distribution of

For values of p near.50, sample sizes as small as 10 permit a normal approximation. With very small (approaching 0) or very large (approaching 1) values of p, much larger samples are needed. Sampling Distribution of

For our example, with n = 30 and p =.72, the normal probability distribution is an acceptable approximation because: Sampling Distribution of for the Proportion of Applicants Wanting On-Campus Housing of Applicants Wanting On-Campus Housing n (1 - p ) = 30(.28) = 8.4 > 5 and np = 30(.72) = 21.6 > 5

Sampling Distribution of for the Proportion of Applicants Wanting On-Campus Housing of Applicants Wanting On-Campus Housing SamplingDistributionof

What is the probability that a simple random sample What is the probability that a simple random sample of 30 applicants will provide an estimate of the population proportion of applicants desiring on-campus housing that is within plus or minus.05 of the actual population proportion? In other words, what is the probability that will be In other words, what is the probability that will be between.67 and.77? Sampling Distribution of for the Proportion of Applicants Wanting On-Campus Housing of Applicants Wanting On-Campus Housing

Sampling Distribution of for the Proportion of Applicants Wanting On-Campus Housing of Applicants Wanting On-Campus Housing Step 1: Calculate the z -value at the upper endpoint of the interval. the interval. z = ( )/.082 =.61 P ( z <.61) =.7291 Step 2: Find the area under the curve to the left of the upper endpoint. upper endpoint.

Cumulative Probabilities for the Standard Normal Distribution the Standard Normal Distribution Sampling Distribution of for the Proportion of Applicants Wanting On-Campus Housing of Applicants Wanting On-Campus Housing

Area =.7291 Sampling Distribution of for the Proportion of Applicants Wanting On-Campus Housing of Applicants Wanting On-Campus Housing SamplingDistributionof

Sampling Distribution of for the Proportion of Applicants Wanting On-Campus Housing of Applicants Wanting On-Campus Housing Step 3: Calculate the z -value at the lower endpoint of the interval. the interval. Step 4: Find the area under the curve to the left of the lower endpoint. lower endpoint. z = ( )/.082 = -.61 P ( z.61) =.2709 = = 1 - P ( z <.61)

Area =.2709 Sampling Distribution of for the Proportion of Applicants Wanting On-Campus Housing of Applicants Wanting On-Campus Housing SamplingDistributionof

P (.67 < <.77) =.4582 Sampling Distribution of for the Proportion of Applicants Wanting On-Campus Housing of Applicants Wanting On-Campus Housing Step 5: Calculate the area under the curve between the lower and upper endpoints of the interval. the lower and upper endpoints of the interval. P (-.61 < z <.61) = P ( z <.61) - P ( z < -.61) = =.4582 The probability that the sample proportion of applicants wanting on-campus housing will be within +/-.05 of the actual population proportion :

Sampling Distribution of for the Proportion of Applicants Wanting On-Campus Housing of Applicants Wanting On-Campus Housing Area =.4582 SamplingDistributionof

Sampling Methods Stratified Random Sampling Cluster Sampling Systematic Sampling Convenience Sampling Judgment Sampling

The population is first divided into groups of The population is first divided into groups of elements called strata. elements called strata. The population is first divided into groups of The population is first divided into groups of elements called strata. elements called strata. Stratified Random Sampling Each element in the population belongs to one and Each element in the population belongs to one and only one stratum. only one stratum. Each element in the population belongs to one and Each element in the population belongs to one and only one stratum. only one stratum. Best results are obtained when the elements within Best results are obtained when the elements within each stratum are as much alike as possible each stratum are as much alike as possible (i.e. a homogeneous group). (i.e. a homogeneous group). Best results are obtained when the elements within Best results are obtained when the elements within each stratum are as much alike as possible each stratum are as much alike as possible (i.e. a homogeneous group). (i.e. a homogeneous group).

Cluster Sampling The population is first divided into separate groups The population is first divided into separate groups of elements called clusters. of elements called clusters. The population is first divided into separate groups The population is first divided into separate groups of elements called clusters. of elements called clusters. Ideally, each cluster is a representative small-scale Ideally, each cluster is a representative small-scale version of the population (i.e. heterogeneous group). version of the population (i.e. heterogeneous group). Ideally, each cluster is a representative small-scale Ideally, each cluster is a representative small-scale version of the population (i.e. heterogeneous group). version of the population (i.e. heterogeneous group). A simple random sample of the clusters is then taken. A simple random sample of the clusters is then taken. All elements within each sampled (chosen) cluster All elements within each sampled (chosen) cluster form the sample. form the sample. All elements within each sampled (chosen) cluster All elements within each sampled (chosen) cluster form the sample. form the sample.

Cluster Sampling Advantage: The close proximity of elements can be Advantage: The close proximity of elements can be cost effective (i.e. many sample observations can be cost effective (i.e. many sample observations can be obtained in a short time). obtained in a short time). Advantage: The close proximity of elements can be Advantage: The close proximity of elements can be cost effective (i.e. many sample observations can be cost effective (i.e. many sample observations can be obtained in a short time). obtained in a short time). Disadvantage: This method generally requires a Disadvantage: This method generally requires a larger total sample size than simple or stratified larger total sample size than simple or stratified random sampling. random sampling. Disadvantage: This method generally requires a Disadvantage: This method generally requires a larger total sample size than simple or stratified larger total sample size than simple or stratified random sampling. random sampling. Example: A primary application is area sampling, Example: A primary application is area sampling, where clusters are city blocks or other well-defined where clusters are city blocks or other well-defined areas. areas. Example: A primary application is area sampling, Example: A primary application is area sampling, where clusters are city blocks or other well-defined where clusters are city blocks or other well-defined areas. areas.

Systematic Sampling If a sample size of n is desired from a population If a sample size of n is desired from a population containing N elements, we might sample one containing N elements, we might sample one element for every n / N elements in the population. element for every n / N elements in the population. If a sample size of n is desired from a population If a sample size of n is desired from a population containing N elements, we might sample one containing N elements, we might sample one element for every n / N elements in the population. element for every n / N elements in the population. We randomly select one of the first n / N elements We randomly select one of the first n / N elements from the population list. from the population list. We randomly select one of the first n / N elements We randomly select one of the first n / N elements from the population list. from the population list. We then select every n / N th element that follows in We then select every n / N th element that follows in the population list. the population list. We then select every n / N th element that follows in We then select every n / N th element that follows in the population list. the population list.

Systematic Sampling This method has the properties of a simple random This method has the properties of a simple random sample, especially if the list of the population sample, especially if the list of the population elements is a random ordering. elements is a random ordering. This method has the properties of a simple random This method has the properties of a simple random sample, especially if the list of the population sample, especially if the list of the population elements is a random ordering. elements is a random ordering. Advantage: The sample usually will be easier to Advantage: The sample usually will be easier to identify than it would be if simple random sampling identify than it would be if simple random sampling were used. were used. Advantage: The sample usually will be easier to Advantage: The sample usually will be easier to identify than it would be if simple random sampling identify than it would be if simple random sampling were used. were used. Example: Selecting every 100 th listing in a telephone Example: Selecting every 100 th listing in a telephone book after the first randomly selected listing book after the first randomly selected listing Example: Selecting every 100 th listing in a telephone Example: Selecting every 100 th listing in a telephone book after the first randomly selected listing book after the first randomly selected listing

Convenience Sampling It is a nonprobability sampling technique. Items are It is a nonprobability sampling technique. Items are included in the sample without known probabilities included in the sample without known probabilities of being selected. of being selected. It is a nonprobability sampling technique. Items are It is a nonprobability sampling technique. Items are included in the sample without known probabilities included in the sample without known probabilities of being selected. of being selected. Example: A professor conducting research might use Example: A professor conducting research might use student volunteers to constitute a sample. student volunteers to constitute a sample. Example: A professor conducting research might use Example: A professor conducting research might use student volunteers to constitute a sample. student volunteers to constitute a sample. The sample is identified primarily by convenience. The sample is identified primarily by convenience.

Convenience Sampling The a dvantage of convenience sampling is that Sample selection and data collection are relatively easy. The disadvantage: It is impossible to determine how representative of the population the sample is.

Judgment Sampling The person most knowledgeable on the subject of the The person most knowledgeable on the subject of the study selects elements of the population that he or study selects elements of the population that he or she feels are most representative of the population. she feels are most representative of the population. The person most knowledgeable on the subject of the The person most knowledgeable on the subject of the study selects elements of the population that he or study selects elements of the population that he or she feels are most representative of the population. she feels are most representative of the population. It is a nonprobability sampling technique. It is a nonprobability sampling technique. Example: A reporter might sample three or four Example: A reporter might sample three or four senators, judging them as reflecting the general senators, judging them as reflecting the general opinion of the senate. opinion of the senate. Example: A reporter might sample three or four Example: A reporter might sample three or four senators, judging them as reflecting the general senators, judging them as reflecting the general opinion of the senate. opinion of the senate.

Judgment Sampling Advantage: It is a relatively easy way of selecting a Advantage: It is a relatively easy way of selecting a sample. sample. Advantage: It is a relatively easy way of selecting a Advantage: It is a relatively easy way of selecting a sample. sample. Disadvantage: The quality of the sample results Disadvantage: The quality of the sample results depends on the judgment of the person selecting the depends on the judgment of the person selecting the sample. sample. Disadvantage: The quality of the sample results Disadvantage: The quality of the sample results depends on the judgment of the person selecting the depends on the judgment of the person selecting the sample. sample.